CP CALC In the circuit in Fig. P29.49, an emf of 90.0 V is

A single loop of wire with an area of 0.0900 m2 is in a uniform magnetic field that has an initial value of 3.80 T, is perpendicular to the plane of the loop, and is decreasing at a constant rate of 0.190 T s. (a) What emf is induced in this loop? (b) If the loop has a resistance of find the current induced in the loop.

In a physics laboratory experiment, a coil with 200 turns enclosing an area of is rotated in 0.040 s from a position where its plane is perpendicular to the earths magnetic field to a position where its plane is parallel to the field. The earths magnetic field at the lab location is (a) What is the total magnetic flux through the coil before it is rotated? After it is rotated? (b) What is the average emf induced in the coil?

Search Coils and Credit Cards. One practical way to measure magnetic field strength uses a small, closely wound coil called a search coil. The coil is initially held with its plane perpendicular to a magnetic field. The coil is then either quickly rotated a quarter-turn about a diameter or quickly pulled out of the field. (a) Derive the equation relating the total charge Q that flows through a search coil to the magnetic-field magnitude B. The search coil has N turns, each with area A, and the flux through the coil is decreased from its initial maximum value to zero in a time \(\Delta t\). The resistance of the coil is R, and the total charge is \(Q=I \Delta t\), where I is the average current induced by the change in flux. (b) In a credit card reader, the magnetic strip on the back of a credit card is rapidly “swiped” past a coil within the reader. Explain, using the same ideas that underlie the operation of a search coil, how the reader can decode the information stored in the pattern of magnetization on the strip. (c) Is it necessary that the credit card be “swiped” through the reader at exactly the right speed? Why or why not?

A closely wound search coil (see Exercise 29.3) has an area of \(3.20 \mathrm{~cm}^{2}\), 120 turns, and a resistance of \(60.0\ \Omega\). It is connected to a charge-measuring instrument whose resistance is \(45.0\ \Omega\). When the coil is rotated quickly from a position parallel to a uniform magnetic field to a position perpendicular to the field, the instrument indicates a charge of \(3.56 \times 10^{-5} \mathrm{C}\). What is the magnitude of the field?

A circular loop of wire with a radius of 12.0 cm and oriented in the horizontal xy-plane is located in a region of uniform magnetic field. A field of 1.5 T is directed along the positive z-direction, which is upward. (a) If the loop is removed from the field region in a time interval of 2.0 ms, find the average emf that will be induced in the wire loop during the extraction process. (b) If the coil is viewed looking down on it from above, is the induced current in the loop clockwise or counterclockwise?

A coil 4.00 cm in radius, containing 500 turns, is placed in a uniform magnetic field that varies with time according to \(B=(0.0120 \mathrm{~T} / \mathrm{s}) t+\left(3.00 \times 10^{-5} \mathrm{~T} / \mathrm{s}^{4}\right) t^{4}\). The coil is connected to a \(600-\Omega\) resistor, and its plane is perpendicular to the magnetic field. You can ignore the resistance of the coil. (a) Find the magnitude of the induced emf in the coil as a function of time. (b) What is the current in the resistor at time t = 5.00 s?

The current in the long, straight wire AB shown in Fig. E29.7 is upward and is increasing steadily at a rate (a) At an instant when the current is i, what are the magnitude and direction of the field at a distance r to the right of the wire? (b) What is the flux through the narrow, shaded strip? (c) What is the total flux through the loop? (d) What is the induced emf in the loop? (e) Evaluate the numerical value of the induced emf if

CALC A flat, circular, steel loop of radius 75 cm is at rest in a uniform magnetic field, as shown in an edge-on view in Fig. E29.8. The field is changing with time, according to (a) Find the emf induced in the loop as a function of time. (b) When is the induced emf equal to of its initial value? (c) Find the direction of the current induced in the loop, as viewed from above the loop.

Shrinking Loop. A circular loop of flexible iron wire has an initial circumference of 165.0 cm, but its circumference is decreasing at a constant rate of due to a tangential pull on the wire. The loop is in a constant, uniform magnetic field oriented perpendicular to the plane of the loop and with magnitude 0.500 T. (a) Find the emf induced in the loop at the instant when 9.0 s have passed. (b) Find the direction of the induced current in the loop as viewed looking along the direction of the magnetic field.

A closely wound rectangular coil of 80 turns has dimensions of 25.0 cm by 40.0 cm. The plane of the coil is rotated from a position where it makes an angle of 37.0 with a magnetic field of 1.10 T to a position perpendicular to the field. The rotation takes 0.0600 s. What is the average emf induced in the coil?

CALC In a region of space, a magnetic field points in the (toward the right). Its magnitude varies with position according to the formula where and b are positive constants, for A flat coil of area A moves with uniform speed from right to left with the plane of its area always perpendicular to this field. (a) What is the emf induced in this coil while it is to the right of the origin? (b) As viewed from the origin, what is the direction (clockwise or counterclockwise) of the current induced in the coil? (c) If instead the coil moved from left to right, what would be the answers to parts (a) and (b)?

Back emf. A motor with a brush-and-commutator arrangement, as described in Example 29.4, has a circular coil with radius 2.5 cm and 150 turns of wire. The magnetic field has magnitude 0.060 T, and the coil rotates at 440 rev/min. (a) What is the maximum emf induced in the coil? (b) What is the average back emf?

The armature of a small generator consists of a flat, square coil with 120 turns and sides with a length of 1.60 cm. The coil rotates in a magnetic field of 0.0750 T. What is the angular speed of the coil if the maximum emf produced is 24.0 mV?

A flat, rectangular coil of dimensions l and w is pulled with uniform speed through a uniform magnetic field B with the plane of its area perpendicular to the field (Fig. E29.14). (a) Find the emf induced in this coil. (b) If the speed and magnetic field are both tripled, what is the induced emf?

A circular loop of wire is in a region of spatially uniform magnetic field, as shown in Fig. E29.15. The magnetic field is directed into the plane of the figure. Determine the direction (clockwise or counterclockwise) of the induced current in the loop when (a) B is increasing; (b) B is decreasing; (c) B is constant with value Explain your reasoning.

The current in Fig. E29.16 obeys the equation where Find the direction (clockwise or counterclockwise) of the current induced in the round coil for

Using Lenzs law, determine the direction of the current in resistor ab of Fig. E29.17 when (a) switch S is opened after having been closed for several minutes; (b) coil B is brought closer to coil A with the switch closed; (c) the resistance of R is decreased while the switch remains closed.

A cardboard tube is wrapped with two windings of insulated wire wound in opposite directions, as shown in Fig. E29.18. Terminals a and b of winding A may be connected to a battery through a reversing switch. State whether the induced current in the resistor R is from left to right or from right to left in the following circumstances: (a) the current in winding A is from a to b and is increasing; (b) the current in winding A is from b to a and is decreasing; (c) the current in winding A is from b to a and is increasing.

A small, circular ring is inside a larger loop that is connected to a battery and a switch, as shown in Fig. E29.19. Use Lenzs law to find the direction of the current induced in the small ring (a) just after switch S is closed; (b) after S has been closed a long time; (c) just after S has been reopened after being closed a long time.

A circular loop of wire with radius and resistance is in a region of spatially uniform magnetic field, as shown in Fig. E29.20. The magnetic field is directed out of the plane of the figure. The magnetic field has an initial value of 8.00 T and is decreasing at a rate of . (a) Is the induced current in the loop clockwise or counterclockwise? (b) What is the rate at which electrical energy is being dissipated by the resistance of the loop?

A circular loop of wire with radius r = 0.0250 m and resistance \(R=0.390\ \Omega\) is in a region of spatially uniform magnetic field, as shown in Fig. E29.21. The magnetic field is directed into the plane of the figure. At t = 0, B= 0. The magnetic field then begins increasing, with \(B(t)=\left(0.380\mathrm{\ T}/\mathrm{s}^3\right)t^3\). What is the current in the loop (magnitude and direction) at the instant when B = 1.33 T?

A rectangular loop of wire with dimensions 1.50 cm by 8.00 cm and resistance is being pulled to the right out of a region of uniform magnetic field. The magnetic field has magnitude and is directed into the B = 3.50 T plane of Fig. E29.22. At he instant when the speed of the loop is 3.00 m s and it is still partially in the field region, what force (magnitude and direction) does the magnetic field exert on the loop?

In Fig. E29.23 a conducting rod of length moves in a magnetic field of magnitude 0.450 T directed into the plane of the figure. The rod moves with speed in the direction shown. (a) What is the potential difference between the ends of the rod? (b) Which point, a or b, is at higher potential? (c) When the charges in the rod are in equilibrium, what are the magnitude and direction of the electric field within the rod? (d) When the charges in the rod are in equilibrium, which point, a or b, has an excess of positive charge? (e) What is the potential difference across the rod if it moves (i) parallel to ab and (ii) directly out of the page?

A rectangle measuring 30.0 cm by 40.0 cm is located inside a region of a spatially uniform magnetic field of 1.25 T, with the field perpendicular to the plane of the coil (Fig. E29.24). The coil is pulled out at a steady rate of traveling perpendicular to the field lines. The region of the field ends abruptly as shown. Find the emf induced in this coil when it is (a) all inside the field; (b) partly inside the field; (c) all outside the field.

Are Motional emfs a Practical Source of Electricity? How fast (in and mph) would a 5.00-cm copper bar have to move at right angles to a 0.650-T magnetic field to generate 1.50 V (the same as a AA battery) across its ends? Does this seem like a practical way to generate electricity?

Motional emfs in Transportation. Airplanes and trains move through the earth's magnetic field at rather high speeds, so it is reasonable to wonder whether this field can have a substantial effect on them. We shall use a typical value of 0.50 G for the earth's field (a) The French TGV train and the Japanese “bullet train” reach speeds of up to 180 mph moving on tracks about 1.5 m apart. At top speed moving perpendicular to the earth's magnetic field, what potential difference is induced across the tracks as the wheels roll? Does this seem large enough to produce noticeable effects? (b) The Boeing 747-400 aircraft has a wingspan of 64.4 m and a cruising speed of 565 mph. If there is no wind blowing (so that this is also their speed relative to the ground), what is the maximum potential difference that could be induced between the opposite tips of the wings? Does this seem large enough to cause problems with the plane?

The conducting rod ab shown in Fig. E29.27 makes contact with metal rails ca and db. The apparatus is in a uniform magnetic field of 0.800 T, perpendicular to the plane of the figure (a) Find the magnitude of the emf induced in the rod when it is moving toward the right with a speed (b) In what direction does the current flow in the rod? (c) If the resistance of the circuit abdc is (assumed to be constant), find the force (magnitude and direction) required to keep the rod moving to the right with a constant speed of You can ignore friction. (d) Compare the rate at which mechanical work is done by the force with the rate at which thermal energy is developed in the circuit.

