Question: Refer to Prob. 7.16, to which
Refer to Prob. 7.16, to which the correct answer was / (a) Find the displacement current density Jd . (b) Integrate it to get the total displacement current, ![Refer to Prob. 7.16, to which the correct answer was
/
(a) Find the displacement current density Jd .
(b) Integrate it to get the total displacement current,
(c) Compare Id and I . (What’s their ratio?) If the outer cylinder were, say, 2 mm in diameter, how high would the frequency have to be, for Id to be 1% of I ? [This problem is designed to indicate why Faraday never discovered displacement currents, and why it is ordinarily safe to ignore them unless the frequency is extremely high.]](../images/refer-to-prob--7-16-to-which-the-correct-259553-1.jpg)
(c) Compare Id and I . (What’s their ratio?) If the outer cylinder were, say, 2 mm in diameter, how high would the frequency have to be, for Id to be 1% of I ? [This problem is designed to indicate why Faraday never discovered displacement currents, and why it is ordinarily safe to ignore them unless the frequency is extremely high.]
> Work out the formulas for u, S, g, and /in the presence of magnetic charge. [Hint: Start with the generalized Maxwell equations (7.44) and Lorentz force law (Eq. 8.44), and follow the derivations in Sections 8.1.2, 8.2.2, and 8.2.3.]
> Picture the electron as a uniformly charged spherical shell, with charge e and radius R, spinning at angular velocity ω. (a) Calculate the total energy contained in the electromagnetic fields. (b) Calculate the total angular momentum contained in the field
> (a) Check the divergence theorem for the function / using as your volume the sphere of radius R, centered at the origin. (b) Do the same for / (If the answer surprises you, look back at Prob. 1.16.)
> A sphere of radius R carries a uniform polarization P and a uniform magnetization M (not necessarily in the same direction). Find the electromagnetic momentum of this configuration. [Answer: (4/9)πμ0 R3(M × P)]
> A point charge q is located at the center of a toroidal coil of rectangular cross section, inner radius a, outer radius a+w, and height h, which carries a total of N tightly-wound turns and current I . (a) Find the electromagnetic momentum p of this confi
> An infinitely long cylindrical tube, of radius a, moves at constant speed v along its axis. It carries a net charge per unit length λ, uniformly distributed over its surface. Surrounding it, at radius b, is another cylinder, moving with the same velocity
> A very long solenoid of radius a, with n turns per unit length, carries a current Is . Coaxial with the solenoid, at radius b ((a, is a circular ring of wire, with resistance R. When the current in the solenoid is (gradually) decreased, a current Ir is i
> Derive Eq. 8.43. [Hint: Use the method of Section 7.2.4, building the two currents up from zero to their final values.]
> Derive Eq. 8.39. [Hint: Treat the lower loop as a magnetic dipole.]
> Imagine an iron sphere of radius R that carries a charge Q and a uniform magnetization M = M zˆ. The sphere is initially at rest. (a) Compute the angular momentum stored in the electromagnetic fields. (b) Suppose the sphere is g
> Calculate the power (energy per unit time) transported down the cables of Ex. 7.13 and Prob. 7.62, assuming the two conductors are held at potential difference V, and carry current I (down one and back up the other).
> An infinite number of different surfaces can be fit to a given boundary line, and yet, in defining the magnetic flux through a loop, / I never specified the particular surface to be used. Justify this apparent oversight.
> A square loop of wire (side a) lies on a table, a distance s from a very long straight wire, which carries a current I , as shown in Fig. 7.18. (a) Find the flux of B through the loop. (b) If someone now pulls the loop directly away from t
> Express the unit vectors rˆ, θˆ, φˆ in terms of xˆ, yˆ, zˆ (that is, derive Eq. 1.64). Check your answers several ways (rˆ · rˆ =? 1, θˆ · φˆ =? 0, rˆ × θˆ =? φˆ,.. .). Also work out the inverse formulas, giving xˆ, yˆ, zˆ in terms of rˆ, θˆ,
> A metal bar of mass m slides frictionlessly on two parallel conducting rails a distance l apart (Fig. 7.17). A resistor R is connected across the rails, and a uniform magnetic field B, pointing into the page, fills the entir
> (a) Show that Maxwell’s equations with magnetic charge (Eq. 7.44) are invariant under the duality transformation where / and α is an arbitrary rotation angle in “E/B-space.” Charge and c
> Prove Alfven’s theorem: In a perfectly conducting fluid (say, a gas of free electrons), the magnetic flux through any closed loop moving with the fluid is constant in time. (The magnetic &ium
> A certain transmission line is constructed from two thin metal “rib- bons,” of width w, a very small distance h
> The magnetic field of an infinite straight wire carrying a steady cur- rent I can be obtained from the displacement current term in the Ampère/Maxwell law, as follows: Picture the current as consisting of a uniform line charge λ moving along the z axis at
> Suppose J(r) is constant in time but ρ(r, t) is not—conditions that might prevail, for instance, during the charging of a capacitor. (a) Show that the charge density at any particular point is a linear function of time: Ï
> A rectangular loop of wire is situated so that one end (height h) is between the plates of a parallel-plate capacitor (Fig. 7.9), oriented parallel to the field E. The other end is way outside, where the field is essentially
> An infinite wire runs along the z axis; it carries a current I (z) that is a function of z (but not of t ), and a charge density λ(t) that is a function of t (but not of z). (a) By examining the charge flowing into a segment dz in a time dt , show that dλ/
> A transformer (Prob. 7.57) takes an input AC voltage of amplitude V1, and delivers an output voltage of amplitude V2, which is determined by the turns ratio (V2/ V1 N2/N1). If N2 > N1, the output voltage is greater than the input voltage. Why doesn’t thi
> Two coils are wrapped around a cylindrical form in such a way that the same flux passes through every turn of both coils. (In practice this is achieved by inserting an iron core through the cylinder; this has the effect of concentrating th
> Find formulas for r,θ,φ in terms of x, y, z (the inverse, in other words, of Eq. 1.62).
