Show that the inductance of a toroid of rectangular cross

(a) The magnetic equator is a line on the surface of Earth on which Earths magnetic field is horizontal. At the magnetic equator, how would you orient a flat sheet of paper so as to create the maximum magnitude of magnetic flux through it? (b) How about the minimum magnitude of magnetic flux

At one of Earths magnetic poles, how would you orient a flat sheet of paper so as to create the maximum magnitude of magnetic flux through it

Show that the following combination of SI units is equivalent to the volt

Show that the following combination of SI units is equivalent to the ohm

A current is induced in a conducting loop that lies in a horizontal plane, and the induced current is clockwise when viewed from above. Which of the following statements could be true? (a) Aconstant magnetic field is directed vertically downward. (b) A constant magnetic field is directed vertically upward. (c) A magnetic field whose magnitude is increasing is directed vertically downward. (d) A magnetic field whose magnitude is decreasing is directed vertically downward. (e) A magnetic field whose magnitude is decreasing is directed vertically upward

Give the direction of the induced current in the circuit, shown on the right in Figure 28-37, when the resistance in the circuit on the left is suddenly (a) increased and (b) decreased. Explain your answer

The planes of the two circular loops in Figure 28-38 are parallel. As viewed from the left, a counterclockwise current exists in loop A. If the magnitude of the current in loop A is increasing, what is the direction of the current induced in loop B? Do the loops attract or repel each other? Explain your answer

A bar magnet moves with constant velocity along the axis of a loop, as shown in Figure 28-39. (a) Make a graph of the magnetic flux through the loop as a function of time. Indicate on the graph when the magnet is halfway through the loop by designating this time Choose the direction of the normal to the flat surface bounded by the loop to be to the right. (b) Make a graph of the induced current in the loop as a function of time. Choose the positive direction for the current to be clockwise as viewed from the left

Abar magnet is mounted on the end of a coiled spring and is oscillating in simple harmonic motion along the axis of a loop, as shown in Figure 28-40. The magnet is in its equilibrium position when its midpoint is in the plane of the loop. (a) Make a graph of the magnetic flux through the loop as a function of time. Indicate when the magnet is halfway through the loop by designating these times and (b) Make a graph of the induced current in the loop as a function of time, choosing the current to be positive when it is clockwise as viewed from above.

Apendulum is fabricated from a thin, flat piece of aluminum. At the bottom of its arc, it passes between the poles of a strong permanent magnet. In Figure 28-41a, the metal sheet is continuous, whereas in Figure 28-41b, there are slots in the sheet. When released from the same angle, the pendulum that has slots swings back and forth many times, but the pendulum that does not have slots stops swinging after no more than one complete oscillation. Explain why.

A bar magnet is dropped inside a long, vertical tube. If the tube is made of metal, the magnet quickly approaches a terminal speed, but if the tube is made of cardboard, the magnet falls with constant acceleration. Explain why the magnet falls differently in the metal tube than it does in the cardboard tube.

Asmall square wire loop lies in the plane of this page, and a constant magnetic field is directed into the page. The loop is moving to the right, which is the direction. Find the direction of the induced current, if any, in the loop if (a) the magnetic field is uniform, (b) the magnetic field strength increases as increases, and (c) the magnetic field strength decreases as x increases.

If the current in an inductor doubles, the energy stored in 25 turns 5.0 cm. the inductor will (a) remain the same, (b) double, (c) quadruple, (d) halve.

Two solenoids are equal in length and radius, and the cores of both are identical cylinders of iron. However, solenoid A has three times the number of turns per unit length as solenoid B. (a) Which solenoid has the larger self-inductance? (b) What is the ratio of the self-inductance of solenoid A to the self-inductance of solenoid B?

True or false: (a) The induced emf in a circuit is equal to the negative of the magnetic flux through the circuit. (b) There can be a nonzero induced emf at an instant when the flux through the circuit is equal to zero. (c) The self-inductance of a solenoid is proportional to the rate of change of the current in the solenoid. (d) The magnetic energy density at some point in space is proportional to the square of the magnitude of the magnetic field at that point. (e) The inductance of a solenoid is proportional to the current in it.

