Figure 32-38 gives the magne- ~ tization curve for a

The magnetic flux through each of five faces of a die (singular of "dice") is given by B = NWb, where N (= 1 to 5) is the number of spots on the face. The flux is positive (outward) for N even and negative (inward) for N odd. What is the flux through the sixth face of the die?

Figure 32-26 shows a closed surface. Along the flat top face, which has a radius of 2.0 cm, a perpendicular magnetic field B of magnitude 0.30 T is directed outward. Along the flat bottom face, a magnetic flux of 0.70 m Wb is directed outward. What are the (a) magnitude and (b) direction (inward or outward) of the magnetic flux through the curved part of the surface?

A Gaussian surface in the Fig. 32-26 Problem 2. shape of a right circular cylinder with end caps has a radius of 12.0 cm and a length of 80.0 cm. Through one end there is an inward magnetic flux of 25.0 ftWb. At the other end there is a uniform magnetic field of 1.60 mT, normal to the surface and directed outward. What are the (a) magnitude and (b) direction (inward or outward) of the net magnetic flux through the curved surface

Two wires, parallel to a z axis and a distance 41' apart, carry equal currents i in opposite directions, as shown in Fig. 32-27. A circular cylinder of radius I' and length L has its axis on the z axis, midway between the wires. Use Gauss' law for y magnetism to derive an expres- Fig. 32-27 Problem 4. sian for the net outward magnetic flux through the half of the cylindrical surface above the x axis. (Hint: Find the flux through the portion of the xz plane that lies within the cylinder.)

The induced magnetic field at radial distance 6.0 mm from the central axis of a circular parallel-plate capacitor is 2.0 X 10-7 T. The plates have radius 3.0 mm. At what rate iEldt is the electric field between the plates changing?

A capacitor with square plates of edge length L is being discharged by a current of 0.75 A Figure 32- 28 is a head-on view of one of the plates from inside the capacitor. A dashed rectangular path is shown. If L = 12 cm, W = 4.0 cm, and H = 2.0 cm, what is the value of 9i B d:f around the dashed path?

Uniform electric flux. Figure 32-29 shows a circular region of radius R = 3.00 cm L L Fig. 32-28 Problem 6. in which a uniform electric flux is directed out of the plane of the page. The total electric flux through the region is given by <I> E = (3.00 m V . m/s )t, where t is in seconds. What is the magnitude of the magnetic field that is induced at radial distances (a) 2.00 cm and (b) 5.00 cm?

Nonuniform electric flux. Figure 32- 29 shows a circular region of radius R = 3.00 cm in which an electric flux is directed out of the plane of the page. The flux encircled by a concentric circle of radius r is given by E,enc = Fig. 32-29 Problems 7 to 10 and 19 to 22. (0.600 V' m/s)(r/R)t, where I' oS Rand t is in seconds. What is the magnitude of the induced magnetic field at radial distances (a) 2.00 cm and (b) 5.00 cm?

Uniform electric field. In Fig. 32-29, a uniform electric field is directed out of the page within a circular region of radius R = 3.00 cm. The field magnitude is given by E = (4.50 X 10-3 Vim' s)t, where t is in seconds. What is the magnitude of the induced magnetic field at radial distances (a) 2.00 cm and (b) 5.00 cm?

Nonuniform electric field. In Fig. 32-29, an electric field is directed out of the page within a circular region of radius R = 3.00 cm. The field magnitude is E = (0.500 Vim' s)(l - rIR)t, where t is in seconds and r is the radial distance (r oS R). What is the magnitude of the induced magnetic field at radial distances (a) 2.00 cm and (b) 5.00 cm?

Suppose that a parallel-plate capacitor has circular plates with radius R = 30 mm and a plate separation of 5.0 mm. Suppose also that a sinusoidal potential difference with a maximum value of 150 V and a frequency of 60 Hz is applied across the plates; that is, V = (150 V) sin[27T(60 Hz)t].