A 1.50-m-long metal bar is pulled to the right at a steady 5.0 m/s perpendicular to a uniform, 0.750-T magnetic field. The bar rides on parallel metal rails connected through a \(25.0-\Omega\) resistor, as shown in Fig. E29.28, so the apparatus makes a complete circuit. You can ignore the resistance of the bar and the rails. (a) Calculate the magnitude of the emf induced in the circuit. (b) Find the direction of the current induced in the circuit (i) using the magnetic force on the charges in the moving bar; (ii) using Faraday's law; (iii) using Lenz's law. (c) Calculate the current through the resistor.

A 0.360-m-long metal bar is pulled to the left by an applied force F. The bar rides on parallel metal rails connected through a \(45.0-\Omega\) resistor, as shown in Fig. E29.29, so the apparatus makes a complete circuit. You can ignore the resistance of the bar and rails. The circuit is in a uniform 0.650-T magnetic field that is directed out of the plane of the figure. At the instant when the bar is moving to the left at 5.90 m/s, (a) is the induced current in the circuit clockwise or counterclockwise and (b) what is the rate at which the applied force is doing work on the bar?

Consider the circuit shown in Fig. E29.29, but with the bar moving to the right with speed . As in Exercise 29.29, the bar has length 0.360 m, , and . (a) Is the induced current in the circuit clockwise or counterclockwise? (b) At an instant when the resistor is dissipating electrical energy at a rate of , what is the speed of the bar?

A 0.250-m-long bar moves on parallel rails that are connected through a resistor, as shown in Fig. E29.31, so the apparatus makes a complete circuit. You can ignore the resistance of the bar and rails. The circuit is in a uniform magnetic field that is directed into the plane of the figure. At an instant when the induced current in the circuit is counterclockwise and equal to 1.75 A, what is the velocity of the bar (magnitude and direction)?

Measuring Blood Flow. Blood contains positive and negative ions and thus is a conductor. A blood vessel, therefore, can be viewed as an electrical wire. We can even picture the flowing blood as a series of parallel conducting slabs whose thickness is the diameter d of the vessel moving with speed . (See Fig. E29.32.) (a) If the blood vessel is placed in a magnetic field B perpendicular to the vessel, as in the figure, show that the motional potential difference induced across it is (b) If you expect that the blood will be flowing at 15 cm s for a vessel 5.0 mm in diameter, what strength of magnetic field will you need to produce a potential difference of 1.0 mV? (c) Show that the volume rate of flow (R) of the blood is equal to (Note: Although the method developed here is useful in measuring the rate of blood flow in a vessel, it is limited to use in surgery because measurement of the potential must be made directly across the vessel.)

A 1.41-m bar moves through a uniform, 1.20-T magnetic field with a speed of (Fig. E29.33). In each case, find the emf induced between the ends of this bar and identify which, if any, end (a or b) is at the higher potential. The bar moves in the direction of (a) the (b) the (c) the (d) How should this bar move so that the emf across its ends has the greatest possible value with b at a higher potential than a, and what is this maximum emf?

A rectangular circuit is moved at a constant velocity of 3.0 m s into, through, and then out of a uniform 1.25-T magnetic field, as shown in Fig. E29.34. The magnetic-field region is considerably wider than 50.0 cm. Find the magnitude and direction (clockwise or counterclockwise) of the current induced in the circuit as it is (a) going into the magnetic field; (b) totally within the magnetic field, but still moving; and (c) moving out of the field. (d) Sketch a graph of the current in this circuit as a function of time, including the preceding three cases.

The magnetic field within a long, straight solenoid with a circular cross section and radius R is increasing at a rate of (a) What is the rate of change of flux through a circle with radius inside the solenoid, normal to the axis of the solenoid, and with center on the solenoid axis? (b) Find the magnitude of the induced electric field inside the solenoid, at a distance from its axis. Show the direction of this field in a diagram. (c) What is the magnitude of the induced electric field outside the solenoid, at a distance from the axis? (d) Graph the magnitude of the induced electric field as a function of the distance r from the axis from to (e) What is the magnitude of the induced emf in a circular turn of radius that has its center on the solenoid axis? (f) What is the magnitude of the induced emf if the radius in part (e) is R? (g) What is the induced emf if the radius in part (e) is 2R?

A long, thin solenoid has 900 turns per meter and radius 2.50 cm. The current in the solenoid is increasing at a uniform rate of What is the magnitude of the induced electric field at a point near the center of the solenoid and (a) 0.500 cm from the axis of the solenoid; (b) 1.00 cm from the axis of the solenoid?

A long, thin solenoid has 400 turns per meter and radius 1.10 cm. The current in the solenoid is increasing at a uniform rate \(d i / d t\). The induced electric field at a point near the center of the solenoid and 3.50 cm from its axis is \(8.00\times10^{-6}\mathrm{V}/\mathrm{m}\). Calculate \(d i / d t\).

A metal ring 4.50 cm in diameter is placed between the north and south poles of large magnets with the plane of its area perpendicular to the magnetic field. These magnets produce an initial uniform field of 1.12 T between them but are gradually pulled apart, causing this field to remain uniform but decrease steadily at 0.250 T/s. (a) What is the magnitude of the electric field induced in the ring? (b) In which direction (clockwise or counterclockwise) does the current flow as viewed by someone on the south pole of the magnet?

A long, straight solenoid with a cross-sectional area of \(8.00 \mathrm{~cm}^{2}\) is wound with 90 turns of wire per centimeter, and the windings carry a current of 0.350 A. A second winding of 12 turns encircles the solenoid at its center. The current in the solenoid is turned off such that the magnetic field of the solenoid becomes zero in 0.0400 s. What is the average induced emf in the second winding?

The magnetic field at all points within the colored circle shown in Fig. E29.15 has an initial magnitude of 0.750 T. (The circle could represent approximately the space inside a long, thin solenoid.) The magnetic field is directed into the plane of the diagram and is decreasing at the rate of (a) What is the shape of the field lines of the induced electric field shown in Fig. E29.15, within the colored circle? (b) What are the magnitude and direction of this field at any point on the circular conducting ring with radius 0.100 m? (c) What is the current in the ring if its resistance is (d) What is the emf between points a and b on the ring? (e) If the ring is cut at some point and the ends are separated slightly, what will be the emf between the ends?

CALC The electric flux through a certain area of a dielectric is The displacement current through that area is 12.9 pA at time ms. Calculate the dielectric constant for the dielectric

A parallel-plate, air-filled capacitor is being charged as in Fig. 29.22. The circular plates have radius 4.00 cm, and at a particular instant the conduction current in the wires is 0.280 A. (a) What is the displacement current density in the air space between the plates? (b) What is the rate at which the electric field between the plates is changing? (c) What is the induced magnetic field between the plates at a distance of 2.00 cm from the axis? (d) At 1.00 cm from the axis?

Displacement Current in a Dielectric. Suppose that the parallel plates in Fig. 29.22 have an area of and are separated by a 2.50-mm-thick sheet of dielectric that completely fills the volume between the plates. The dielectric has dielectric constant 4.70. (You can ignore fringing effects.) At a certain instant, the potential difference between the plates is 120 V and the conduction current equals 6.00 mA. At this instant, what are (a) the charge q on each plate; (b) the rate of change of charge on the plates; (c) the displacement current in the dielectric?

CALC In Fig. 29.22 the capacitor plates have area and separation 2.00 mm. The plates are in vacuum. The charging current has a constant value of 1.80 mA. At the charge on the plates is zero. (a) Calculate the charge on the plates, the electric field between the plates, and the potential difference between the plates when (b) Calculate the time rate of change of the electric field between the plates. Does vary in time? (c) Calculate the displacement current density between the plates, and from this the total displacement current iD. How do and compare?

Displacement Current in a Wire. A long, straight, copper wire with a circular cross-sectional area of \(2.1\mathrm{\ mm}^2\) carries a current of 16 A. The resistivity of the material is \(2.0\times10^{-8}\ \Omega\cdot\mathrm{m}\). (a) What is the uniform electric field in the material? (b) If the current is changing at the rate of 4000 A/s, at what rate is the electric field in the material changing? (c) What is the displacement current density in the material in part (b)? (Hint: Since K for copper is very close to 1, use \(\epsilon=\epsilon_{0}\).) (d) If the current is changing as in part (b), what is the magnitude of the magnetic field 6.0 cm from the center of the wire? Note that both the conduction current and the displacement current should be included in the calculation of B. Is the contribution from the displacement current significant?

At temperatures near absolute zero, approaches 0.142 T for vanadium, a type-I superconductor. The normal phase of vanadium has a magnetic susceptibility close to zero. Consider a long, thin vanadium cylinder with its axis parallel to an external magnetic field in the -direction. At points far from the ends of the cylinder, by symmetry, all the magnetic vectors are parallel to the x-axis. At temperatures near absolute zero, what are the resultant magnetic field and the magnetization inside and outside the cylinder (far from the ends) for (a) and (b)

The compound \(\mathrm{SiV}_{3}\) is a type-II superconductor. At temperatures near absolute zero the two critical fields are \(B_{\mathrm{c1}}=55.0\ \mathrm{mT}\) and \(B_{\mathrm{c2}}=15.0\ \mathrm{T}\) The normal phase of \(\mathrm{SiV}_{3}\) has a magnetic susceptibility close to zero. A long, thin \(\mathrm{SiV}_{3}\) cylinder has its axis parallel to an external magnetic field \(\vec{\boldsymbol{B}}_{0}\) in the +x-direction. At points far from the ends of the cylinder, by symmetry, all the magnetic vectors are parallel to the x-axis. At a temperature near absolute zero, the external magnetic field is slowly increased from zero. What are the resultant magnetic field \(\vec{\boldsymbol{B}}\) and the magnetization \(\vec{\boldsymbol{M}}\) inside the cylinder at points far from its ends (a) just before the magnetic flux begins to penetrate the material, and (b) just after the material becomes completely normal?

CALC A Changing Magnetic Field. You are testing a new data-acquisition system. This system allows you to record a graph of the current in a circuit as a function of time. As part of the test, you are using a circuit made up of a 4.00-cm-radius, 500-turn coil of copper wire connected in series to a resistor. Copper has resistivity and the wire used for the coil has diameter 0.0300 mm. You place the coil on a table that is tilted from the horizontal and that lies between the poles of an electromagnet. The electromagnet generates a vertically upward magnetic field that is zero for equal to for and equal to 0.240 T for (a) Draw the graph that should be produced by your data-acquisition system. (This is a full-featured system, so the graph will include labels and numerical values on its axes.) (b) If you were looking vertically downward at the coil, would the current be flowing clockwise or counterclockwise?