CONTEXT-RICH Your baseball teammates, having just studied this chapter, are concerned about generating enough voltage to shock them while swinging aluminum bats at fastballs. Estimate the maximum possible motional emf measured between the ends of an aluminum baseball bat during a swing. Do you think your team should switch to wooden bats to avoid electrocution?

Compare the energy density stored in Earths electric field near its surface to that stored in Earths magnetic field near its surface.

Aphysics teacher does the following emf demonstration. She has two students hold a long wire connected to a voltmeter. The wire is held slack, so that it sags with a large arc in it. When she says, Start, the students begin rotating the wire as if they were playing jump rope. The students stand apart, and the sag in the wire is about The motional emf from the jump rope is then measured on the voltmeter. (a) Estimate a reasonable value for the maximum angular speed at which the students can rotate the wire. (b) From this, estimate the maximum motional emf in the wire. Hint: What field is involved in creating the induced emf?

(a) Estimate the maximum possible motional emf between the wingtips of a typical commercial airliner in flight. (b) Estimate the magnitude of the electric field between the wingtips.

A uniform magnetic field of magnitude is in the direction. A square coil that has 5.00-cm-long sides has a single turn and makes an angle with the axis, as shown in Figure 28-42. Find the magnetic flux through the coil when is (a) (b) (c) 60, and (d) 90.

Acircular coil has and a radius of It is at the equator, where Earths magnetic field is , north. The axis of the coil is the line that passes through the center of the coil and is perpendicular to the plane of the coil. Find the magnetic flux through the coil when the axis of the coil is (a) vertical, (b) horizontal with the axis pointing north, (c) horizontal with the axis pointing east, and (d) horizontal with the axis making an angle of with north.

Amagnetic field of is perpendicular to the plane of a 14-turn square coil with sides long. (a) Find the magnetic flux through the coil. (b) Find the magnetic flux through the coil if the magnetic field makes an angle of with the normal to the plane of the coil.

A uniform magnetic field is perpendicular to the base of a hemisphere of radius Calculate the magnetic flux (in terms of and through the spherical surface of the hemisphere.

Find the magnetic flux through a 400-turn solenoid that has a length equal to has a radius equal to and carries a current of

Find the magnetic flux through a 800-turn solenoid that has a length equal to has a radius equal to and carries a current of

Acircular coil has has a radius of and is in a uniform magnetic field of in the direction. Find the flux through the coil when the unit normal to the plane of the coil is (a) (b) (c) (d) and (e)

A long solenoid has turns per unit length, has a radius and carries a current A circular coil with radius and with total turns is coaxial with the solenoid and equidistant from its ends. (a) Find the magnetic flux through the coil if (b) Find the magnetic flux through the coil if

(a) Compute the magnetic flux through the rectangular loop shown in Figure 28-43. (b) Evaluate your answer for and

A long, cylindrical conductor with a radius and a length carries a current . Find the magnetic flux per unit length through the area indicated in Figure 28-44.

The flux through a loop is given by where is in webers and is in seconds. (a) Find the induced emf as a function of time. (b) Find both and at t _ 4.0 s, and t _ 6.0 s.

The flux through a loop is given by where is in webers and is in seconds. (a) Sketch graphs of magnetic flux and induced emf as a function of time. (b) At what time(s) is the flux minimum? What is the induced emf at that (those) time(s)? (c) At what time(s) is the flux zero? What is (are) the induced emf(s) at those time(s)?

A solenoid that has a length equal to a radius equal to and 400 turns is in a region where a magnetic field of exists and makes an angle of with the axis of the solenoid. (a) Find the magnetic flux through the solenoid. (b) Find the magnitude of the average emf induced in the solenoid if the magnetic field is reduced to zero in

A100-turn circular coil has a diameter of and a resistance of and the two ends of the coil are connected together. The plane of the coil is perpendicular to a uniform magnetic field of magnitude The direction of the field is reversed. (a) Find the total charge that passes through a cross section of the wire. If the reversal takes find (b) the average current and (c) the average emf during the reversal.