> (a) Use the Neumann formula (Eq. 7.23) to calculate the mutual inductance of the configuration in Fig. 7.37, assuming a is very small /Compare your answer to Prob. 7.22. (b) For the general case (not assuming a is small), show that where
> In the discussion of motional emf (Sect. 7.1.3) I assumed that the wire loop (Fig. 7.10) has a resistance R; the current generated is then I= v Bh/R. But what if the wire is made out of perfectly conducting material, so that R is zero? In that case, the
> A circular wire loop (radius r , resistance R) encloses a region of uniform magnetic field, B, perpendicular to its plane. The field (occupying the shaded region in Fig. 7.56) increases linearly with time (B= α
> The current in a long solenoid is increasing linearly with time, so the flux is proportional to t : Ф =αt . Two voltmeters are connected to diametrically opposite points ( A and B), together with resistors (R1 and
> An atomic electron (charge q) circles about the nucleus (charge Q) in an orbit of radius r ; the centripetal acceleration is provided, of course, by the Coulomb attraction of opposite charges. Now a small magnetic field dB is slowly turned on, perpendicu
> An infinite wire carrying a constant current / direction is moving in the y direction at a constant speed v. Find the electric field, in the quasistatic approximation, at the instant the wire coincides with the z axis (Fig. 7.54). /
> Electrons undergoing cyclotron motion can be sped up by increasing the magnetic field; the accompanying electric field will impart tangential acceleration. This is the principle of the betatron. One would like to keep the radius of the orbit constant dur
> A battery of emf ε and internal resistance r is hooked up to a variable “load” resistance, R. If you want to deliver the maximum possible power to the load, what resistance R should you choose? (You can’t change E and r , of course.)
> (a) Referring to Prob. 5.52(a) and Eq. 7.18, show that for Faraday-induced electric fields. Check this result by taking the divergence and curl of both sides. (b) A spherical shell of radius R carries a uniform surface charge σ . It spins abou
> Refer to Prob. 7.11 (and use the result of Prob. 5.42): How long does is take a falling circular ring (radius a, mass m, resistance R) to cross the bottom of the magnetic field B, at its (changing) terminal velocity?
> (a) Show that / (b) Show that
> A perfectly conducting spherical shell of radius a rotates about the z axis with angular velocity ω, in a uniform magnetic field Calculate the emf developed between the “north pole” and the e
> If a magnetic dipole levitating above an infinite superconducting plane (Prob. 7.45) is free to rotate, what orientation will it adopt, and how high above the surface will it float?