A parallel-plate capacitor with circular plates of radius 40 mm is being discharged by a current of 6.0 AAt what radius (a) inside and (b) outside the capacitor gap is the magnitude of the induced magnetic field equal to 75% of its maximum value? (c) What is that maximum value?

of a parallel-plate capacitor with a 2.0 ftF capacitance be changed to produce a displacement current of 1.5 A?

A parallel-plate capacitor with circular plates of radius R is being charged. Show that the magnitude of the current density of the displacement current is Id = Bo(dEldt) for I' oS R.

Prove that the displacement current in a parallel-plate capacitor of capacitance C can be written as id = C(dVldt), where V is the potential difference between the plates.

A parallel-plate capacitor with circular plates of radius 0.10 m is being discharged. A circular loop of radius 0.20 m is concentric with the capacitor and halfway between the plates. The displacement current through the loop is 2.0 A. At what rate is the electric field between the plates changing?

A silver wire has resistivity p = 1.62 X 10-8 f1 . m and a cross-sectional area of 5.00 mm2. The current in the wire is uniform and changing at the rate of 2000 A/s when the current is 100 A. (a) What is the magnitude of the (uniform) electric field in the wire when the current in the wire is 100 A? (b) What is the displacement current in the wire at that time? (c) What is the ratio of the magnitude of the magnetic field due to the displacement current to that due to the current at a distance r from the wire?

The circuit in Fig. 32-30 consists of switch S, a 12.0 V ideal battery, a 20.0 Mf1 resistor, and an air- filled capacitor. The capacitor has parallel circular plates of radius 5.00 cm, separated by 3.00 mm. At time t = 0, switch S is closed to begin charging Fig. 32-30 Problem 18. the capacitor. The electric field between the plates is uniform. At t = 250 j.LS, what is the magnitude of the magnetic field within the capacitor, at radial distance 3.00 cm?

Uniform displacement-current density. Figure 32-29 shows a circular region of radius R = 3.00 cm in which a displacement current is directed out of the page. The displacement current has a uniform density of magnitude Jd = 6.00 A/m2 What is the magnitude of the magnetic field due to the displacement current at radial distances (a) 2.00 cm and (b) 5.00 cm?

Uniform displacement current. Figure 32-29 shows a circular region of radius R = 3.00 cm in which a uniform displacement current id = 0.500 A is out of the page. What is the magnitude of the magnetic field due to the displacement current at radial distances (a) 2.00 cm and (b) 5.00 cm?

Nonuniform displacement-current density. Figure 32-29 shows a circular region of radius R = 3.00 cm in which a displacement current is directed out of the page. The magnitude of the density of this displacement current is Jd = (4.00 A/m2)(1 - rIR), where r is the radial distance (r:s R). What is the magnitude of the magnetic field due to the displacement current at (a) r = 2.00 cm and (b) r = 5.00 cm?

Nonuniform displacement current. Figure 32-29 shows a circular region of radius R = 3.00 cm in which a displacement current id is directed out of the page. The magnitude of the displacement current is given by id = (3.00 A)(rIR), where r is the radial distance (r :S R). What is the magnitude of the magnetic field due to id at radial distances (a) 2.00 cm and (b) 5.00 cm?

In Fig. 32-31, a parallel-plate capacitor has square plates of edge length L = 1.0 m. A current of 2.0 A charges the capacitor, producing a uniform electric field E between the plates, with E perpendicular to the plates. (a) What is the displacement current id through the region between the plates? (b) What is dE/dt in this region? (c) What is the displacement current encircled by the square dashed path of edge length d = 0.50 m? (d) What is g; B dS' around this square dashed path?

The magnitude of the electric field between the two circular parallel plates in Fig. 32-32 is E = (4.0 X 105) - (6.0 X 104t), with E in volts per meter and t in seconds. At t = 0, E is upward. The plate area is 4.0 X 10-2 m2 For t 2: 0, what are the (a) magnitude and (b) direction (up or down) of the displacement current between the plates and (c) is Fig. 32-32 Problem 24. the direction of the induced magnetic field clockwise or counterclockwise in the figure?