At the equator, a 1000-turn coil that has a cross-sectional area of and a resistance of is aligned so that its plane is perpendicular to Earths magnetic field of (a) If the coil is flipped over in what is the average induced current in it during the (b) How much charge flows through a cross section of the coil wire during the

ENGINEERING APPLICATION Acurrent integrator measures the current as a function of time and integrates (adds) the current to find the total charge passing through it. (Because the integrator calculates the integral of the current or Acircular coil that has and a radius equal to is connected to such an instrument. The total resistance of the circuit is The plane of the coil is originally aligned perpendicular to Earths magnetic field at some point. When the coil is rotated through about an axis that is in the plane of the coil a charge of passes through the current integrator. Calculate the magnitude of Earths magnetic field at that point.

A 30.0-cm-long rod moves steady at in a plane that is perpendicular to a magnetic field of The velocity of the rod is perpendicular to its length. Find (a) the magnetic force on an electron in the rod, (b) the electrostatic field in the rod, and (c) the potential difference between the ends of the rod.

A 30.0-cm-long rod moves in a plane that is perpendicular to a magnetic field of The velocity of the rod is perpendicular to its length. Find the speed of the rod if the potential difference between the ends is

In Figure 28-45, let the magnetic field strength be the rod speed be the rod length be and the resistance of the resistor be (The resistance of the rod and rails are negligible.) Find (a) the induced emf in the circuit, (b) the induced current in the circuit (including direction), and (c) the force needed to move the rod with constant speed (assuming negligible friction). Find (d) the power delivered by the force found in Part (c) and (e) the rate of Joule heating in the resistor.

A 10-cm by 5.0-cm rectangular loop (Figure 28-46) that has a resistance equal to moves at a constant speed of through a region that has a uniform 1.7-T magnetic field directed out of the page as shown. The front of the loop enters the field region at time (a) Graph the flux through the loop as a function of time. (b) Graph the induced emf and the current in the loop as functions of time. Neglect any self-inductance of the loop and construct your graphs to include the interval

A uniform 1.2-T magnetic field is in the direction. A conducting rod of length lies parallel to the axis and oscillates in the direction with displacement given by where has units of (a) Find an expression for potential difference between the ends of the rod as a function of time? (b) What is the maximum potential difference between the ends the rod?

In Figure 28- 47, the rod has a mass and a resistance The rails are horizontal, frictionless, and have negligible resistances. The distance between the rails is An ideal battery that has an emf is connected between points and so that the current in the rod is downward. The rod is released from rest at (a) Derive an expression for the force on the rod as a function of the speed. (b) Show that the speed of the rod approaches a terminal speed and find an expression for the terminal speed. (c) What is the current when the rod is moving at its terminal speed?

A uniform magnetic field is established perpendicular to the plane of a loop that has a radius equal to and a resistance equal to The magnitude of the field is increasing at a rate of Find (a) the magnitude of the induced emf in the loop, (b) the induced current in the loop, and (c) the rate of Joule heating in the loop.

In Figure 28- 48, a conducting rod that has a mass and a negligible resistance is free to slide without friction along two parallel frictionless rails that have negligible resistances separated by a distance and connected by a resistance The rails are attached to a long inclined plane that makes an angle with the horizontal. There is a magnetic field directed upward as shown. (a) Show that there is a retarding force directed up the incline given by (b) Show that the terminal speed of the rod is

A conducting rod of length rotates at constant angular speed about one end, in a plane perpendicular to a uniform magnetic field (Figure 28-49). (a) Show that the potential difference between the ends of the rod is (b) Let the angle u between the rotating rod and the dashed line be given by Show that the area of the pie-shaped region swept out by the rod during time is (d) Compute the flux through that area, and apply (Faradays law) to show that the motional emf is given by

A 2.00-cm by 1.50-cm rectangular coil has and rotates in a region that has magnetic field of (a) What is the maximum emf generated when the coil rotates at (b) What must its angular speed be to generate a maximum emf of

The coil of Problem at in a magnetic field. If the maximum emf generated by the coil is ,what is the magnitude of the magnetic field?

When the current in an 8.00-H coil is equal to and is increasing at find (a) the magnetic flux through the coil and (b) the induced emf in the coil.

A 300-turn solenoid has a radius equal to and a length equal to a 1000-turn solenoid has a radius equal to and is also long. The two solenoids are coaxial, with one nested completely inside the other. What is their mutual inductance?