CP CALC In the circuit shown in Fig. P29.49 the capacitor has capacitance and is initially charged to 100 V with the polarity shown. The resistor has resistance At time the switch is closed. The small circuit is not connected in any way to the large one. The wire of the small circuit has a resistance of and contains 25 loops. The large circuit is a rectangle 2.0 m by 4.0 m, while the small one has dimensions and The distance c is 5.0 cm. (The figure is not drawn to scale.) Both circuits are held stationary. Assume that only the wire nearest the small circuit produces an appreciable magnetic field through it. (a) Find the current in the large circuit after S is closed. (b) Find the current in the small circuit after S is closed. (Hint: See Exercise 29.7.) (c) Find the direction of the current in the small circuit. (d) Justify why we can ignore the magnetic field from all the wires of the large circuit except for the wire closest to the small circuit.

In the circuit in Fig. P29.49, an emf of 90.0 V is added in series with the capacitor and the resistor, and the capacitor is initially uncharged. The emf is placed between the capacitor and the switch, with the positive terminal of the emf adjacent to the capacitor. Otherwise, the two circuits are the same as in 29.49. The switch is closed at t = 0. When the current in the large circuit is 5.00 A, what are the magnitude and direction of the induced current in the small circuit?

CALC A very long, straight solenoid with a crosssectional area of is wound with 90.0 turns of wire per centimeter. Starting at , the current in the solenoid is increasing according to . A secondary winding of 5 turns encircles the solenoid at its center, such that the secondary winding has the same cross- sectional area as the solenoid. What is the magnitude of the emf induced in the secondary winding at the instant that the current in the solenoid is 3.20 A?

A flat coil is oriented with the plane of its area at right angles to a spatially uniform magnetic field. The magnitude of this field varies with time according to the graph in Fig. P29.52. Sketch a qualitative (but accurate!) graph of the emf induced in the coil as a function of time. Be sure to identify the times and on your graph.

In Fig. P29.53 the loop is being pulled to the right at constant speed A constant current I flows in the long wire, in the direction shown. (a) Calculate the magnitude of the net emf induced in the loop. Do this two ways: (i) by using Faradays law of induction (Hint: See Exercise 29.7) and (ii) by looking at the emf induced in each segment of the loop due to its motion. (b) Find the direction (clockwise or counterclockwise) of the current induced in the loop. Do this two ways: (i) using Lenzs law and (ii) using the magnetic force on charges in the loop. (c) Check your answer for the emf in part (a) in the following special cases to see whether it is physically reasonable: (i) The loop is stationary; (ii) the loop is very thin, so (iii) the loop gets very far from the wire.

Suppose the loop in Fig. P29.54 is (a) rotated about the y-axis; (b) rotated about the x-axis; (c) rotated about an edge parallel to the z-axis. What is the maximum induced emf in each case if and

As a new electrical engineer for the local power company, you are assigned the project of designing a generator of sinusoidal ac voltage with a maximum voltage of 120 V. Besides plenty of wire, you have two strong magnets that can produce a constant uniform magnetic field of 1.5 T over a square area of 10.0 cm on a side when they are 12.0 cm apart. The basic design should consist of a square coil turning in the uniform magnetic field. To have an acceptable coil resistance, the coil can have at most 400 loops. What is the minimum rotation rate (in rpm) of the coil so it will produce the required voltage?

Make a Generator? You are shipwrecked on a deserted tropical island. You have some electrical devices that you could operate using a generator but you have no magnets. The earths magnetic field at your location is horizontal and has magnitude and you decide to try to use this field for a generator by rotating a large circular coil of wire at a high rate. You need to produce a peak emf of 9.0 V and estimate that you can rotate the coil at 30 rpm by turning a crank handle. You also decide that to have an acceptable coil resistance, the maximum number of turns the coil can have is 2000. (a) What area must the coil have? (b) If the coil is circular, what is the maximum translational speed of a point on the coil as it rotates? Do you think this device is feasible? Explain

A flexible circular loop 6.50 cm in diameter lies in a magnetic field with magnitude 1.35 T, directed into the plane of the page as shown in Fig. P29.57. The loop is pulled at the points indicated by the arrows, forming a loop of zero area in 0.250 s. (a) Find the average induced emf in the circuit. (b) What is the direction of the current in R: from a to b or from b to a? Explain your reasoning.

CALC A conducting rod with length , mass , and resistance moves without friction on metal rails as shown in Fig. 29.11. A uniform magnetic field with magnitude is directed into the plane of the figure. The rod is initially at rest, and then a constant force with magnitude and directed to the right is applied to the bar. How many seconds after the force is applied does the bar reach a speed of ?

Terminal Speed. A conducting rod with length L, mass m, and resistance R moves without friction on metal rails as shown in Fig. 29.11. A uniform magnetic field is directed into the plane of the figure. The rod starts from rest and is acted on by a constant force directed to the right. The rails are infinitely long and have negligible resistance. (a) Graph the speed of the rod as a function of time. (b) Find an expression for the terminal speed (the speed when the acceleration of the rod is zero).

CP CALC Terminal Speed. A bar of length m is free to slide without friction on horizontal rails, as shown in Fig. P29.60. There is a uniform magnetic field directed into the plane of the figure. At one end of the rails there is a battery with emf and a switch. The bar has mass 0.90 kg and resistance and all other resistance in the circuit can be ignored. The switch is closed at time (a) Sketch the speed of the bar as a function of time. (b) Just after the switch is closed, what is the acceleration of the bar? (c) What is the acceleration of the bar when its speed is (d) What is the terminal speed of the bar?

Antenna emf. A satellite, orbiting the earth at the equator at an altitude of 400 km, has an antenna that can be modeled as a 2.0-m-long rod. The antenna is oriented perpendicular to the earths surface. At the equator, the earths magnetic field is essentially horizontal and has a value of ignore any changes in B with altitude. Assuming the orbit is circular, determine the induced emf between the tips of the antenna

emf in a Bullet. At the equator, the earths magnetic field is approximately horizontal, is directed toward the north, and has a value of (a) Estimate the emf induced between the top and bottom of a bullet shot horizontally at a target on the equator if the bullet is shot toward the east. Assume the bullet has a length of 1 cm and a diameter of 0.4 cm and is traveling at Which is at higher potential: the top or bottom of the bullet? (b) What is the emf if the bullet travels south? (c) What is the emf induced between the front and back of the bullet for any horizontal velocity?

CALC A very long, cylindrical wire of radius R carries a current uniformly distributed across the cross section of the wire. Calculate the magnetic flux through a rectangle that has one side of length W running down the center of the wire and another side of length R, as shown in Fig. P29.63 (see Exercise 29.7).

CALC A circular conducting ring with radius lies in the xy-plane in a region of uniform magnetic field In this expression, and is constant, t is time, is the unit vector in the -direction, and and is constant. At points a and b (Fig. P29.64) there is a small gap in the ring with wires leading to an external circuit of resistance There is no magnetic field at the location of the external circuit. (a) Derive an expression, as a function of time, for the total magnetic flux through the ring. (b) Determine the emf induced in the ring at time What is the polarity of the emf? (c) Because of the internal resistance of the ring, the current through R at the time given in part (b) is only 3.00 mA. Determine the internal resistance of the ring. (d) Determine the emf in the ring at a time What is the polarity of the emf? (e) Determine the time at which the current through R reverses its direction.

The long, straight wire shown in Fig. P29.65a carries constant current I. A metal bar with length L is moving at constant velocity \(\vec{\boldsymbol{v}}\) as shown in the figure. Point a is a distance d from the wire. (a) Calculate the emf induced in the bar. (b) Which point, a or b, is at higher potential? (c) If the bar is replaced by a rectangular wire loop of resistance R (Fig. P29.65b), what is the magnitude of the current induced in the loop?

The cube shown in Fig. P29.66, 50.0 cm on a side, is in a uniform magnetic field of 0.120 T, directed along the positive y-axis. Wires A, C, and D move in the directions indicated, each with a speed of (Wire A moves parallel to the xy-plane, C moves at an angle of below the xy-plane, and D moves parallel to the xz-plane.) What is the potential difference between the ends of each wire?

A slender rod, 0.240 m long, rotates with an angular speed of 8.80 rad/s about an axis through one end and perpendicular to the rod. The plane of rotation of the rod is perpendicular to a uniform magnetic field with a magnitude of 0.650 T. (a) What is the induced emf in the rod? (b) What is the potential difference between its ends? (c) Suppose instead the rod rotates at 8.80 rad/s about an axis through its center and perpendicular to the rod. In this case, what is the potential difference between the ends of the rod? Between the center of the rod and one end?

A Magnetic Exercise Machine. You have designed a new type of exercise machine with an extremely simple mechanism (Fig. E29.28). A vertical bar of silver (chosen for its low resistivity and because it makes the machine look cool) with length is free to move left or right without friction on silver rails. The entire apparatus is placed in a horizontal, uniform magnetic field of strength 0.25 T. When you push the bar to the left or right, the bars motion sets up a current in the circuit that includes the bar. The resistance of the bar and the rails can be neglected. The magnetic field exerts a force on the current-carrying bar, and this force opposes the bars motion. The health benefit is from the exercise that you do in working against this force. (a) Your design goal is that the person doing the exercise is to do work at the rate of 25 watts when moving the bar at a steady What should be the resistance R? (b) You decide you want to be able to vary the power required from the person, to adapt the machine to the persons strength and fitness. If the power is to be increased to 50 W by altering R while leaving the other design parameters constant, should R be increased or decreased? Calculate the value of R for 50 W. (c) When you start to construct a prototype machine, you find it is difficult to produce a 0.25-T magnetic field over such a large area. If you decrease the length of the bar to 0.20 m while leaving B, and R the same as in part (a), what will be the power required of the person?

CP CALC A rectangular loop with width L and a slide wire with mass m are as shown in Fig. P29.69. A uniform magnetic field is directed perpendicular to the plane of the loop into the plane of the figure. The slide wire is given an initial speed of and then released. There is no friction between the slide wire and the loop, and the resistance of the loop is negligible in comparison to the resistance R of the slide wire. (a) Obtain an expression for F, the magnitude of the force exerted on the wire while it is moving at speed (b) Show that the distance x that the wire moves before coming to rest is

A 25.0-cm-long metal rod lies in the xy-plane and makes an angle of with the positive x-axis and an angle of with the positive y-axis. The rod is moving in the -direction with a speed of . The rod is in a uniform magnetic field (a) What is the magnitude of the emf induced in the rod? (b) Indicate in a sketch which end of the rod is at higher potential.