As a parallel-plate capacitor with circular plates 20 cm in diameter is being charged, the current density of the displacement current in the region between the plates is uniform and has a magnitude of 20 Nm2 (a) Calculate the magnitude B of the magnetic field at a distance r = 50 mm from the axis of symmetry of this region. (b) Calculate dEldtin this region.

A capacitor with parallel circular plates of radius R = 1.20 cm is discharging via a current of 12.0 A. Consider a loop of radius RI3 that is centered on the central axis between the plates. (a) How much displacement current is encircled by the loop? The maximum induced magnetic field has a magnitude of 12.0 mT. At what radius (b) inside and (c) outside the capacitor gap is the magnitude of the induced magnetic field 3.00 mT?

In Fig. 32-33, a uniform electric field E collapses. The vertical axis scale is set by Es = 6.0 X 105 N/C, and the horizontal axis scale is set by ts = 12.0 fLs. Calculate the magnitUde of the displacement current through a 1.6 m2 area perpendicular to the field during each of the time intervals a, b, and c shown on the graph. (Ignore the behavior at the ends of the intervals.)

Figure 32-34a shows the current i that is produced in a wire of resistivity 1.62 X 10-8 f1 . m. The magnitude of the current versus time t is shown in Fig. 32-34b. The vertical axis scale is set by is = 10.0 A, and the horizontal axis scale is set by ts = 50.0 ms. Point P is at radial distance 9.00 mm from the wire's center. Determine the magnitude of the magnetic field Bi at point P due to the actual current i in the wire at (a) t = 20 ms, (b) t = 40 ms, and (c) t = 60 ms. Next, assume that the electric field driving the current is confined to the wire. Then determine the magnitude of the magnetic field Bid at point P due to the displacement current i" in the wire at (d) t = 20 ms, (e) t = 40 ms, and (f) t = 60 ms. At point P at t = 20 s, what is the direction (into or out of the page) of (g) Bi and (h) Bid?

In Fig. 32-35, a capacitor with circular plates of radius R = 18.0 cm is connected to a source of emf 'i!S = 'i!Sm sin wt, where 'i!S111 = 220 V and w = 130 rad/s. The maximum value of the displacement current is id = 7.60 J.LA. Neglect fringing of the electric field at the edges of the plates. (a) What is the maximum value of the current i in the circuit? (b) What is the maximum value of d<p E/dt, where <P E is the electric flux through the region between the plates? (c) What is the separation d between the plates? (d) Find the maximum value of the magnitude of E between the plates at a distance r = 11.0 cm from the center.

Assume the average value of the vertical component of Earth's magnetic field is 43 J.LT (downward) for all of Arizona, which has an area of 2.95 X H)5 km2 What then are the (a) magnitude and (b) direction (inward or outward) of the net magnetic flux through the rest of Earth's surface (the entire surface excluding Arizona)?

In New Hampshire the average horizontal component of Earth's magnetic field in 1912 was 16 J.LT, and the average inclination or "dip" was 73. What was the corresponding magnitude of Earth's magnetic field?

Figure 32-36a is a one-axis graph along which two of the allowed energy values (levels) of an atom are plotted. When the atom is E;, ---- - -- placed in a magnetic field of 0.500 T, the graph changes to that of Fig. 32- 36b because of the energy associ- E1 E1 ated with 71orb' E. (We neglect 71s.) Level E1 is unchanged, but level E2 (a) (b) splits into a (closely spaced) triplet of levels. What are the allowed val- Fig. 32-36 Problem 32. ues of me associated with (a) energy level El and (b) energy level E2? (c) In joules, what amount of energy is represented by the spacing between the triplet levels?

If an electron in an atom has an orbital angular momentum with m = 0, what are the components (a) Lorb,z and (b) J.Lorb,z? If the atom is in an external magnetic field lJ that has magnitude 35 mT and is directed along the z axis, what are (c) the energy Uorb associated with l10rb and (d) the energy Uspin associated with I1s? If, instead, the electron has m = -3, what are (e) Lorb,Z' (f) J.Lorb,z' (g) Uorb, and (h) Uspin?