An insulated wire that has a resistance of and a length of will be used to construct a resistor. First, the wire is bent in half and then the doubled wire is wound on a cylindrical form (Figure 28-50) to create a 25.0-cm-long helix that has a diameter equal to Find both the resistance and the inductance of this wire-wound resistor. SSM

You are given a length of wire that has radius and are told to wind it into an inductor in the shape of a helix that has a circular cross section of radius The windings are to be as close together as possible without overlapping. Show that the self-inductance of this inductor is

Using the result of 50, calculate the self-inductance of an inductor wound from 10 cm of wire that has a diameter of 1.0 mm into a coil that has a radius of 0.25 cm.

In Figure 28-51, circuit 2 has a total resistance of After switch is closed, the current in Circuit 1 increasesreaching a value of 5.00 A after a long time. A charge of passes through the galvanometer in Circuit 2 during the time that the current in Circuit 1 is increasing. What is the mutual inductance between the two coils?

Show that the inductance of a toroid of rectangular cross section, as shown in Figure 28-52, is given by where is the total number of turns, is the inside radius, is the outside radius, and H is the height of the toroid. SSM

Acoil that has a self-inductance of and a resistance of is connected to an ideal 24.0-V battery. (a) What is the steady-state current? (b) How much energy is stored in the inductor when the steady-state current is established?

In a plane electromagnetic wave, the magnitudes of the electric fields and magnetic fields are related by where is the speed of light. Show that when the electric and the magnetic energy densities are equal.

A solenoid has a cross-sectional area equal to and a length equal to The solenoid carries a current of (a) Calculate the magnetic energy stored in the solenoid using where (b) Divide your answer in Part (a) by the volume of the region inside the solenoid to find the magnetic energy per unit volume in the solenoid. (c) Check your Part (b) result by computing the magnetic energy density from where

A long cylindrical wire has a radius equal to and carries a current of uniformly distributed over its cross-sectional area. Find the magnetic energy per unit length within the wire.

A toroid that has a mean radius equal to and circular loops with radii equal to is wound with a superconducting wire. The wire has a length equal to and carries a current of (a) What is the number of turns of the wire? (b) What is the magnetic field strength and magnetic energy density at the mean radius? (c) Estimate the total energy stored in this toroid by assuming that the energy density is uniformly distributed in the region inside the toroid.

A circuit consists of a coil that has a resistance equal to and a self-inductance equal to an open switch, and an ideal 100-V batteryall connected in series. At the switch is closed. Find the current and its rate of change at times (a) (b) t _ 0.100 ms, (c) t _ 0.500 ms, and (d) t _ 1.00 ms. SSM

In the circuit shown in Figure 28-53, the throw of the make-before-break switch has been at contact for a long time and the current in the coil is equal to At the throw is quickly moved to contact The total resistance of the coil and the resistor is Find the current when (a) and (b) t _ 10.0 ms.

In the circuit shown in Figure 28-54, let and The switch, which was initially open, is closed at time At time find (a) the rate at which the battery supplies energy, (b) the rate of Joule heating in the resistor, and (c) the rate at which energy is being stored in the inductor. SSM

How many time constants must elapse before the current in an circuit (Figure 28-54) that is initially zero reaches (a) 90 percent, (b) 99 percent, and (c) 99.9 percent of its steady-state value?

A circuit consists of a 4.00-mH coil, a resistor, a 12.0-V ideal battery, and an open switchall connected in series. After the switch is closed: (a) What is the initial rate of increase of the current? (b) What is the rate of increase of the current when the current is equal to half its steady-state value? (c) What is the steady-state value of the current? (d) How long does it take for the current to reach 99 percent of its steady state value?