The magnetic field at all points within a circular region of radius R, is uniform in space and directed into the plane of the page as shown in Fig. P29.71. (The region could be a cross section inside the windings of a long, straight solenoid.) If the magnetic field is increasing at a rate what are the magnitude and direction of the force on a stationary positive point charge q located at points a, b, and c? (Point a is a distance r above the center of the region, point b is a distance r to the right of the center, and point c is at the center of the region.)

An airplane propeller of total length L rotates around its center with angular speed \(\omega\) in a magnetic field that is perpendicular to the plane of rotation. Modeling the propeller as a thin, uniform bar, find the potential difference between (a) the center and either end of the propeller and (b) the two ends. (c) If the field is the earth’s field of 0.50 G and the propeller turns at 220 rpm and is 2.0 m long, what is the potential difference between the middle and either end? It this large enough to be concerned about?

A dielectric of permittivity \(3.5 \times 10^{-11} \mathrm{~F} / \mathrm{m}\) completely fills the volume between two capacitor plates. For t > 0 the electric flux through the dielectric is \(\left(8.0 \times 10^{3} \mathrm{~V} \cdot \mathrm{m} / \mathrm{s}^{3}\right) t^{3}\). The dielectric is ideal and nonmagnetic; the conduction current in the dielectric is zero. At what time does the displacement current in the dielectric equal \(21 \mu \mathrm{A}\)?

A capacitor has two parallel plates with area A separated by a distance d. The space between plates is filled with a material having dielectric constant K. The material is not a perfect insulator but has resistivity The capacitor is initially charged with charge of magnitude on each plate that gradually discharges by conduction through the dielectric. (a) Calculate the conduction current density in the dielectric. (b) Show that at any instant the displacement current density in the dielectric is equal in magnitude to the conduction current density but opposite in direction, so the total current density is zero at every instant.

CALC A rod of pure silicon (resistivity is carrying a current. The electric field varies sinusoidally with time according to where and the frequency (a) Find the magnitude of the maximum conduction current density in the wire. (b) Assuming find the maximum displacement current density in the wire, and compare with the result of part (a). (c) At what frequency would the maximum conduction and displacement densities become equal if (which is not actually the case)? (d) At the frequency determined in part (c), what is the relative phase of the conduction and displacement currents?

A square, conducting, wire loop of side L, total mass m, and total resistance R initially lies in the horizontal xy-plane, with corners at and There is a uniform, upward magnetic field in the space within and around the loop. The side of the loop that extends from to is held in place on the x-axis; the rest of the loop is free to pivot around this axis. When the loop is released, it begins to rotate due to the gravitational torque. (a) Find the net torque (magnitude and direction) that acts on the loop when it has rotated through an angle from its original orientation and is rotating downward at an angular speed (b) Find the angular acceleration of the loop at the instant described in part (a). (c) Compared to the case with zero magnetic field, does it take the loop a longer or shorter time to rotate through Explain. (d) Is mechanical energy conserved as the loop rotates downward? Explain.

A metal bar with length L, mass m, and resistance R is placed on frictionless metal rails that are inclined at an angle above the horizontal. The rails have negligible resistance. A uniform magnetic field of magnitude B is directed downward as shown in Fig. P29.77. The bar is released from rest and slides down the rails. (a) Is the direction of the current induced in the bar from a to b or from b to a? (b) What is the terminal speed of the bar? (c) What is the induced current in the bar when the terminal speed has been reached? (d) After the terminal speed has been reached, at what rate is electrical energy being converted to thermal energy in the resistance of the bar? (e) After the terminal speed has been reached, at what rate is work being done on the bar by gravity? Compare your answer to that in part (d).

Problem 1DQ A sheet of copper is placed between the poles of an electromagnet with the magnetic field perpendicular to the sheet. When the sheet is pulled out, a considerable force is required, and the force required increases with speed. Explain. Is a force required also when the sheet is inserted between the poles? Explain.

Problem 1E A single loop of wire with an area of 0.0900 m2 is in a uniform magnetic field that has an initial value of 3.80 T, is perpendicular to the plane of the loop, and is decreasing at a constant rate of 0.190 T/s. (a) What emf is induced in this loop? (b) If the loop has a resistance of 0.600 ?, find the current induced in the loop.

In Fig. 29.8, if the angular speed \(\omega\) of the loop is doubled, then the frequency with which the induced current changes direction doubles, and the maximum emf also doubles. Why? Does the torque required to turn the loop change? Explain. Equation Transcription: Text Transcription: omega

Problem 2E In a physics laboratory experiment, a coil with 200 turns enclosing an area of 12 cm2 is rotated in 0.040 s from a position where its plane is perpendicular to the earth’s magnetic field to a position where its plane is parallel to the field. The earth’s magnetic field at the lab location is 6.0 × 10-5 T. (a) What is the total magnetic flux through the coil before it is rotated? After it is rotated? (b) What is the average emf induced in the coil?

Two circular loops lie side by side in the same plane. One is connected to a source that supplies an increasing current; the other is a simple closed ring. Is the induced current in the ring in the same direction as the current in the loop connected to the source, or opposite? What if the current in the first loop is decreasing? Explain.

Problem 3E Search Coils and Credit Cards. One practical way to measure magnetic field strength uses a small, closely wound coil called a search coil. The coil is initially held with its plane perpendicular to a magnetic field. The coil is then either quickly rotated a quarter-turn about a diameter or quickly pulled out of the field. (a) Derive the equation relating the total charge Q that flows through a search coil to the magnetic-field magnitude B. The search coil has N turns, each with area A, and the flux through the coil is decreased from its initial maximum value to zero in a time ?t. The resistance of the coil is R, and the total charge is Q = I?t, where I is the average current induced by the change in flux. (b) In a credit card reader, the magnetic strip on the back of a credit card is rapidly “swiped” past a coil within the reader. Explain, using the same ideas that underlie the operation of a search coil, how the reader can decode the information stored in the pattern of magnetization on the strip. (c) Is it necessary that the credit card be “swiped” through the reader at exactly the right speed? Why or why not?

For Eq. (29.6), show that if \(v\) is in meters per second, B in teslas, and L in meters, then the units of the right-hand side of the equation are joules per coulomb or volts (the correct SI units for \(\varepsilon\)). Equation Transcription: Text Transcription: v epsilon

Problem 4E A closely wound search coil (see Exercise 29.3) has an area of 3.20 cm2, 120 turns, and a resistance of 60.0 ?. It is connected to a charge-measuring instrument whose resistance is 45.0 ?. When the coil is rotated quickly from a position parallel to a uniform magnetic field to a position perpendicular to the field, the instrument indicates a charge of 3.56 × 10-5 C. What is the magnitude of the field? 29.3 .. Search Coils and Credit Cards. One practical way to measure magnetic field strength uses a small, closely wound coil called a search coil. The coil is initially held with its plane perpendicular to a magnetic field. The coil is then either quickly rotated a quarter-turn about a diameter or quickly pulled out of the field. (a) Derive the equation relating the total charge Q that flows through a search coil to the magnetic-field magnitude B. The search coil has N turns, each with area A, and the flux through the coil is decreased from its initial maximum value to zero in a time ?t. The resistance of the coil is R, and the total charge is Q = I?t, where I is the average current induced by the change in flux. (b) In a credit card reader, the magnetic strip on the back of a credit card is rapidly “swiped” past a coil within the reader. Explain, using the same ideas that underlie the operation of a search coil, how the reader can decode the information stored in the pattern of magnetization on the strip. (c) Is it necessary that the credit card be “swiped” through the reader at exactly the right speed? Why or why not?

Problem 5DQ A long, straight conductor passes through the center of a metal ring, perpendicular to its plane. If the current in the conductor increases, is a current induced in the ring? Explain.

Problem 5E A circular loop of wire with a radius of 12.0 cm and oriented in the horizontal xy-plane is located in a region of uniform magnetic field. A field of 1.5 T is directed along the positive z-direction, which is upward. (a) If the loop is removed from the field region in a time interval of 2.0 ms, find the average emf that will be induced in the wire loop during the extraction process. (b) If the coil is viewed looking down on it from above, is the induced current in the loop clockwise or counterclockwise?

Problem 6DQ A student asserted that if a permanent magnet is dropped down a vertical copper pipe, it eventually reaches a terminal velocity even if there is no air resistance. Why should this be? Or should it?

A coil 4.00 cm in radius, containing 500 turns, is placed in a uniform magnetic field that varies with time according to \(B=(0.0120\mathrm{\ T}/\mathrm{s})t+\left(3.00\times10^{-5}\mathrm{\ T}/\mathrm{s}^4\right)t^4\). The coil is connected to a \(600-\Omega\) resistor, and its plane is perpendicular to the magnetic field. You can ignore the resistance of the coil. (a) Find the magnitude of the induced emf in the coil as a function of time. (b) What is the current in the resistor at time \(t=5.00\) s?

Problem 7DQ An airplane is in level flight over Antarctica, where the magnetic field of the earth is mostly directed upward away from the ground. As viewed by a passenger facing toward the front of the plane, is the left or the right wingtip at higher potential? Does your answer depend on the direction the plane is flying?

CALC The current in the long, straight wire \(A B\) shown in Fig. E29.7 is upward and is increasing steadily at a rate \(d i / d t\) (a) At an instant when the current is \(i\), what are the magnitude and direction of the field \(\vec{B}\) at a distance \(r\) to the right of the wire? (b) What is the flux \(d \Phi_{B}\) through the narrow, shaded strip? (c) What is the total flux through the loop? (d) What is the induced emf in the loop? (e) Evaluate the numerical value of the induced emf if \(a=12.0 \mathrm{~cm}, b=36.0 \mathrm{~cm}\), \(L=24.0 \mathrm{~cm}\), and \(d i / d t=9.60 \mathrm{~A} / \mathrm{s}\).

Consider the situation in Exercise 29.19. In part (a), find the direction of the force that the large circuit exerts on the small one. Explain how this result is consistent with Lenz’s law.

CALC A flat, circular, steel loop of radius 75 cm is at rest in a uniform magnetic field, as shown in an edge-on view in Fig. E29.8. The field is changing with time, according to \(B(t)=(1.4 \mathrm{~T}) e^{-\left(0.057 s^{-1}\right) t}\). (a) Find the emf induced in the loop as a function of time. (b) When is the induced emf equal to \(\frac{1}{10}\) of its initial value? (c) Find the direction of the current induced in the loop, as viewed from above the loop. Equation Transcription: Text Transcription: B(t)=(1.4 T)e^-(0.057 s^-1)t 1 over 10 Vector B 60deg

Problem 9DQ A metal rectangle is close to a long, straight, current-carrying wire, with two of its sides parallel to the wire. If the current in the long wire is decreasing, is the rectangle repelled by or attracted to the wire? Explain why this result is consistent with Lenz’s law.