> A familiar demonstration of superconductivity (Prob. 7.44) is the levitation of a magnet over a piece of superconducting material. This phenomenon can be analyzed using the method of images.31 Treat the magnet as a perfect dipole m, a height z above the
> In a perfect conductor, the conductivity is infinite, so E=0 (Eq. 7.3), and any net charge resides on the surface (just as it does for an imperfect conductor, in electrostatics). (a) Show that the magnetic field is constant (∂B/∂t=0), inside a perfect cond
> The magnetic field outside a long straight wire carrying a steady current I is The electric field inside the wire is uniform: where ρ is the resistivity and a is the radius (see Exs. 7.1 and 7.3). Question: What i
> A rare case in which the electrostatic field E for a circuit can actually be calculated is the following:28 Imagine an infinitely long cylindrical sheet, of uniform resistivity and radius a. A slot (corresponding to the batt
> Two long, straight copper pipes, each of radius a, are held a distance 2d apart (see Fig. 7.50). One is at potential V0, the other at V0. The space surrounding the pipes is filled with weakly conducting material of conductivity σ
> Sea water at frequency / Hz has permittivity / permeability μ=μ0, and resistivity ρ=0.23 Ω m. What is the ratio of conduction current to displacement current? [Hint: Consider a parallel-plate capacitor immersed in sea water and driven by a voltage V0
> Suppose the conductivity of the material separating the cylinders in Ex. 7.2 is not uniform; specifically, σ(s)=k/s, for some constant k. Find the resistance between the cylinders. [Hint: Because σ is a function of position, Eq. 7.5 does not hold, the cha
> Suppose a magnetic monopole qm passes through a resistanceless loop of wire with self-inductance L. What current is induced in the loop?27
> Check Corollary 1 by using the same function and boundary line as in Ex. 1.11, but integrating over the five faces of the cube in Fig. 1.35. The back of the cube is open.
> Assuming that “Coulomb’s law” for magnetic charges (qm ) reads work out the force law for a monopole qm moving with velocity v through electric and magnetic fields E and B.26
> Suppose / (The theta function is defined in Prob. 1.46b). Show that these fields satisfy all of Maxwell’s equations, and determine ρ and J. Describe the physical situation that gives rise to these fields.
> The preceding problem was an artificial model for the charging capacitor, designed to avoid complications associated with the current spreading out over the surface of the plates. For a more realistic model, imagine thin wires that connect
> A fat wire, radius a, carries a constant current I , uniformly distributed over its cross section. A narrow gap in the wire, of width w
> An infinite cylinder of radius R carries a uniform surface charge σ . We propose to set it spinning about its axis, at a final angular velocity ωf . How much work will this take, per unit length? Do it two ways, and compare your answers: (a) Find the magne
> Two tiny wire loops, with areas a1 and a2, are situated a displacement r apart (Fig. 7.42). (a) Find their mutual inductance. [Hint: Treat them as magnetic dipoles, and use Eq. 5.88.] Is your formula consistent with Eq. 7.24? (b) Suppose a current I1 is
> Suppose the circuit in Fig. 7.41 has been connected for a long time when suddenly, at time t = 0, switch S is thrown from A to B, bypassing the battery. (a) What is the current at any subsequent time t ? (b) What is the total energy delivered to the resi
> A long cable carries current in one direction uniformly distributed over its (circular) cross section. The current returns along the surface (there is a very thin insulating sheath separating the currents). Find the self-inductance per unit length.
> (a) Two metal objects are embedded in weakly conducting material of conductivity σ (Fig. 7.6). Show that the resistance between them is related to the capacitance of the arrangement by / (b) Suppose you connected a battery between 1 and 2, an
> Test Stokes’ theorem for the function v = (xy) xˆ + (2yz) yˆ + (3zx) zˆ, using the triangular shaded area of Fig. 1.34.
> Calculate the energy stored in the toroidal coil of Ex. 7.11, by applying Eq. 7.35. Use the answer to check Eq. 7.28.
> Find the energy stored in a section of length l of a long solenoid (radius R, current I , n turns per unit length), (a) using Eq. 7.30 (you found L in Prob. 7.24); (b) using Eq. 7.31 (we worked out A in Ex. 5.12); (c) using Eq. 7.35; (d) using Eq. 7.3
> A capacitor C is charged up to a voltage V and connected to an inductor L, as shown schematically in Fig. 7.39. At time t= 0, the switch S is closed. Find the current in the circuit as a function of time. How does your answer change if a resistor R is in
> An alternating current I (t)= I0 cos(ωt ) (amplitude 0.5 A, frequency 60 Hz) flows down a straight wire, which runs along the axis of a toroidal coil with rectangular cross section (inner radius 1 cm, outer radius 2 cm, height 1 cm, 1000 turns). The coil
> Try to compute the self-inductance of the “hairpin” loop shown in Fig. 7.38. (Neglect the contribution from the ends; most of the flux comes from the long straight section.) You’ll run
> Find the self-inductance per unit length of a long solenoid, of radius R, carrying n turns per unit length.
> A square loop of wire, of side a, lies midway between two long wires, 3a apart, and in the same plane. (Actually, the long wires are sides of a large rectangular loop, but the short ends are so far away that they can be neglected.) A clock- wise current
> A small loop of wire (radius a) is held a distance z above the center of a large loop (radius b), as shown in Fig. 7.37. The planes of the two loops are parallel, and perpendicular to the common axis. (a) Suppose current I flows in the big loop. Find the
> Imagine a uniform magnetic field, pointing in the z direction and filling all space / A positive charge is at rest, at the origin. Now somebody turns off the magnetic field, thereby inducing an electric field. In what direction does the charge move?16
> Where is ∂B/∂t nonzero, in Figure 7.21(b)? Exploit the analogy between Faraday’s law and Ampère’s law to sketch (qualitatively) the electric field
> Test the divergence theorem for the function v = (xy) xˆ + (2yz) yˆ+ (3zx) zˆ. Take as your volume the cube shown in Fig. 1.30, with sides of length 2.