Acircuit consists of a large electromagnet that has an inductance of and a resistance of a dc 250-V power source, and an open switchall connected in series. How long after the switch is closed is the current equal to (a) and (b)

SPREADSHEET Given the circuit shown in Figure 28-55, assume that the inductor has negligible internal resistance and that the switch S has been closed for a long time so that a steady current exists in the inductor. (a) Find the battery current, the current in the resistor, and the current in the inductor. (b) Find the potential drop across the inductor immediately after the switch S is opened. (c) Using a spreadsheet program, make graphs of the current in the inductor and the potential drop across the inductor as functions of time, for the period during which the switch is open. SSM

Given the circuit shown in Figure 28-56, the inductor has negligible internal resistance and the switch has been open for a long time. The switch is then closed. (a) Find the current in the battery, the current in the resistor, and the current in the inductor immediately after the switch is closed. (b) Find the current in the battery, the current in the resistor, and the current in the inductor a long time after the switch is closed. After being closed for a long time the switch is now opened. (c) Find the current in the battery, the current in the resistor, and the current in the inductor immediately after the switch is opened. (d) Find the current in the battery, the current in the resistor, and the current in the inductor after the switch is opened for a long time.

An inductor, two resistors, a make-before-break switch, and a battery are connected as shown in Figure 28-57. The switch throw has been at contact for a long time and the current in the inductor is Then, at the throw is quickly moved to contact During the next the current in the inductor drops to (a) What is the time constant for this circuit? (b) If the resistance R is equal to 0.40 , what is the value of the inductance L?

A circuit consists of a coil that has a self-inductance equal to and an internal resistance equal to an ideal 12.0-V battery, and an open switchall connected in series (Figure 28-58). At the switch is closed. Find the time when the rate at which energy is dissipated in the coil equals the rate at which magnetic energy is stored in the coil.

In the circuit shown in Figure 28-54, let and The switch is closed at time During the time from to find (a) the amount of energy supplied by the battery, (b) the amount of energy dissipated in the resistor, and (c) the amount of energy delivered to the inductor. Hint: Find the energy transfer rates as functions of time and integrate. SSM

A100-turn coil has a radius of and a resistance of (a) The coil is in a uniform magnetic field that is perpendicular to the plane of the coil. What rate of change of the magnetic field strength will induce a current of in the coil? (b) What rate of change of the magnetic field strength is required if the magnetic field makes an angle of 20 with the normal to the plane of the coil?

ENGINEERING APPLICATION Figure 28-59 shows a schematic drawing of an ac generator. The basic generator consists of a rectangular loop of dimensions and and has turns connected to slip rings. The loop rotates (driven by a gasoline engine) at an angular speed of in a uniform magnetic field (a) Show that the induced potential difference between the two slip rings is given by (b) If and at what angular frequency must the coil rotate to generate an emf whose maximum value is

ENGINEERING APPLICATION Prior to 1960, magnetic field strengths were usually measured by a rotating coil gaussmeter. The device uses a small multiturn coil rotating at a high speed on an axis perpendicular to the magnetic field. The coil is connected to an ac voltmeter by means of slip rings, like those shown in Figure 28-59. In one specific design, the rotating coil has and an area of The coil rotates at If the magnetic field strength is find the maximum induced emf in the coil and the orientation of the normal to the plane of the coil relative to the field for which the maximum induced emf occurs. 180 rev>min. 0.450 T, 400 turns 1.40 cm2. 100 V? SSM

Show that the equivalent self-inductance for two inductors that have self-inductances and and are connected in series is given by if there is no flux linkage between the two inductors. (Saying there is no flux linkage between them is equivalent to saying that the mutual inductance between them is zero.)

Show that the equivalent self-inductance for two in ductors that have self-inductances and and are connected in parallel is given by if there is no flux linkage between the two inductors. (Saying there is no flux linkage between them is equivalent to saying that the mutual inductance between them is equal to zero.)

A circuit consists of a 12-V battery, a switch, and a lightbulb all connected in series. It is known that the lightbulb requires a minimum current of in order to produce a visible glow. In the circuit, that particular bulb draws when the switch has been closed for a long time. Now, an inductor is put in series with the bulb and the rest of the circuit. If the lightbulb begins to glow after the switch is closed, how large is the selfinductance of the inductor? Ignore any heating time of the filament and assume the glow is observed as soon as the current in the filament reaches the 0.10-A threshold.

Your friend decides to generate electrical power by rotating a coil of wire around an axis in the plane of the coil and through its center. The coil is perpendicular to Earths magnetic field in a region where the field strength is equal to The loops of the coil have a radius of and the coil has negligible resistance. (a) If your friend turns the coil at a rate of what peak current will exist in a resistor that is connected across the terminals of the coil? (b) The average of the square of the current will equal half of the square of the peak current. What will be the average power delivered to the resistor? Is this an economical way to generate power? Hint: Energy has to be expended to keep the coil rotating.

