A small current element at the origin has a length of and

Sketch the field lines for the electric dipole and the magnetic dipole shown in Figure 27-47. How do the field lines differ in appearance close to the center of each dipole

Two wires lie in the plane of the page and carry equal currents in opposite directions, as shown in Figure 27-48. At a point midway between the wires, the magnetic field is (a) zero, (b) into the page, (c) out of the page, (d) toward the top or bottom of the page, (e) toward one of the two wires

Parallel wires 1 and 2 carry currents and respectively, where The two currents are in the same direction. The magnitudes of the magnetic force by current 1 on wire 2 and by current 2 on wire 1 are and respectively. These magnitudes are related by (a) (b) (c) (d) (e

Make a field-line sketch of the magnetic field due to the currents in the pair of identical coaxial coils shown in Figure 27-49. Consider two cases: (a) the currents in the coils have the same magnitude and have the same direction and (b) the currents in the coils have the same magnitude and have the opposite directions

Discuss the differences and similarities between Gausss law for magnetism and Gausss law for electricity

Explain how you would modify Gausss law if scientists discovered that single, isolated magnetic poles actually existed

You are facing directly into one end of a long solenoid and the magnetic field inside of the solenoid points away from you. From your perspective, is the direction of the current in the solenoid coils clockwise or counterclockwise? Explain your answer

Opposite ends of a helical metal spring are connected to the terminals of a battery. Do the spacings between the coils of the spring tend to increase, decrease, or remain the same when the battery is connected? Explain your answer

The current density is constant and uniform in a long, straight wire that has a circular cross section. True or false: (a) The magnitude of the magnetic field produced by the wire is greatest at the surface of the wire. (b) The magnetic field strength in the region surrounding the wire varies inversely with the square of the distance from the wires central axis. (c) The magnetic field is zero at all points on the wires central axis. (d) The magnitude of the magnetic field inside the wire increases linearly with the distance from the wires central axis.

If the magnetic susceptibility of a material is positive, (a) paramagnetic effects or ferromagnetic effects must be greater than diamagnetic effects, (b) diamagnetic effects must be greater than paramagnetic effects, (c) diamagnetic effects must be greater than ferromagnetic effects, (d) ferromagnetic effects must be greater than paramagnetic effects, (e) paramagnetic effects must be greater than ferromagnetic effects.

Of the four gases listed in Table 27-1, which are diamagnetic and which are paramagnetic?

When a current is passed through the wire in Figure 27-50, will the wire tend to bunch up or will it tend to form a circle? Explain your answer.

At time a particle has a charge of is located in the plane at and has a velocity equal to Find the magnetic field in the plane at (a) the origin, (b) (c) and (d) x _ 0, y _ 4.0 m. SSM

At time a particle has a charge of is located in the plane at and has a velocity equal to Find the magnetic field in the plane at (a) (b) and (c)

A proton has a velocity of and is located in the plane at at some time Find the magnetic field in the plane at (a) (b) and (c)

In a prequantum-mechanical model of the hydrogen atom, an electron orbits a proton at a radius of m. According to this model, what is the magnitude of the magnetic field at the proton due to the orbital motion of the electron? Neglect any motion of the proton.

Two equal point charges are, at some instant, located at (0, 0, 0) and at (0, 0). They are both moving with speed in the direction (assume . Find the ratio of the magnitude of the magnetic force to the magnitude of the electric force on each charge.

A small current element at the origin has a length of and carries a current of in the direction. Find the magnetic field due to the current element (a) on the axis at (b) on the axis at (c) on the axis at and (d) on the axis at

A small current element at the origin has a length of and carries a current of in the direction. Find the magnitude and direction of the magnetic field due to this current element at the point .

A small current element at the origin has a length of and carries a current of in the direction. Find the magnitude of the magnetic field due to this current element and indicate its direction on a diagram at (a) and (b)

A single conducting loop has a radius equal to and carries a current equal to What is the magnitude of the magnetic field on the line through the center of the loop and perpendicular to the plane of the loop (a) at the center of the loop, (b) from the center, (c) from the center, and (d) from the center?