What is the energy difference between parallel and antiparallel alignment of the z component of an electron's spin magnetic dipole moment with an external magnetic field of magnitude 0.25 T, directed parallel to the z axis?

What is the measured component of the orbital magnetic dipole moment of an electron with (a) me = 1 and (b) me = -2?

An electron is placed in a magnetic field E that is directed along a z axis. The energy difference between parallel and antiparallel alignments of the z component of the electron's spin magnetic moment with E is 6.00 X 10-25 1. What is the magnitude of E?

Figure 32-37 shows a loop model (loop L) for a diamagnetic material. (a) Sketch the magnetic field lines within and about the material due to the bar magnet. What is the direction of (b) the loop's net magnetic dipole moment 11, (c) the conventional current i in the loop (clockwise or counterclockwise in the figure), and (d) the magnetic force on the loop?

Assume that an electron of mass m and charge magnitude e moves in a circular orbit of radius r about a nucleus. A uniform magnetic field E is then established perpendicular to the plane of the orbit. Assuming also that the radius of the orbit does not change and that the change in the speed of the electron due to field E is small, find an expression for the change in the orbital magnetic dipole moment of the electron due to the field.

A sample of the paramagnetic salt to which the magnetization curve of Fig. 32-14 applies is to be tested to see whether it obeys Curie's law. The sample is placed in a uniform 0.50 T magnetic field that remains constant throughout the experiment. The magnetization M is then measured at temperatures ranging from 10 to 300 K. Will it be found that Curie's law is valid under these conditions?

A sample of the paramagnetic salt to which the magnetization curve of Fig. 32-14 applies is held at room temperature (300 K). At what applied magnetic field will the degree of magnetic saturation of the sample be (a) 50% and (b) 90%? (c) Are these fields attainable in the laboratory?

A magnet in the form of a cylindrical rod has a length of 5.00 cm and a diameter of 1.00 cm. It has a uniform magnetization of 5.30 X 103 A/m. What is its magnetic dipole moment?

A 0.50 T magnetic field is applied to a paramagnetic gas whose atoms have an intrinsic magnetic dipole moment of 1.0 X 10-23 J/T. At what temperature will the mean kinetic energy of translation of the atoms equal the energy required to reverse such a dipole end for end in this magnetic field?

An electron with kinetic energy Ke travels in a circular path that is perpendicular to a uniform magnetic field, which is in the positive direction of a z axis. The electron's motion is subject only to the force due to the field. (a) Show that the magnetic dipole moment of the electron due to its orbital motion has magnitUde J.L = Ke/B and that it is in the direction opposite that of E. What are the (b) magnitude and (c) direction of the magnetic dipole moment of a positive ion with kinetic energy Kj under the same circumstances? (d) An ionized gas consists of 5.3 X 1021 electrons/m3 and the same number density of ions. Take the average electron kinetic energy to be 6.2 X 10-20 J and the average ion kinetic energy to be 7.6 X 10-21 1. Calculate the magnetization of the gas when it is in a magnetic field of 1.2 T.

Figure 32-38 gives the magne- ~ tization curve for a paramagnetic ma- i terial. The vertical axis scale is set by ~ a = 0.15, and the horizontal axis scale is set by b = 0.2 T/K. Let J.Lsam be the measured net magnetic moment of a sample of the material and J.Lmax be o b the maximum possible net magnetic Fig. 32-38 Problem 44. moment of that sample. According to Curie's law, what would be the ratio J.L,anJ J.Lmax were the sample placed in a uniform magnetic field of magnitude 0.800 T, at a temperature of2.00 K?