Problem 9E Shrinking Loop. A circular loop of flexible iron wire has an initial circumference of 165.0 cm, but its circumference is decreasing at a constant rate of 12.0 cm/s due to a tangential pull on the wire. The loop is in a constant, uniform magnetic field oriented perpendicular to the plane of the loop and with magnitude 0.500 T. (a) Find the emf induced in the loop at the instant when 9.0 s have passed. (b) Find the direction of the induced current in the loop as viewed looking along the direction of the magnetic field.

Your physics study partner asks you to consider a parallel-plate capacitor that has a dielectric completely filling the volume between the plates. He then claims that Eqs. (29.13) and (29.14) show that the conduction current in the dielectric equals the displacement current in the dielectric. Do you agree? Explain.

The current in Fig. E29.16 obeys the equation \(I(t)=I_{0} e^{-b t}\), where \(b>0\). Find the direction (clockwise or counterclockwise) of the current induced in the round coil for \(t>0\). Equation Transcription: Text Transcription: I(t)=I_0e^-bt b>0 t>0

Problem 17DQ Match the mathematical statements of Maxwell’s equations as given in Section 29.7 to these verbal statements. (a) Closed electric field lines are evidently produced only by changing magnetic flux. (b) Closed magnetic field lines are produced both by the motion of electric charge and by changing electric flux. (c) Electric field lines can start on positive charges and end on negative charges. (d) Evidently there are no magnetic monopoles on which to start and end magnetic field lines.

Using Lenz’s law, determine the direction of the current in resistor ab of Fig. E29.17 when (a) switch S is opened after having been closed for several minutes; (b) coil B is brought closer to coil A with the switch closed; (c) the resistance of R is decreased while the switch remains closed.

If magnetic monopoles existed, the right-hand side of Eq. (29.21) would include a term proportional to the current of magnetic monopoles. Suppose a steady monopole current is moving in a long straight wire. Sketch the electric field lines that such a current would produce.

A cardboard tube is wrapped with two windings of insulated wire wound in opposite directions, as shown in Fig. E29.18. Terminals \(a\) and \(b\) of winding A may be connected to a battery through a reversing switch. State whether the induced current in the resistor R is from left to right or from right to left in the following circumstances: (a) the current in winding A is from \(a\) to \(b\) and is increasing; (b) the current in winding A is from \(b\) to \(a\) and is decreasing; (c) the current in winding A is from \(b\) to \(a\) and is increasing. Equation Transcription: Text Transcription: a b a b b a

A small, circular ring is inside a larger loop that is connected to a battery and a switch, as shown in Fig. E29.19. Use Lenz’s law to find the direction of the current induced in the small ring (a) just after switch S is closed; (b) after S has been closed a long time; (c) just after S has been reopened after being closed a long time.

A circular loop of wire with radius \(r=0.0480 \mathrm{~m}\) and resistance \(R=0.160 \Omega\) is in a region of spatially uniform magnetic field, as shown in Fig. E29.20. The magnetic field is directed out of the plane of the figure. The magnetic field has an initial value of \(8.00 \mathrm{~T}\) and is decreasing at a rate of \(d B / d t=-0.680 \mathrm{~T} / \mathrm{s}\). (a) Is the induced current in the loop clockwise or counterclockwise? (b) What is the rate at which electrical energy is being dissipated by the resistance of the loop? Equation Transcription: Text Transcription: r =0.0480 m R =0.160 omega 8.00 ???? dB/dt =-0.680 ????/s

CALC A circular loop of wire with radius \(r=0.025\) m and resistance \(R=0.390\) \(\Omega\) is in a region of spatially uniform magnetic field, as shown in Fig. E29.21. The magnetic field is directed into the plane of the figure. At \(t=0\), \(B=0\). The magnetic field then begins increasing, with \(B(t)=\left(0.380 \mathrm{~T} / \mathrm{s}^{3}\right) t^{3}\). What is the current in the loop (magnitude and direction) at the instant when \(B=1.33 \mathrm{~T}\)? Equation Transcription: Text Transcription: r=0.025 R=0.390 ohms t=0 B=0 B(t)=(0.380 T/s^3)t^3 B=1.33 Vector B

Are Motional emfs a Practical Source of Electricity? How fast (in m/s and mph) would a 5.00-cm copper bar have to move at right angles to a 0.650-T magnetic field to generate 1.50 V (the same as a AA battery) across its ends? Does this seem like a practical way to generate electricity?

Problem 26E Motional emfs in Transportation. Airplanes and trains move through the earth’s magnetic field at rather high speeds, so it is reasonable to wonder whether this field can have a substantial effect on them. We shall use a typical value of 0.50 G for the earth’s field. (a) The French TGV train and the Japanese “bullet train” reach speeds of up to 180 mph moving on tracks about 1.5 m apart. At top speed moving perpendicular to the earth’s magnetic field, what potential difference is induced across the tracks as the wheels roll? Does this seem large enough to produce noticeable effects? (b) The Boeing 747-400 aircraft has a wingspan of 64.4 m and a cruising speed of 565 mph. If there is no wind blowing (so that this is also their speed relative to the ground), what is the maximum potential difference that could be induced between the opposite tips of the wings? Does this seem large enough to cause problems with the plane?

The conducting rod \(a b\) shown in Fig. E29.27 makes contact with metal rails \(ca\) and \(d b\). The apparatus is in a uniform magnetic field of 0.800 ????, perpendicular to the plane of the figure (a) Find the magnitude of the emf induced in the rod when it is moving toward the right with a speed \(7.50 \mathrm{~m} / \mathrm{s}\). (b) In what direction does the current flow in the rod? (c) If the resistance of the circuit \(a b d c\) is 1.50 \(\Omega\)(assumed to be constant), find the force (magnitude and direction) required to keep the rod moving to the right with a constant speed of \(7.50 \mathrm{~m} / \mathrm{s}\). You can ignore friction. (d) Compare the rate at which mechanical work is done by the force \((F v)\) with the rate at which thermal energy is developed in the circuit \(\left(I^{2} R\right)\). Equation Transcription: Text Transcription: ab ca db 7.50 m/s abdc Ohms 7.50 m/s (Fv) (I^2R) Vector B

A 1.50-m-long metal bar is pulled to the right at a steady \(5.0 \mathrm{~m} / \mathrm{s}\) perpendicular to a uniform, 0.750-???? magnetic field. The bar rides on parallel metal rails connected through a 25.0-\(\Omega\) resistor, as shown in Fig. E29.28, so the apparatus makes a complete circuit. You can ignore the resistance of the bar and the rails. (a) Calculate the magnitude of the emf induced in the circuit. (b) Find the direction of the current induced in the circuit (i) using the magnetic force on the charges in the moving bar; (ii) using Faraday’s law; (iii) using Lenz’s law. (c) Calculate the current through the resistor. Equation Transcription: Text Transcription: 5.0 m/s Ohms Vector B

A 0.360-m-long metal bar is pulled to the left by an applied force ????. The bar rides on parallel metal rails connected through a 45.0 \(\Omega\) resistor, as shown in Fig. E29.29, so the apparatus makes a complete circuit. You can ignore the resistance of the bar and rails. The circuit is in a uniform 0.650-???? magnetic field that is directed out of the plane of the figure. At the instant when the bar is moving to the left at \(5.90 \mathrm{~m} / \mathrm{s}\), (a) is the induced current in the circuit clockwise or counterclockwise and (b) what is the rate at which the applied force is doing work on the bar? Equation Transcription: Text Transcription: Ohms 5.90 m/s Vector B

Consider the circuit shown in Fig. E29.29, but with the bar moving to the right with speed . As in Exercise 29.29, the bar has length 0.360 m, \(R=45.0\) \(\Omega\), and \(B=0.650\) ????. (a) Is the induced current in the circuit clockwise or counterclockwise? (b) At an instant when the 45.0-\(\Omega\)resistor is dissipating electrical energy at a rate of \(0.840 \mathrm{~J} / \mathrm{s}\), what is the speed of the bar? Equation Transcription: Text Transcription: R=45.0 Ohms B=0.650 Ohms 0.840 J/s

Displacement Current in a Dielectric. Suppose that the parallel plates in Fig. 29.22 have an area of 3.00 \(cm^{2}\) and are separated by a 2.50-mm-thick sheet of dielectric that completely fills the volume between the plates. The dielectric has dielectric constant 4.70. (You can ignore fringing effects.) At a certain instant, the potential difference between the plates is 120 V and the conduction current \(i_{c}\) equals 6.00 mA. At this instant, what are (a) the charge q on each plate; (b) the rate of change of charge on the plates; (c) the displacement current in the dielectric?

CALC In Fig. 29.22 the capacitor plates have area 5.00 \(\mathrm{cm}^{2}\) and separation 2.00 mm. The plates are in vacuum. The charging current \(i_{C}\) has a constant value of 1.80 mA. At \(t=0\) the charge on the plates is zero. (a) Calculate the charge on the plates, the electric field between the plates, and the potential difference between the plates when \(t=0.500\) \(\mu \mathrm{s}\). (b) Calculate \(d E / d t\), the time rate of change of the electric field between the plates. Does \(d E / d t\) vary in time? (c) Calculate the displacement current density \(j_{D}\) between the plates, and from this the total displacement current \(i_{D}\). How do \(i_{C}\) and \(i_{D}\) compare? Equation Transcription: Text Transcription: cm^2 i_c t=0 t=0.500 mu s dE/dt dE/dt j_D i_D i_C i_D

Problem 45E Displacement Current in a Wire. A long,straight, copper wire with a circular cross-sectional area of 2.1 mm2 carries a current of 16 A. The resistivity of the material is 2.0 × 10?8 ? ? m. (a) What is the uniform electric field in the material? (b) If the current is changing at the rate of 4000 A/s, at what rate is the electric field in the material changing? (c) What is the displacement current density in the material in part(b)? (Hint: Since K for copper is very close to 1, use ? = ?0.) (d) If the current is changing as in part (b), what is the magnitude of the magnetic field 6.0 cm from the center of the wire? Note that both the conduction current and the displacement current should be included in the calculation of B. Is the contribution from the displacement current significant?