SPREADSHEET A pair of identical coils, each having a radius of are separated by a distance equal to their radii, that is, Called Helmholtz coils, they are coaxial and carry equal currents in directions such that their axial fields are in the same direction. A feature of Helmholtz coils is that the resultant magnetic field in the region between the coils is very uniform. Assume the current in each is and there are 250 turns for each coil. Using a spreadsheet program, calculate and graph the magnetic field as a function of the distance from the center of the coils along the common axis, for Over what range of does the field vary by less than 20%?

Apair of Helmholtz coils that have radii have their axes along the axis (see Problem 22). One coil is in the plane and the second coil is in the plane. Show that on the axis at and (Note: These results show that the magnitude and direction of the magnetic field in the region to either side of the midpoint are approximately equal to the magnitude and direction of the magnetic field at the midpoint.)

ENGINEERING APPLICATION Anti-Helmholtz coils are used in many physics applications, such as laser cooling and trapping, where a field with a uniform gradient is desired. These coils have the same construction as a Helmholtz coil, except that the currents have opposite directions, so that the axial fields are in opposite directions, and the coil separation is rather than Graph the magnetic field as a function of the axial distance from the center of the coils, for an anti-Helmholtz coil using the same parameters as in Problem 22. Over what interval of the axis is within one percent of its value at the midpoint between the coils?

If the currents are both in the direction, find the magnetic field at the following points on the axis: (a) (b) (c) and (d)

SPREADSHEET Using a spreadsheet program or graphing calculator, graph versus when both currents are in the direction.

The current in the wire at is in the direction and the current in the wire at is in the direction. Find the magnetic field at the following points on the axis: (a) (b) (c) and (d)

SPREADSHEET The current in the wire at is in the direction and the current in the wire at is in the direction. Using a spreadsheet program or graphing calculator, graph B versus y.

Find the magnetic field on the axis at if (a) the currents are both in the direction and (b) the current in the wire at is in the direction and the current in the wire at is in the direction.

Find the magnitude of the force per unit length exerted by one wire on the other.

Two long, straight parallel wires apart carry equal currents. The wires repel each other with a force per unit length of (a) Are the currents parallel or antiparallel? Explain your answer. (b) Determine the current in each wire.

The current in the wire shown in Figure 27-52 is 8.0 A. Find the magnetic field at point P.

As a student technician, you are preparing a lecture demonstration on magnetic suspension. You have a 16-cm-long straight, rigid wire that will be suspended by flexible conductive lightweight leads above a long, straight wire. Currents that are equal but are in opposite directions will be established in the two wires so the 16-cm wire floats a distance above the long wire with no tension in its suspension leads. If the mass of the 16-cm wire is and if (the distance between the central axes of the two wires) is what should their common current be?

Three long, parallel straight wires pass through the vertices of an equilateral triangle that has sides equal to as shown in Figure 27-53. A dot indicates that the direction of the current is out of the page and a cross indicates that the direction of the current is into the page. If each current is find (a) the magnetic field at the location of the upper wire due to the currents in the two lower wires and (b) the force per unit length on the upper wire.

Rework Problem 34 with the current in the lower right corner of Figure 27-53 reversed.

An infinitely long wire lies along the axis and carries current in the direction. A second infinitely long wire lies along the axis and carries current in the direction. At what points in the plane is the resultant magnetic field zero?

An infinitely long wire lies along the axis and carries a current of in the direction. A second infinitely long wire is parallel to the z axis and intersects the x axis at x _ 10.0 cm. (a) Find the current in the second wire if the magnetic field is zero at (b) What is the magnetic field at

Three long parallel wires are at the corners of a square, as shown in Figure 27-54. The wires each carry a current Find the magnetic field at the unoccupied corner of the square when (a) all the currents are into the page, (b) and are into the page and is out, and (c) and are into the page and is out. Your answers should be in terms of I and L.