Consider a solid containing N atoms per unit volume, each atom having a magnetic dipole moment 71. Suppose the direction of 71 can be only parallel or antiparallel to an externally ap- plied magnetic field B (this will be the case if /1 is due to the spin of a single electron). According to statistical mechanics, the probability of an atom being in a state with energy U is proportional to e- UlkT, where T is the temperature and k is Boltzmann's constant. Thus, because energy U is - /1- B, the fraction of atoms whose dipole moment is parallel to B is proportional to ellBlkT and the fraction of atoms whose dipole moment is antiparallel to B is proportional to e-IlBlkT. (a) Show that the magnitude of the magnetization of this solid is M = NfL tanh(fLBlkT). Here tanh is the hyperbolic tangent function: tanh(x) = (eX - e-X)/(e' + e-X). (b) Show that the result given in (a) reduces to M = NfL2BlkTfor fLB ~ kT. (c) Show that the result of (a) reduces to M = NfL for fLB ~ kT. (d) Show that both (b) and (c) agree qualitatively with Fig. 32-14.

You place a magnetic compass on a horizontal surface, allow the needle to settle, and then give the compass a gentle wiggle to cause the needle to oscillate about its equilibrium position. The oscillation frequency is 0.312 Hz. Earth's magnetic field at the location of the compass has a horizontal component of 18.0 fLT. The needle has a magnetic moment of 0.680 mJ/T. What is the needle's rotational inertia about its (vertical) axis of rotation?

The magnitude of the magnetic dipole moment of Earth is 8.0 X 1022 liT. (a) If the origin of this magnetism were a magnetized iron sphere at the center of Earth, what would be its radius? (b) What fraction of the volume of Earth would such a sphere occupy? Assume complete alignment of the dipoles. The density of Earth's inner core is 14 g/cm3 The magnetic dipole moment of an iron atom is 2.1 X 10-23 liT. (Note: Earth's inner core is in fact thought to be in both liquid and solid forms and partly iron, but a permanent magnet as the source of Earth's magnetism has been ruled out by several considerations. For one, the temperature is certainly above the Curie point.)

The magnitude of the dipole moment associated with an atom of iron in an iron bar is 2.1 X 10-23 J/T. Assume that all the atoms in the bar, which is 5.0 cm long and has a cross-sectional area of 1.0 cm2, have their dipole moments aligned. (a) What is the dipole moment of the bar? (b) What torque must be exerted to hold this magnet perpendicular to an external field of magnitude 1.5 T? (The density of iron is 7.9 g/cm3.)

The exchange coupling mentioned in Section 32-11 as being responsible for ferromagnetism is not the mutual magnetic interaction between two elementary magnetic dipoles. To show this, calculate (a) the magnitude of the magnetic field a distance of 10 nm away, along the dipole axis, from an atom with magnetic dipole moment 1.5 X 10-23 lIT (cobalt), and (b) the minimum energy required to turn a second identical dipole end for end in this field. (c) By comparing the latter with the mean translational kinetic energy of 0.040 e V, what can you conclude?

A magnetic rod with length 6.00 cm, radius 3.00 mm, and (uniform) magnetization 2.70 X 103 Aim can turn about its center like a compass needle. It is placed in a uniform magnetic field B of magnitude 35.0 mT, such that the directions of its dipole moment and B make an angle of 68.0. (a) What is the magnitude of the torque on the rod due to B? (b) What is the change in the orientation energy of the rod if the angle changes to 34.00?

The saturation magnetization Mmax of the ferromagnetic metal nickel is 4.70 X 105 AIm. Calculate the magnetic dipole moment of a single nickel atom. (The density of nickel is 8.90 g/cm3, and its molar mass is 58.71 g/mol.)

Measurements in mines and boreholes indicate that Earth's interior temperature increases with depth at the average rate of 30 CO/km. Assuming a surface temperature of 10C, at what depth does iron cease to be ferromagnetic? (The Curie temperature of iron varies very little with pressure.)

A Rowland ring is formed of ferromagnetic material. It is circular in cross section, with an inner radius of 5.0 cm and an outer radius of 6.0 cm, and is wound with 400 turns of wire. (a) What current must be set up in the windings to attain a toroidal field of magnitude Bo = 0.20 mT? (b) A secondary coil wound around the toroid has 50 turns and resistance 8.0 n. If, for this value of Bo, we have BM = 800Bo, how much charge moves through the secondary coil when the current in the toroid windings is turned on?