At temperatures near absolute zero, \(B_{\mathrm{c}}\) approaches 0.142 T for vanadium, a type-I superconductor. The normal phase of vanadium has a magnetic susceptibility close to zero. Consider a long, thin vanadium cylinder with its axis parallel to an external magnetic field \(\overrightarrow{\boldsymbol{B}}_{0}\) in the \(+x \text {-direction }\). At points far from the ends of the cylinder, by symmetry, all the magnetic vectors are parallel to the x-axis. At temperatures near absolute zero, what are the resultant magnetic field \(\overrightarrow{\boldsymbol{B}}\) and the magnetization \(\overrightarrow{\boldsymbol{M}}\) inside and outside the cylinder (far from the ends) for (a) \(\overrightarrow{\boldsymbol{B}}_{0}=(0.130 \mathrm{~T}) \hat{\boldsymbol{\imath}}\) and (b) \(\overrightarrow{\boldsymbol{B}}_{0}=(0.260 \mathrm{~T}) \hat{\boldsymbol{\imath}}\)?

The compound \(\mathrm{SiV}_{3}\) is a type-II superconductor. At temperatures near absolute zero the two critical fields are \(B_{c 1}=55.0\) m???? and \(B_{c 2}=15.0\) ????.The normal phase of \(\mathrm{SiV}_{3}\) has a magnetic susceptibility close to zero. A long, thin \(\mathrm{SiV}_{3}\) cylinder has its axis parallel to an external magnetic field \(\vec{B}_{0}\) in the \(+x\)-direction. At points far from the ends of the cylinder, by symmetry, all the magnetic vectors are parallel to the \(x\)-axis. At a temperature near absolute zero, the external magnetic field is slowly increased from zero. What are the resultant magnetic field \(\vec{B}\) and the magnetization \(\vec{M}\) inside the cylinder at points far from its ends (a) just before the magnetic flux begins to penetrate the material, and (b) just after the material becomes completely normal? Equation Transcription: Text Transcription: SiV_3 B_c1=55.0 B_c2=15.0 SiV_3 SiV_3 Vector B_0 +x x Vector B Vector M

Problem 48P A Changing Magnetic Field. You are testing a new data-acquisition system. This system allows you to record a graph of the current in a circuit as a function of time. As part of the test, you are using a circuit made up of a 4.00-cm-radius, 500-turn coil of copper wire connected in series to a 600 ? resistor. Copper has resistivity 1.72 × 10?8 ? ? m, and the wire used for the coil has diameter 0.0300 mm. You place the coil on a table that is tilted 30.0° from the horizontal and that lies between the poles of an electromagnet. The electromagnet generates a vertically upward magnetic field that is zero for t < 0, equal to (0.120 T) × (1 ? cos ?t) for 0 ? t ? 1.00 s, and equal to 0.240 T for t > 1.00 s. (a) Draw the graph that should be produced by your data-acquisition system. (This is a full-featured system, so the graph will include labels and numerical values on its axes.) (b) If you were looking vertically downward at the coil, would the current be flowing clockwise or counterclockwise?

CP CALC In the circuit shown in Fig. P29.49 the capacitor has capacitance \(C=20\) \(\mu \mathrm{F}\) and is initially charged to 100 V with the polarity shown. The resistor \(R_{0}\) has resistance 10 \(\Omega\). At time \(t=0\) the switch is closed. The small circuit is not connected in any way to the large one. The wire of the small circuit has a resistance of 1.0 \(\Omega / \mathrm{m}\) and contains 25 loops. The large circuit is a rectangle 2.0 m by 4.0 m, while the small one has dimensions \(a=10.0 \mathrm{~cm}\) and \(b=20.0 \mathrm{~cm}\). The distance c is 5.0 cm. (The figure is not drawn to scale.) Both circuits are held stationary. Assume that only the wire nearest the small circuit produces an appreciable magnetic field through it. (a) Find the current in the large circuit 200 \(\mu \mathrm{s}\) after S is closed. (b) Find the current in the small circuit 200 \(\mu \mathrm{s}\) after S is closed. (Hint: See Exercise 29.7.) (c) Find the direction of the current in the small circuit. (d) Justify why we can ignore the magnetic field from all the wires of the large circuit except for the wire closest to the small circuit. Equation Transcription: Text Transcription: C=20 muF R_0 Ohms t=0 ohms/m a=10.0 cm b=20.0 cm Mu s Mu s R_0

CP CALC In the circuit in Fig. P29.49, an emf of 90.0 V is added in series with the capacitor and the resistor, and the capacitor is initially uncharged. The emf is placed between the capacitor and the switch, with the positive terminal of the emf adjacent to the capacitor. Otherwise, the two circuits are the same as in Problem 29.49. The switch is closed at \(t=0\). When the current in the large circuit is 5.00 A, what are the magnitude and direction of the induced current in the small circuit? Equation Transcription: Text Transcription: t=0

Problem 51P CALC A very long, straight solenoid with a cross-sectional area of 2.00 cm2 is wound with 90.0 turns of wire per centimeter. Starting at t = 0, the current in the solenoid is increasing according to i(t) = (0.160 A/s2)t2. A secondary winding of 5 turns encircles the solenoid at its center, such that the secondary winding has the same cross-sectional area as the solenoid. What is the magnitude of the emf induced in the secondary winding at the instant that the current in the solenoid is 3.20 A?

Problem 55P As a new electrical engineer for the local power company, you are assigned the project of designing a generator of sinusoidal ac voltage with a maximum voltage of 120 V. Besides plenty of wire. you have two strong magnets that can produce a constant uniform magnetic field or 1.5 T over a square area of 10.0 cm on a side when they are 12.0 cm apart. The basic design should consist of a square coil turning in the uniform magnetic field. To have an acceptable coil resistance, the coil can have an most 400 loops. What is the minimum rotation rate (in rpm) of the coil so it will produce the required voltage?

Problem 56P Make a Generator? You are shipwrecked on a deserted tropical island. You have some electrical devices that you could operate using a generator but you have no magnets. The earth’s magnetic field at your location is horizontal and has magnitude 8.0 × 10-5 T, and you decide to try to use this field for a generator by rotating a large circular coil of wire at a high rate. You need to produce a peak emf of 9.0 V and estimate that you can rotate the coil at 30 rpm by turning a crank handle. You also decide that to have an acceptable coil resistance, the maximum number of turns the coil can have is 2000. (a) What area must the coil have? (b) If the coil is circular, what is the maximum translational speed of a point on the coil as it rotates? Do you think this device is feasible? Explain.

A flexible circular loop 6.50 cm in diameter lies in a magnetic field with magnitude 1.35 T, directed into the plane of the page as shown in Fig. P29.57. The loop is pulled at the points indicated by the arrows, forming a loop of zero area in 0.250 s. (a) Find the average induced emf in the circuit. (b) What is the direction of the current in R: from \(a\) to \(b\) or from \(b\) to \(a\)? Explain your reasoning. Equation Transcription: Text Transcription: a b b a

CALC A conducting rod with length \(L=0.200\) m, mass \(m=0.120\) kg, and resistance \(R=80.0\) \(\Omega\) moves without friction on metal rails as shown in Fig. 29.11. A uniform magnetic field with magnitude \(B=1.50\) T is directed into the plane of the figure. The rod is initially at rest, and then a constant force with magnitude \(F=1.90\) N and directed to the right is applied to the bar. How many seconds after the force is applied does the bar reach a speed of \(25.0 \mathrm{~m} / \mathrm{s}\)? Equation Transcription: Text Transcription: L=0.200 m=0.120 R=80.0 Ohms B=1.50 F=1.90 25.0 m/s

Terminal Speed. A conducting rod with length L, mass m, and resistance R moves without friction on metal rails as shown in Fig. 29.11. A uniform magnetic field \(\vec{B}\) is directed into the plane of the figure. The rod starts from rest and is acted on by a constant force \(\vec{F}\) directed to the right. The rails are infinitely long and have negligible resistance. (a) Graph the speed of the rod as a function of time. (b) Find an expression for the terminal speed (the speed when the acceleration of the rod is zero). Equation Transcription: Text Transcription: Vector B Vector F

CP CALC Terminal Speed. A bar of length \(L=0.36\) m is free to slide without friction on horizontal rails, as shown in Fig. P29.60. There is a uniform magnetic field \(B=1.5\) T directed into the plane of the figure. At one end of the rails there is a battery with emf \(\varepsilon=12\) V and a switch. The bar has mass 0.90 kg and resistance 5.0 \(\Omega\), and all other resistance in the circuit can be ignored. The switch is closed at time \(t=0\). (a) Sketch the speed of the bar as a function of time. (b) Just after the switch is closed, what is the acceleration of the bar? (c) What is the acceleration of the bar when its speed is \(2.0 \mathrm{~m} / \mathrm{s}\)? (d) What is the terminal speed of the bar? Equation Transcription: Text Transcription: L=0.36 B=1.5 Epsilon=12 Ohms t=0 2.0 m/s Vector B Epsilon

Problem 73P CALC A dielectric of permittivity 3.5 × 10-11 F/m completely fills the volume between two capacitor plates. For t > 0 the electric flux through the dielectric is (8.0 × 103 V ? m/s3)t3. The dielectric is ideal and nonmagnetic; the conduction current in the dielectric is zero. At what time does the displacement current in the dielectric equal 21 µA?

Problem 74P CP CALC A capacitor has two parallel plates with area A separated by a distance d. The space between plates is filled with a material having dielectric constant K. The material is not a perfect insulator but has resistivity r. The capacitor is initially charged with charge of magnitude Q0 on each plate that gradually discharges by conduction through the dielectric. (a) Calculate the conduction current density jC(t) in the dielectric. (b) Show that at any instant the displacement current density in the dielectric is equal in magnitude to the conduction current density but opposite in direction, so the total current density is zero at every instant.

Problem 75P A rod of pure silicon (resistivity ? = 2300 ? ? m) is carrying a current. The electric field varies sinusoidally with time according to E = E0 sin ?t. where E0 = 0.450 V/m, ? = 2?f, and the frequency f = 120 Hz. (a) Find the magnitude of the maximum conduction current density in the wire. (b) Asuuming ? = ?0, find the maximum displacement current density in the wire, and compare with the result of the part (a). (c) At what frequency f would the maximum conduction on displacement densities become equal. If ? = ?0 (which is not actually the case)? (d) At the frequency determined in part (c). What is the relative phase of the conduction and displacements currents?