Figure 28-60a shows an experiment designed to measure the acceleration due to gravity. A large plastic tube is encircled by a wire, which is arranged in single loops separated by a distance of A strong magnet is dropped through the top of the loop. As the magnet falls through each loop the voltage rises; then the voltage rapidly falls through zero to a large negative value and then returns to zero. The shape of the voltage signal is shown in Figure 28-60b. (a) Explain the basic physics behind the generation of this voltage pulse. (b) Explain why the tube cannot be made of a conductive material. (c) Qualitatively explain the shape of the voltage signal in Figure 28-60b. (d) The times at which the voltage crosses zero as the magnet falls through each loop in succession are given in the table on the next page. Use these data to calculate a value for g. SSM

The rectangular coil shown in Figure 28-61 has 80 turns, is \(25 \mathrm{~cm}\) wide, is \(30 \mathrm{~cm}\) long, and is located in a magnetic field of \(0.14 \mathrm{~T}\) directed out of the page, as shown. Only half of the coil is in the region of the magnetic field. The resistance of the coil is \(24 \Omega\). Find the magnitude and the direction of the induced current if the coil is moving with a velocity of \(2.0 \mathrm{~m} / \mathrm{s}\) (a) to the right, (b) up the page, (c) to the left, and (d) down the page.

A long solenoid has turns per unit length and carries a current that varies with time according to The solenoid has a circular cross section of radius Find the induced electric field, at points near the plane equidistant from the ends of the solenoid, as a function of both the time and the perpendicular distance from the axis of the solenoid for (a) and (b)

Acoaxial cable consists of two very thin-walled conducting cylinders of radii and (Figure 28-62). The currents in the inner and outer cylinders are equal in magnitude but opposite in direction. (a) Use Ampres law to find the magnetic field as a function of the perpendicular distance from the central axis of the cable for (1) (2) and (3) (b) Show that the magnetic energy density in the region between the cylinders is given by (c) Show that the total magnetic energy in a cable volume of length is given by (d) Use the result in Part (c) and the relationship between magnetic energy, current, and inductance to show that the self-inductance per unit length of the cable arrangement is given by L>_ _ (m0 >2p) ln(r2 >r1).

A coaxial cable consists of two very thin-walled conducting cylinders of radii and (Figure 28-63). The currents in the inner and outer cylinders are equal in magnitude but opposite in direction. Compute the flux through a rectangular area of sides and between the conductors shown in Figure 28-63. Use the relationship between flux and current to show that the self-inductance per unit length of the cable is given by L>_ _ (m0 .

SPREADSHEET Figure 28-64 shows a rectangular loop of wire that is wide, is long, and lies in the vertical plane which is perpendicular to a region that has a uniform magnetic field. The magnitude of the uniform magnetic field is and the direction of the magnetic field is into the page. The portion of the loop not in the magnetic field is long. The resistance of the loop is and its mass is The loop is released from rest at (a) What are the magnitude and direction of the induced current when the loop has a downward speed (b) What is the force that acts on the loop as a result of the current? (c) What is the net force acting on the loop? (d) Write out Newtons second law for the loop. (e) Obtain an expression for the speed of the loop as a function of time. (f) Integrate the expression obtained in Part (e) to find the distance the loop falls as a function of time. (g) Using a spreadsheet program, make a graph of the position of the loop as a function of time (letting at the start) for values of between and (i.e., when the loop leaves the magnetic field). (h) At what time does the loop completely leave the field region? Compare this to the time it would have taken if there were no field.

A coil that has turns and an area is suspended from the ceiling by a wire that provides a linear restoring torque that has a torsion constant The two ends of the coil are connected to each other, the coil has resistance and the moment of inertia of the coil is The plane of the coil is vertical and parallel to a uniform horizontal magnetic field when the wire is not twisted (i.e., The coil is twisted about a vertical axis through its center by a small angle and released. The coil then undergoes damped harmonic oscillation. Show that its angle with its equilibrium position will vary with time according to where t _ RI>(NBA)2, v _ 1k>I and v_ _ v021 _ (2v0t)_2.