Using the approximations given in Problem 61, find (a) the altitude above Earth's surface where the magnitude of its magnetic field is 50.0% of the surface value at the same latitude; (b) the maximum magnitude of the magnetic field at the core-mantle boundary, 2900 km below Earth's surface; and the (c) magnitude and (d) inclination of Earth's magnetic field at the north geographic pole. (e) Suggest why the values you calculated for ( c) and (d) differ from measured values.

Earth has a magnetic dipole moment of 8.0 X 1022 liT. (a) What current would have to be produced in a single turn of wire extending around Earth at its geomagnetic equator if we wished to set up such a dipole? Could such an arrangement be used to cancel out Earth's magnetism (b) at points in space well above Earth's surface or (c) on Earth's surface?

A charge q is distributed uniformly around a thin ring of radius r. The ring is rotating about an axis through its center and perpendicular to its plane, at an angular speed w. (a) Show that the magnetic moment due to the rotating charge has magnitude fL = !qwr2. (b) What is the direction of this magnetic moment if the charge is positive?

A magnetic compass has its needle, of mass 0.050 kg and length 4.0 cm, aligned with the horizontal component of Earth's magnetic field at a place where that component has the value B" = 16 fLT. After the compass is given a momentary gentle shake, the needle oscillates with angular frequency w = 45 rad/s. Assuming that the needle is a uniform thin rod mounted at its center, find the magnitude of its magnetic dipole moment.

The capacitor in Fig. 32-7 is being charged with a 2.50 A current. The wire radius is 1.50 mm, and the plate radius is 2.00 cm. Assume that the current i in the wire and the displacement current id in the capacitor gap are both uniformly distributed. What is the magnitude of the magnetic field due to i at the following radial distances from the wire's center: (a) 1.00 mm (inside the wire), (b) 3.00 mm (outside the wire), and (c) 2.20 cm (outside the wire)? What is the magnitude of the magnetic field due to id at the following radial distances from the central axis between the plates: (d) 1.00 mm (inside the gap), (e) 3.00 mm (inside the gap), and (f) 2.20 cm (outside the gap)? (g) Explain why the fields at the two smaller radii are so different for the wire and the gap but the fields at the largest radius are not.

A parallel-plate capacitor with circular plates of radius R = 16 mm and gap width d = 5.0 mm has a uniform electric field between the plates. Starting at time t = 0, the potential difference between the two plates is V = (100 V)e-tIT, where the time constant 7 = 12 ms. At radial distance r = 0.80R from the central axis, what is the magnetic field magnitude (a) as a function of time for t 2: 0 and (b) at time t = 37?

A magnetic flux of 7.0 m Wb is directed outward through the flat bottom face of the closed surface shown in Fig. 32-39. Along the flat top face (which has a radius of 4.2 cm) there is a OAO T magnetic field 11 directed perpendicular to the face. What are the (a) magnitude and (b) direction (inward or outward) of the magnetic flux through the curved part of the surface?

The magnetic field of Earth can be approximated as the magnetic field of a dipole. The horizontal and vertical components of this field at any distance r from Earth's center are given by B /-toM \ B /-toM \ h = 4171'3 cos "m' v = 27Tr 3 SIn """ where Am is the magnetic latitude (this type of latitude is measured from the geomagnetic equator toward the north or south geomagnetic pole). Assume that Earth's magnetic dipole moment has magnitude M = 8.00 X 1022 A m2. (a) Show that the magnitude of Earth's field at latitude Am is given by (b) Show that the inclination i of the magnetic field is related to the magnetic latitude Am by tan i = 2 tan Am

Use the results displayed in Problem 61 to predict the (a) magnitude and (b) inclination of Earth's magnetic field at the geomagnetic equator, the (c) magnitude and (d) inclination at geomagnetic latitude 60.0, and the (e) magnitude and (f) inclination at the north geomagnetic pole.