CP CALC A square, conducting, wire loop of side L, total mass m, and total resistance R initially lies in the horizontal \(x y\)-plane, with corners at \((x, y, z)=(0,0,0)\), \((0,L,0)\), \((L,0,0)\), and \((L,L,0)\). There is a uniform, upward magnetic field \(\vec{B}=B \widehat{k}\) in the space within and around the loop. The side of the loop that extends from \((0,0,0)\) to \((L,0,0)\) to is held in place on the \(x\)-axis; the rest of the loop is free to pivot around this axis. When the loop is released, it begins to rotate due to the gravitational torque. (a) Find the net torque (magnitude and direction) that acts on the loop when it has rotated through an angle \(\phi\) from its original orientation and is rotating downward at an angular speed \(\omega\) (b) Find the angular acceleration of the loop at the instant described in part (a). (c) Compared to the case with zero magnetic field, does it take the loop a longer or shorter time to rotate through \(90^{\circ}\)? Explain. (d) Is mechanical energy conserved as the loop rotates downward? Explain. Equation Transcription: Text Transcription: xy (x,y,z)=(0,0,0) (0,L,0) (L,0,0) (L,L,0) Vector B=Bk hat (0,0,0) (L,0,0) x phi omega 90deg

A metal bar with length L, mass m, and resistance R is placed on frictionless metal rails that are inclined at an angle \(\phi\) above the horizontal. The rails have negligible resistance. A uniform magnetic field of magnitude B is directed downward as shown in Fig. P29.77. The bar is released from rest and slides down the rails. (a) Is the direction of the current induced in the bar from \(a\) to \(b\) or from \(b\) to \(a\)? (b) What is the terminal speed of the bar? (c) What is the induced current in the bar when the terminal speed has been reached? (d) After the terminal speed has been reached, at what rate is electrical energy being converted to thermal energy in the resistance of the bar? (e) After the terminal speed has been reached, at what rate is work being done on the bar by gravity? Compare your answer to that in part (d). Equation Transcription: Text Transcription: phi a b b a Vector B phi Vector B

Problem 10DQ A square conducting loop is in a region of uniform, constant magnetic field. Can the loop be rotated about an axis along one side and no emf be induced in the loop? Discuss, in terms of the orientation of the rotation axis relative to the magnetic-field direction.

Problem 10E A closely wound rectangular coil of 80 turns has dimensions of 25.0 cm by 40.0 cm. The plane of the coil is rotated from a position where it makes angle of 37.0° with a magnetic field of 1.10 T to a position perpendicular to the field. The rotation takes 0.0600 s. What is the average emf induced in the coil.

Example discusses the external force that must be applied to the slidewire to move it at constant speed. If there were a break in the left-hand end of the U-shaped conductor, how much force would be needed to move the slidewire at constant speed? As in the example, you can ignore friction.

Problem 11E CALC In a region of space, a magnetic field points in the +x-direction (toward the right). Its magnitude varies with position according to the formula Bx = B0 + bx, where B0 and b are positive constants, for x ? 0. A flat coil of area A moves with uniform speed v from right to left with the plane of its area always perpendicular to this field. (a) What is the emf induced in this coil while it is to the right of the origin? (b) As viewed from the origin, what is the direction (clockwise or counterclockwise) of the current induced in the coil? (c) If instead the coil moved from left to right, what would be the answers to parts (a) and (b)?

In the situation shown in Fig. 29.17, would it be appropriate to ask how much energy an electron gains during a complete trip around the wire loop with current \(I^{\prime}\)? Would it be appropriate to ask what potential difference the electron moves through during such a complete trip? Explain your answers.

Back emf. A motor with a brush-and-commutator arrangement, as described in Example , has a circular coil with radius and 150 turns of wire. The magnetic field has magnitude , and the coil rotates at 440 rev/min. (a) What is the maximum emf induced in the coil? (b) What is the average back emf?

Problem 13DQ A metal ring is oriented with the plane of its area perpendicular to a spatially uniform magnetic field that increases at a steady rate. If the radius of the ring is doubled, by what factor do (a) the emf induced in the ring and (b) the electric field induced in the ring change?

Problem 13E The armature of a small generator consists of a flat, square coil with 120 turns and sides with a length of 1.60 cm. The coil rotates in a magnetic field of 0.0750 T. What is the angular speed of the coil if the maximum emf produced is 24.0 mV?

A type-II superconductor in an external field between \(B_{c1}\) and \(B_{c2}\) has regions that contain magnetic flux and have resistance, and also has superconducting regions. What is the resistance of a long, thin cylinder of such material? Text Transcription: B_c1 B_c2

A flat, rectangular coil of dimensions and is pulled with uniform speed through a uniform magnetic field with the plane of its area perpendicular to the field (Fig. E29.14). (a) Find the emf induced in this coil. (b) If the speed and magnetic field are both tripled, what is the induced emf?

Problem 15DQ Can one have a displacement current as well as a conduction current within a conductor? Explain.

A circular loop of wire is in a region of spatially uniform magnetic field, as shown in Fig. E29.15. The magnetic field is directed into the plane of the figure. Determine the direction (clockwise or counterclockwise) of the induced current in the loop when (a) is increasing; (b) is decreasing; (c) is constant with value . Explain your reasoning.

A rectangular loop of wire with dimensions by and resistance \(R=0.600 \Omega\) is being pulled to the right out of a region of uniform magnetic field. The magnetic field has magnitude and is directed into the plane of Fig. E29.22. At the instant when the speed of the loop is and it is still partially in the field region, what force (magnitude and direction) does the magnetic field exert on the loop? Equation transcription: Text transcription: R=0.600 \Omega

In Fig. E29.23 a conducting rod of length moves in a magnetic field \(\vec{B}\) of magnitude directed into the plane of the figure. The rod moves with speed \(v=5.00 \mathrm{~m} / \mathrm{s}\) in the direction shown. (a) What is the potential difference between the ends of the rod? (b) Which point, or , is at higher potential? (c) When the charges in the rod are in equilibrium, what are the magnitude and direction of the electric field within the rod? (d) When the charges in the rod are in equilibrium, which point, or , has an excess of positive charge? (e) What is the potential difference across the rod if it moves (i) parallel to and (ii) directly out of the page? Equation transcription: Text transcription: \vec{B} v=5.00{~m} / s}

A rectangle measuring by is located inside a region of a spatially uniform magnetic field of , with the field perpendicular to the plane of the coil (Fig. E29.24). The coil is pulled out at a steady rate of traveling perpendicular to the field lines. The region of the field ends abruptly as shown. Find the emf induced in this coil when it is (a) all inside the field; (b) partly inside the field; (c) all outside the field. \(30.0 \mathrm{~cm}\) \(40.0 \mathrm{~cm}\) \(2.00 \mathrm{~cm} / \mathrm{s}\) Equation transcription: Text transcription: 30.0{~cm} 40.0{~cm} 2.00{~cm} /{s}

A 0.250-m-long bar moves on parallel rails that are connected through a resistor, as shown in Fig. E29.31, so the apparatus makes a complete circuit. You can ignore the resistance of the bar and rails. The circuit is in a uniform magnetic field that is directed into the plane of the figure. At an instant when the induced current in the circuit is counterclockwise and equal to , what is the velocity of the bar (magnitude and direction)?

BIO Measuring Blood Flow. Blood contains positive and negative ions and thus is a conductor. A blood vessel, therefore, can be viewed as an electrical wire. We can even picture the flowing blood as a series of parallel conducting slabs whose thickness is the diameter \(d\) of the vessel moving with speed \(v\). (See Fig. E29.32.) (a) If the blood vessel is placed in a magnetic field \(B\) perpendicular to the vessel, as in the figure, show that the motional potential difference induced across it is \(\varepsilon=v B d\). (b) If you expect that the blood will be flowing at \(15 \mathrm{~cm} / \mathrm{s}\) for a vessel \(5.0 \mathrm{~mm}\) in diameter, what strength of magnetic field will you need to produce a potential difference of \(1.0 \mathrm{mV}\) ? (c) Show that the volume rate of flow \((R)\) of the blood is equal to \(R= \pi Ed /4B\) (Note: Although the method developed here is useful in measuring the rate of blood flow in a vessel, it is limited to use in surgery because measurement of the potential \(\varepsilon\) must be made directly across the vessel.)

A 1.41-m bar moves through a uniform, -T magnetic field with a speed of (Fig. E29.33). In each case, find the emf induced between the ends of this bar and identify which, if any, end or is at the higher potential. The bar moves in the direction of (a) the -axis; (b) the -axis; (c) the -axis. (d) How should this bar move so that the emf across its ends has the greatest possible value with at a higher potential than , and what is this maximum emf?

A rectangular circuit is moved at a constant velocity of into, through, and then out of a uniform -T magnetic field, as shown in Fig. E29.34. The magnetic-field region is considerably wider than . Find the magnitude and direction (clockwise or counterclockwise) of the current induced in the circuit as it is (a) going into the magnetic field; (b) totally within the magnetic field, but still moving; and (c) moving out of the field. (d) Sketch a graph of the current in this circuit as a function of time, including the preceding three cases. \(3.0 \mathrm{~m} / \mathrm{s}\) \(12.5 \Omega\) \(75.0 \mathrm{~cm}\) \(50.0 \mathrm{~cm}\) Equation transcription: Text transcription: 3.0{~m} /{s} 12.5 Omega 75.0{~cm} 50.0{~cm}

Problem 35E The magnetic field within a long, straight solenoid with a circular cross section and radius R is increasing at a rate of dB/dt. (a) What is the rate of change of flux through a circle with radius r1 inside the solenoid, normal to the axis of the solenoid, and with center on the solenoid axis? (b) Find the magnitude of the induced electric field inside the solenoid, at a distance r1 from its axis. Show the direction of this field in a diagram. (c) What is the magnitude of the induced electric field outside the solenoid, at a distance r2 from the axis? (d) Graph the magnitude of the induced electric field as a function of the distance r from the axis from r = 0 to r = 2R. (e) What is the magnitude of the induced emf in a circular turn of radius R/2 that has its center on the solenoid axis? (f) What is the magnitude of the induced emf if the radius in part (e) is R? (g) What is the induced emf if the radius in part (e) is 2R?

A long, thin solenoid has 900 turns per meter and radius 2.50 cm. The current in the solenoid is increasing at a uniform rate of 60.0 A/s. What is the magnitude of the induced electric field at a point near the center of the solenoid and (a) 0.500 cm from the axis of the solenoid; (b) 1.00 cm from the axis of the solenoid?

Problem 37E A long, thin solenoid has 400 turns per meter and radius 1.10 cm. The current in the solenoid is increasing at a uniform rate di/dt. The induced electric field at a point near the center of the solenoid and 3.50 cm from its axis is 8.00 × 10-6 V/m. Calculate di/dt.

A metal ring 4.50 cm in diameter is placed between the north and south poles of large magnets with the plane of its area perpendicular to the magnetic field. These magnets produce an initial uniform field of 1.12 T between them but are gradually pulled apart, causing this field to remain uniform but decrease steadily at 0.250 T/s. (a) What is the magnitude of the electric field induced in the ring? (b) In which direction (clockwise or counterclockwise) does the current flow as viewed by someone on the south pole of the magnet?