Four long, straight parallel wires each carry current In a plane perpendicular to the wires, the wires are at the corners of a square of side length Find the magnitude of the force per unit length on one of the wires if (a) all the currents are in the same direction and (b) the currents in the wires at adjacent corners are oppositely directed.

Five long, straight, current-carrying wires are parallel to the axis, and each carries a current in the direction. The wires each are a distance from the axis. Two of the wires intersect the axis, one at and the other at Another wire intersects the axis at One of the remaining wires intersects the plane at the point and the last remaining wire intersects the plane at the point Find the magnetic field on the axis.

A solenoid that has length radius and 300 turns carries a current of Find the magnitude of the magnetic field on the axis of the solenoid (a) at the center of the solenoid and (b) at one end of the solenoid.

A solenoid is long, has a radius of and has 600 turns. It carries a current of What is the magnitude of the magnetic field inside the solenoid and far from either end?

A solenoid has turns per unit length, has a radius and carries a current Its axis coincides with the axisone end at and the other end at Show that the magnitude of the magnetic field at a point on the axis is given by where the angles are related to the geometry by and

In Problem 43, an expression for the magnitude of the magnetic field along the axis of a solenoid is given. For and the angles and are very small, so the small-angle approximations and are highly accurate. (a) Draw a diagram and use it to show that, for these conditions, the angles can be approximated as and (b) Using these approximations, show that the magnetic field at points on the axis where can be written as where is the distance to the near end of the solenoid, is the distance to the far end, and the quantity is defined by where is the magnitude of the magnetic moment of the solenoid.

A long, straight, thin-walled cylindrical shell of radius carries a current parallel to the central axis of the shell. Find the magnetic field (including direction) both inside and outside the shell.

In Figure 27-55, one current is into the page, the other current is out of the page, and each curve is a circular path. (a) Find for each path, assuming that each integral is to be evaluated in the counterclockwise direction. (b) Which path, if any, can be used to find the combined magnetic field of the currents?

Show that a uniform magnetic field that has no fringing field, such as that shown in Figure 27-56, is impossible because it violates Ampres law. Do this calculation by applying Ampres law to the rectangular curve shown by the dashed lines. SSM

SPREADSHEET A coaxial cable consists of a solid conducting cylinder that has a radius equal to and a conducting cylindrical shell that has an inner radius equal to and an outer radius equal to The solid cylinder carries a current of parallel to the central axis. The cylindrical shell carries a current of in the opposite direction. Assume that the current densities are uniformly distributed in both conductors. (a) Using a spreadsheet program or graphing calculator, graph the magnitude of the magnetic field as a function of the radial distance from the central axis for (b) What is the magnitude of the field for

A long cylindrical shell has an inner radius and an outer radius and carries a current parallel to the central axis. Assume that within the material of the shell the current density is uniformly distributed. Find an expression for the magnitude of the magnetic field for (a) (b) and (c)

Figure 27-57 shows a solenoid that has turns per unit length and carries a current Apply Ampres law to the rectangular curve shown in the figure to derive an expression for the magnitude of the magnetic field. Assume that inside the solenoid the magnetic field is uniform and parallel with the central axis, and that outside the solenoid there is no magnetic field.

A tightly wound 1000-turn toroid has an inner radius of and an outer radius of and carries a current of The toroid is centered at the origin with the centers of the individual turns in the plane. In the plane: (a) What is the magnetic field strength at a distance of from the origin? (b) What is the magnetic field strength at a distance of from the origin?

A thin conducting sheet in the plane carries current in the _x direction (Figure 27-58a). The sheet extends inz definitely in all directions and the current is uniformly distributed throughout the sheet. To find the direction of the magnetic field at point consider the field due only to the currents and in the two narrow strips shown. The strips are identical, so (a) What is the direction of the magnetic field at point due to just and ? Explain your answer using a sketch. (b) What is the direction of the magnetic field at point due to the entire sheet? Explain your answer. (c) What is the direction of the field at a point to the right of point (where )? Explain your answer. (d) What is the direction of the field at a point below the sheet (where )? Explain your answer using a sketch. (e) Apply Ampres law to the rectangular curve (Figure 27-58b) to show that the magnetic field strength at point is given by where is the current per unit length along the axis.