A parallel-plate capacitor with circular plates of radius 55.0 mm is being charged. At what radius (a) inside and (b) outside the capacitor gap is the magnitude of the induced magnetic field equal to 50.0% of its maximum value?

A sample of the paramagnetic salt to which the magnetization curve of Fig. 32-14 applies is immersed in a uniform magnetic field of 2.0 T. At what temperature will the degree of magnetic saturation of the sample be (a) 50 % and (b) 90 % ?

A parallel-plate capacitor with circular plates of radius R is being discharged. The displacement current through a central circular area, parallel to the plates and with radius R12, is 2.0 A. What is the discharging current?

Figure 32-40 gives the variation of an electric field that is perpendicular to a circular area of 2.0 m2 During the time period shown, what is the greatest displacement current through the area? 6 G 4 "'- b 2 I-i-+-+-i--,~-r-f--"'-L-,-I ~ 246 8 t (l1S) 12 Fig. 32-40 Problem 66

In Fig. 32-41, a parallel-plate capacitor is being discharged by a current i = 5.0 A. The plates are square with edge length L = 8.0 mm. (a) What is the rate at which the electric field between the plates is changing? (b) What is the value of p 11 . d-s around the dashed path, where H = 2.0 mm and W = 3.0 mm?

What is the measured component of the orbital magnetic dipole moment of an electron with the values(a)me=3and(b)me=-4?

In Fig. 32-42, a bar magnet lies near a paper cylinder. (a) Sketch the magnetic field lines that pass through the surface of the cylinder. (b) What is the sign of 11 . dA for every area dA on the surface? (c) Does this contradict Gauss' law for magnetism? Explain. Fig. 32-42 Problem 69.

In the lowest energy state of the hydrogen atom, the most probable distance of the single electron from the central proton (the nucleus) is 5.2 X 10-11 m. (a) Compute the magnitude of the proton's electric field at that distance. The component fLs,z of the proton's spin magnetic dipole moment measured on a z axis is 1A X 10-26 JIT. (b) Compute the magnitude of the proton's magnetic field at the distance 5.2 X 10-11 m on the z axis. (Hint: Use Eq. 29-27.) (c) What is the ratio of the spin magnetic dipole moment of the electron to that of the proton?

Figure 32-37 shows a loop model (loop L) for a paramagnetic material. (a) Sketch the field lines through and about the material due to the magnet. What is the direction of (b) the loop's net magnetic dipole moment /1, (c) the conventional current i in the loop (clockwise or counterclockwise in the figure), and (d) the magnetic force acting on the loop?

Two plates (as in Fig. 32-7) are being discharged by a constant current. Each plate has a radius of 4.00 cm. During the discharging, at a point between the plates at radial distance 2.00 cm from the central axis, the magnetic field has a magnitude of 12.5 nT. (a) What is the magnitude of the magnetic field at radial distance 6.00 cm? (b) What is the current in the wires attached to the plates?

If an electron in an atom has orbital angular momentum with me values limited by :+:3, how many values of (a) Lorb,z and (b) fLorb,z can the electron have? In terms of h, m, and e, what is the greatest allowed magnitude for (c) Lorb,z and (d) fLocb,z? (e) What is the greatest allowed magnitude for the z component of the electron's net angular momentum (orbital plus spin)? (f) How many values (signs included) are allowed for the z component of its net angular momentum?

A parallel-plate capacitor with circular plates is being charged. Consider a circular loop centered on the central axis and located between the plates. If the loop radius of 3.00 cm is greater than the plate radius, what is the displacement current between the plates when the magnetic field along the loop has magnitude 2.00 fLT?

Suppose that :+:4 are the limits to the values of me for an electron in an atom. (a) How many different values of the electron's fLocb,z are possible? (b) What is the greatest magnitude of those possible values? Next, if the atom is in a magnetic field of magnitude 0.250 T, in the positive direction of the z axis, what are (c) the maximum energy and (d) the minimum energy associated with those possible values of fLocb,z?