Problem 39E A long, straight solenoid with a cross-sectional area of 8.00 cm2 is wound with 90 turns of wire per centimeter, and the windings carry a current of 0.350 A. A second winding of 12 turns encircles the solenoid at its center. The current in the solenoid is turned off such that the magnetic field of the solenoid becomes zero in 0.0400 s. What is the average induced emf in the second winding?

The magnetic field \(\vec{B}\) at all points within the colored circle shown in Fig. E29.15 has an initial magnitude of . (The circle could represent approximately the space inside a long, thin solenoid.) The magnetic field is directed into the plane of the diagram and is decreasing at the rate of . (a) What is the shape of the field lines of the induced electric field shown in Fig. E29.15, within the colored circle? (b) What are the magnitude and direction of this field at any point on the circular conducting ring with radius (c) What is the current in the ring if its resistance is \(4.00 \Omega\) is the emf between points and on the ring? (e) If the ring is cut at some point and the ends are separated slightly, what will be the emf between the ends? Equation transcription: Text transcription: vec{B} 4.00 Omega

Problem 41E The electric flux through a certain area of a dielectric is (8.76 × 103 V ? m/s4) t4. The displacement current through that area is 12.9 pA at time t = 26.1 ms. Calculate the dielectric constant for the dielectric.

A parallel-plate, air-filled capacitor is being charged as in Fig. The circular plates have radius , and at a particular instant the conduction current in the wires is A. (a) What is the displacement current density \(j_{D}\) in the air space between the plates? (b) What is the rate at which the electric field between the plates is changing? (c) What is the induced magnetic field between the plates at a distance of from the axis? (d) At from the axis? Equation transcription: Text transcription: j{D}

A flat coil is oriented with the plane of its area at right angles to a spatially uniform magnetic field. The magnitude of this field varies with time according to the graph in Fig. P29.52. Sketch a qualitative (but accurate!) graph of the emf induced in the coil as a function of time. Be sure to identify the times \(t_{1}, t_{2}\), and \(t_{3}\) on your graph. Text transcription: Text transcription: t{1}, t{2} t{3}

In Fig. P29.53 the loop is being pulled to the right at constant speed v. A constant current I flows in the long wire, in the direction shown. (a) Calculate the magnitude of the net \(\text { emf } \varepsilon\) induced in the loop. Do this two ways: (i) by using Faraday's law of induction (Hint: See Exercise and (ii) by looking at the emf induced in each segment of the loop due to its motion. (b) Find the direction (clockwise or counterclockwise) of the current induced in the loop. Do this two ways: (i) using Lenz's law and (ii) using the magnetic force on charges in the loop. (c) Check your answer for the emf in part (a) in the following special cases to see whether it is physically reasonable: (i) The loop is stationary; (ii) the loop is very thin, so \(a \rightarrow 0\); (iii) the loop gets very far from the wire. Equation Transcription: Text Transcription: emf epsilon a rightarrow 0

Suppose the loop in Fig. P29.54 is (a) rotated about the -axis; (b) rotated about the -axis; (c) rotated about an edge parallel to the -axis. What is the maximum induced emf in each case if \(A=600 \mathrm{~cm}^{2}, \omega=35.0 r a d / s\), and Equation transcription: Text transcription: A=600{~cm}^{2}, omega=35.0 r a d / s

Problem 61P Antenna emf. A satellite, orbiting the earth at the equator at an altitude of 400 km, has an antenna that can be modeled as a 2.0-m-long rod. The antenna is oriented perpendicular to the earth’s surface. At the equator, the earth’s magnetic field is essentially horizontal and has a value of 8.0 × 10?5 T; ignore any changes in B with altitude. Assuming the orbit is circular, determine the induced emf between the tips of the antenna.

emf in a Bullet. At the equator, the earth’s magnetic field is approximately horizontal, is directed toward the north, and has a value of \(8\times 10^{-5}\mathrm{\ T}\). (a) Estimate the emf induced between the top and bottom of a bullet shot horizontally at a target on the equator if the bullet is shot toward the east. Assume the bullet has a length of 1 cm and a diameter of 0.4 cm and is traveling at 300 m/s. Which is at higher potential: the top or bottom of the bullet? (b) What is the emf if the bullet travels south? (c) What is the emf induced between the front and back of the bullet for any horizontal velocity?

CALC A very long, cylindrical wire of radius carries a current \(I_{0}\) uniformly distributed across the cross section of the wire. Calculate the magnetic flux through a rectangle that has one side of length running down the center of the wire and another side of length , as shown in Fig. P29.63 (see Exercise 29.7). Equation transcription: Text transcription: I{0}

CALC A circular conducting ring with radius \(r_{0}=0.0420\) m lies in the xy-plane in a region of uniform magnetic field \(\overrightarrow{\boldsymbol{B}}=B_{0}\left[1-3\left(t / t_{0}\right)^{2}+\right. \left.2\left(t / t_{0}\right)^{3}\right] \hat{\boldsymbol{k}}\). In this expression, \(t_0=0.0100 \mathrm {\ s}\) and is constant, t is time, \(\hat{\boldsymbol{k}}\) is the unit vector in the \(+z \text {-direction}\), and \(B_0=0.0800 \mathrm {\ T}\) and is constant. At points a and b (Fig. P29.64) there is a small gap in the ring with wires leading to an external circuit of resistance \(R=12.0\ \Omega\). There is no magnetic field at the location of the external circuit. (a) Derive an expression, as a function of time, for the total magnetic flux \(\Phi_{B}\) through the ring. (b) Determine the emf induced in the ring at time \(t=5.00 \times 10^{-3} \mathrm {\ s}\). What is the polarity of the emf? (c) Because of the internal resistance of the ring, the current through R at the time given in part (b) is only 3.00 mA. Determine the internal resistance of the ring. (d) Determine the emf in the ring at a time \(t=1.21 \times 10^{-2} \mathrm {\ s}\). What is the polarity of the emf? (e) Determine the time at which the current through R reverses its direction.

CALC The long, straight wire shown in Fig. P29.65a carries constant current A metal bar with length is moving at constant velocity \(\vec{v}\), as shown in the figure. Point is a distance from the wire. (a) Calculate the emf induced in the bar. (b) Which point, or , is at higher potential? (c) If the bar is replaced by a rectangular wire loop of resistance (Fig. P29.65b), what is the magnitude of the current induced in the loop? Equation transcription: Text transcription: vec{v}

The cube shown in Fig. on a side, is in a uniform magnetic field of , directed along the positive -axis. Wires , and move in the directions indicated, each with a speed of (Wire moves parallel to the -plane, moves at an angle of \(45.0^{0}\) below the -plane, and moves parallel to the -plane.) What is the potential difference between the ends of each wire? Equation transcription: Text transcription: 45.0^{0}

Problem 67P CALC A slender rod, 0.240 m long, rotates with an angular speed of 8.80 rad/s about an axis through one end and perpendicular to the rod. The plane of rotation of the rod is perpendicular to a uniform magnetic field with a magnitude of 0.650 T. (a) What is the induced emf in the rod? (b) What is the potential difference between its ends? (c) Suppose instead the rod rotates at 8.80 rad/s about an axis through its center and perpendicular to the rod. In this case, what is the potential difference between the ends of the rod? Between the center of the rod and one end?

A Magnetic Exercise Machine. You have designed a new type of exercise machine with an extremely simple mechanism (Fig. E29.28). A vertical bar of silver (chosen for its low resistivity and because it makes the machine look cool) with length L = 3.0 m is free to move left or right without friction on silver rails. The entire apparatus is placed in a horizontal, uniform magnetic field of strength 0.25 T. When you push the bar to the left or right, the bar’s motion sets up a current in the circuit that includes the bar. The resistance of the bar and the rails can be neglected. The magnetic field exerts a force on the current-carrying bar, and this force opposes the bar’s motion. The health benefit is from the exercise that you do in working against this force. (a) Your design goal is that the person doing the exercise is to do work at the rate of 25 watts when moving the bar at a steady 2.0 m/s. What should be the resistance R? (b) You decide you want to be able to vary the power required from the person, to adapt the machine to the person’s strength and fitness. If the power is to be increased to 50 W by altering R while leaving the other design parameters constant, should R be increased or decreased? Calculate the value of R for 50 W. (c) When you start to construct a prototype machine, you find it is difficult to produce a 0.25-T magnetic field over such a large area. If you decrease the length of the bar to 0.20 m while leaving B, v and R the same as in part (a), what will be the power required of the person?

CP CALC A rectangular loop with width and a slide wire with mass are as shown in Fig. P29.69. A uniform magnetic field \(\vec{B}\) is directed perpendicular to the plane of the loop into the plane of the figure. The slide wire is given an initial speed of \(V_{0}\) and then released. There is no friction between the slide wire and the loop, and the resistance of the loop is negligible in comparison to the resistance of the slide wire. (a) Obtain an expression for , the magnitude of the force exerted on the wire while it is moving at speed . (b) Show that the distance that the wire moves before coming to rest is \(x=m v_{0} R / a^{2} B^{2}\). Equation transcription: Text transcription: \vec{B} V{0} x=m v{0} R / a^{2} B^{2}

A -cm-long metal rod lies in the -plane and makes an angle of with the positive -axis and an angle of \(53.1^{0}\) with the positive -axis. The rod is moving in the -direction with a speed of .The rod is in a uniform magnetic field \(\vec{B}=(0.120 T) \hat{i}-(0.220 T) \bar{J}-(0.09007) \bar{K}\) (a) What is the magnitude of the emf induced in the rod? (b) Indicate in a sketch which end of the rod is at higher potential. Equation transcription: Text transcription: 53.1^{0} vec{B}=(0.120 T)hat{i}-(0.220 T) bar{J}-(0.09007) bar{K}

The magnetic field \(\vec{B}\), at all points within a circular region of radius , is uniform in space and directed into the plane of the page as shown in Fig. P29.71. (The region could be a cross section inside the windings of a long, straight solenoid.) If the magnetic field is increasing at a rate \(d B / d t\), what are the magnitude and direction of the force on a stationary positive point charge located at points , and (Point is a distance above the center of the region, point is a distance to the right of the center, and point is at the center of the region.) Equation transcription: Text transcription: vec{B} d B / d t

Problem 72P CALC An airplane propeller of total length L rotates around its center with angular speed ? in a magnetic field that is perpendicular to the plane of rotation. Modeling the propeller as a thin, uniform bar, find the potential difference between (a) the center and either end of the propeller and (b) the two ends. (c) If the field is the earth’s field of 0.50 G and the propeller turns at 220 rpm and is 2.0 m long, what is the potential difference between the middle and either end? It this large enough to be concerned about?