A tightly wound solenoid is long, has and carries a current of so that its axial field is in the direction. Find and at the center when (a) there is no core in the solenoid and (b) there is a soft iron core that has a magnetization of

A long tungsten-core solenoid carries a current. (a) If the core is removed while the current is held constant, does the magnetic field strength in the region inside the solenoid decrease or increase? (b) By what percentage does the magnetic field strength in the region inside the solenoid decrease or increase?

As a liquid fills the interior volume of a solenoid that carries a constant current, the magnetic field inside the solenoid decreases by Determine the magnetic susceptibility of the liquid.

A long thin solenoid carrying a current of has per centimeter of length. What is the magnetic field strength in the region occupied by the interior of the solenoid when the interior is (a) a vacuum, (b) filled with aluminum, and (c) filled with silver?

Acylinder of iron, initially unmagnetized, is cooled to What is the magnetization of the cylinder at that temperature due to the influence of Earths magnetic field of Assume a magnetic moment of 2.00 Bohr magnetons per atom.

A cylinder of silver at a temperature of has a magnetization equal to 0.075% of its saturation magnetization. Assume a magnetic moment of one Bohr magneton per atom. The density of silver is (a) What value of applied magnetic field parallel to the central axis of the cylinder is required to reach this magnetization? (b) What is the magnetic field strength at the center of the cylinder?

During a solid-state physics lab, you are handed a cylindrically shaped sample of unknown magnetic material. You and your lab partners place the sample in a long solenoid that has turns per unit length and a current I. The values for magnetic field B within the material versus are given below. Use these values to plot versus and versus where is the field due to the current and is the relative permeability of the sample.

Nickel has a density of and a molar mass of Nickels saturation magnetization is Calculate the magnetic moment of a nickel atom in Bohr magnetons.

Repeat Problem 60 for cobalt, which has a density of a molar mass of and a saturation magnetization of

Show that Curies law predicts that the magnetic susceptibility of a paramagnetic substance is given by

In a simple model of paramagnetism, we can consider that some fraction of the atoms have their magnetic moments aligned with the external magnetic field and that the rest of the atoms are randomly oriented and therefore do not contribute to the magnetic field. (a) Use this model and Curies law to show that at temperature and external magnetic field the fraction of aligned atoms is given by (b) Calculate this fraction for a sample temperature of and an external field of Assume that has a value of

Assume that the magnetic moment of an aluminum atom is The density of aluminum is and its molar mass is (a) Calculate the value of the saturation magnetization and the saturation magnetic field for aluminum. (b) Use the result of Problem 62 to calculate the magnetic susceptibility at (c) Explain why the result for Part (b) is larger than the value listed in Table 27-1.

A toroid has turns, carries a current has a mean radius and has a cross-sectional radius where (Figure 27-59). When the toroid is filled with material, it is called a Rowland ring. Find and in such a ring, assuming a magnetization that is everywhere parallel to B SSM

A toroid is filled with liquid oxygen that has a magnetic susceptibility of The toroid has and carries a current of Its mean radius is and the radius of its cross section is (a) What is the magnetization? (b) What is the magnetic field? (c) What is the percentage change in the magnetic field produced by the liquid oxygen?

The centers of the turns of a toroid form a circle with a radius of 14.0 cm. The cross-sectional area of each turn is \(3.00 \ \mathrm {cm}^2\). It is wound with 5278 turns of fine wire, and the wire carries a current of 4.00 A. The core is filled with a paramagnetic material of magnetic susceptibility \(2.90 \times 10^{-4}\). (a) What is the magnitude of the magnetic field within the substance? (b) What is the magnitude of the magnetization? (c) What would the magnitude of the magnetic field be if there were no paramagnetic core present?

For annealed iron, the relative permeability has its maximum value of approximately 5500 at Find the magnitude of the magnetization and magnetic field in annealed iron when is maximum.

The saturation magnetization for annealed iron occurs when Find the permeability and the relative permeability of annealed iron at saturation. (See Table 27-2.)

The coercive force (which is a misnomer because it is really a magnetic field value) is defined as the applied magnetic field needed to bring the magnetic field back to zero along the hysteresis curve (which is point in Figure 27-44). For a certain permanent bar magnet, the coercive force is known to be The bar magnet is to be demagnetized by placing it inside a 15.0-cm-long solenoid that has What minimum current is needed in the solenoid to demagnetize the magnet?

A long thin solenoid has and carries a current of The solenoid is filled with iron and the magnetic field is measured to be (a) Neglecting end effects, what is the magnitude of the applied magnetic field? (b) What is the magnetization? (c) What is the relative permeability?

When the current in Problem 71 is the magnetic field is measured to be (a) Neglecting end effects, what is the applied magnetic field? (b) What is the magnetization? (c) What is the relative permeability?

A toroid has turns, carries a current has a mean radius and has a cross-sectional radius where (Figure 27-59). The core of the toroid is filled with iron. When the current is the magnetic field in the region where the iron is has a magnitude of (a) What is the magnetization? (b) Find the values for the relative permeability, the permeability, and the magnetic susceptibility for this iron sample.

The centers of the turns of a toroid form a circle with a radius of The cross-sectional area of each turn is It is wound with of fine wire, and the wire carries a current of The core is filled with soft iron, which has a relative permeability of 500. What is the magnetic field strength in the core?

A long, straight wire that has a radius of is coated with an insulating ferromagnetic material that has a thickness of and a relative magnetic permeability of 400. The coated wire is in air and the wire itself is nonmagnetic. The wire carries a current of (a) Find the magnetic field in the region occupied by the inside of the wire as a function of the perpendicular distance, from the central axis of the wire. (b) Find the magnetic field in the region occupied by the inside of the ferromagnetic material as a function of the perpendicular distance, from the central axis of the wire. (c) Find the magnetic field in the region surrounding the wire and coating as a function of the perpendicular distance, from the central axis of the wire. (d) What must the magnitudes and directions of the amperian currents be on the surfaces of the ferromagnetic material to account for the magnetic fields observed?

Find the magnetic field at point P in Figure 27-60.

Using Figure 27-61, find the magnetic field (in terms of the parameters given in the figure) at point the common center of the two arcs. SSM

Awire of length is wound into a circular coil of N turns and carries a current Show that the magnetic field strength in the region occupied by the center of the coil is given by

A very long wire carrying a current is bent into the shape shown in Figure 27-62. Find the magnetic field at point P.

A power cable carrying is below Earths surface, but the cables direction and precise position are unknown. Explain how you could locate the cable using a compass. Assume that you are at the equator, where Earths magnetic field is horizontal and 0.700 G due north.

Along, straight wire carries a current of as shown in Figure 27-63. A rectangular coil that has two sides parallel to the straight wire has sides that are long and long. The side nearest to the wire is from the wire. The coil carries a current of (a) Find the force on each segment of the rectangular coil due to the current in the long, straight wire. (b) What is the net force on the coil? SSM

The closed loop shown in Figure 27-64 carries a current of in the counterclockwise direction. The radius of the outer arc is and that of the inner arc is Find the magnetic field at point P.

A closed circuit consists of two semicircles of radii and that are connected by straight segments, as shown in Figure 27-65. Acurrent of exists in this circuit and has a clockwise direction. Find the magnetic field at point P.

A very long, straight wire carries a current of An electron outside the wire is from the central axis of the wire and is moving with a speed of Find the force on the electron when it moves (a) directly away from the wire, (b) parallel to the wire in the direction of the current, and (c) perpendicular to the central axis of wire and tangent to a circle that is coaxial with the wire.

SPREADSHEET A current of is uniformly distributed over the cross section of a long, straight wire of radius Using a spreadsheet program, graph the magnetic field strength as a function of (the distance from the central axis of the wire) for 0 _ R _ 10R0 .

A50-turn coil of radius carries a current of and a concentric 20-turn coil of radius carries a current of The planes of the two coils are perpendicular. Find the magnitude of the torque exerted by the large coil on the small coil. (Neglect any variation in magnetic field due to the current in the large coil over the region occupied by the small coil.)

The magnetic needle of a compass is a uniform rod with a length of a radius of and a density of The needle is free to rotate in a horizontal plane, where the horizontal component of Earths magnetic field is When disturbed slightly, the compass executes simple harmonic motion about its midpoint with a frequency of (a) What is the magnetic dipole moment of the needle? (b) What is the magnetization of the needle? (c) What is the amperian current on the surface of the needle?

A relatively inexpensive ammeter, called a tangent galvanometer, can be made using Earth’s magnetic field. A plane circular coil that has N turns and a radius R is oriented so the magnetic field \(B_c\) it produces in the center of the coil is either east or west. A compass is placed at the center of the coil. When there is no current in the coil, assume the compass needle points due north. When there is a current in the coil (I) , the compass needle points in the direction of the resultant magnetic field at an angle \(\theta\) to the north. Show that the current I is related to \(\theta\) and to the horizontal component of Earth’s magnetic field \(B_e\) by \(I=\frac{2RB_e}{\mu_0N} \tan \ \theta\).

Earths magnetic field is about at the magnetic poles, and is pointed vertically downward at the magnetic pole in the northern hemisphere. If the magnetic field were due to an electric current circulating in a loop at the radius of the inner iron core of Earth (approximately , (a) what would be the magnitude of the current required? (b) What direction would this current havethe same as Earths spin, or opposite? Explain your answer.

A long, narrow bar magnet has its magnetic moment parallel to its long axis and is suspended at its centerin essence becoming a frictionless compass needle. When the magnet is placed in a magnetic field it lines up with the field. If it is displaced by a small angle and released, show that the magnet will oscillate about its equilibrium position with frequency given by where is the moment of inertia about the point of suspension.

An infinitely long, straight wire is bent, as shown in Figure 27-66. The circular portion has a radius of and its center is a distance from the straight part. Find the value of such that the magnetic field at the center of the circular portion is zero.

(a) Find the magnetic field strength at point on the perpendicular bisector of a wire segment carrying current as shown in Figure 27-67. (b) Use your result from Part (a) to find the magnetic field strength at the center of a regular polygon of sides. (c) Show that when is very large, your result approaches that for the magnetic field strength at the center of a circle.

The current in a long cylindrical conductor of radius varies with distance from the axis of the cylinder according to the relation Find the magnetic field at the following perpendicular distances from the wires central axis: (a) (b) and (c)

Figure 27-68 shows a square loop that has 20-cm-long sides and is in the plane with its center at the origin. The loop carries a current of An infinitely long wire that is parallel to the axis and carries a current of intersects the axis at . The directions of the currents are shown in the figure. (a) Find the net torque on the loop. (b) Find the net force on the loop.

Acurrent balance is constructed in the following way: A straight 10.0-cm-long section of wire is placed on top of the pan of an electronic balance (Figure 27-69). This section of wire is connected in series with a power supply and a long straight horizontal section of wire that is parallel to it and positioned directly above it. The distance between the central axes of the two wires is The power supply provides a current in the wires. When the power supply is switched on, the reading on the balance increases by What is the current in the wire? SSM

Consider the current balance of Problem 95. If the sensitivity of the balance is what is the minimum current detectable using this current balance?

A nonconducting disk has radius carries a uniform surface charge density and rotates with angular speed (a) Consider an annular strip that has a radius a width and a charge Show that the current produced by this rotating strip is given by (b) Use your result from Part (a) to show that the magnetic field strength at the center of the disk is given by the exxppression (c) Use your result from Part (a) to find an expression for the magnetic field strength at a point on the central axis of the disk a distance from its center.

A square loop that has sides of length lies in the plane with its center at the origin. The loop carries a current (a) Derive an expression for the magnetic field strength at any point on the axis. (b) Show that for much larger than your result from Part (a) becomes where is the magnitude of the magnetic moment of the loop.