> A capacitor C has been charged up to potential V0; at time t=0, it is connected to a resistor R, and begins to discharge (Fig. 7.5a). (a) Determine the charge on the capacitor as a function of time, Q(t). What is the current through the resistor, I (t)?
> A toroidal coil has a rectangular cross section, with inner radius a, outer radius a+w, and height h. It carries a total of N tightly wound turns, and the current is increasing at a constant rate (dI/dt=k). If w and h are both much less than a, find the e
> A square loop, side a, resistance R, lies a distance s from an infinite straight wire that carries current I (Fig. 7.29). Now someone cuts the wire, so I drops to zero. In what direction does the induced current in the square loop ï&n
> A long solenoid of radius a, carrying n turns per unit length, is looped by a wire with resistance R, as shown in Fig. 7.28. (a) If the current in the solenoid is increasing at a constant rate (dI/dt= k), what current flows in the loop, an
> An alternating current I=I0 cos (ωt) flows down a long straight wire, and returns along a coaxial conducting tube of radius a. (a) In what direction does the induced electric field point (radial, circumferential, or longitudinal)? (b) Assuming that the fiel
> A long solenoid with radius a and n turns per unit length carries a time-dependent current I (t) in the φˆ direction. Find the electric field (magnitude and direction) at a distance s from the axis (both inside and outside the solenoid), in the quasistati
> As a lecture demonstration a short cylindrical bar magnet is dropped down a vertical aluminum pipe of slightly larger diameter, about 2 meters long. It takes several seconds to emerge at the bottom, whereas an otherwise identical piece of unmagnetized ir
> A square loop of wire, with sides of length a, lies in the first quadrant of the xy plane, with one corner at the origin. In this region, there is a nonuniform time-dependent magnetic field / (where k is a constant). Find the emf induced in the loop.
> A long solenoid, of radius a, is driven by an alternating current, so that the field inside is sinusoidal: / A circular loop of wire, of radius a/2 and resistance R, is placed inside the solenoid, and coaxial with it. Find the current induced in the loop
> A square loop is cut out of a thick sheet of aluminum. It is then placed so that the top portion is in a uniform magnetic field B, and is allowed to fall under gravity (Fig. 7.20). (In the diagram, shading indicates the fiel
> Check the fundamental theorem for gradients, using T = x 2 + 4xy + 2yz3, the points a = (0, 0, 0), b = (1, 1, 1), and the three paths in Fig. 1.28: (a) (0, 0, 0) → (1, 0, 0) → (1, 1, 0) → (1, 1, 1); (
> A square loop (side a) is mounted on a vertical shaft and rotated at angular velocity ω (Fig. 7.19). A uniform magnetic field B points to the right. Find the E(t) for this alternating current generator.
> Two concentric metal spherical shells, of radius a and b, respectively, are separated by weakly conducting material of conductivity σ (Fig. 7.4a). (a) If they are maintained at a potential difference V, what current flows from o
> A short circular cylinder of radius a and length L carries a “frozen-in” uniform magnetization M parallel to its axis. Find the bound current, and sketch the magnetic field of the cylinder. (Make three sketches: one for L ((a, one for L
> A long circular cylinder of radius R carries a magnetization M = ks2 φˆ, where k is a constant, s is the distance from the axis, and φˆ is the usual azimuthal unit vector (Fig. 6.13). Find the
> An infinitely long circular cylinder carries a uniform magnetization M parallel to its axis. Find the magnetic field (due to M) inside and outside the cylinder.
> Of the following materials, which would you expect to be paramagnetic and which diamagnetic: aluminum, copper, copper chloride (CuCl2), carbon, lead, nitrogen (N2), salt (NaCl), sodium, sulfur, water? (Actually, copper is slightly diamagnetic; otherwise
> A uniform current density J = J0 zˆ fills a slab straddling the yz plane, from x = −a to x = +a. A magnetic dipole m = m0 xˆ is situated at the origin. (a) Find the force on the dipole, using Eq. 6.3. (b) Do the same for a dipole pointing in the y directi
> Derive Eq. 6.3. [Here’s one way to do it: Assume the dipole is an infinitesimal square, of side ε (if it’s not, chop it up into squares, and apply the argument to each one). Choose axes as s
> Find the force of attraction between two magnetic dipoles, m1 and m2, oriented as shown in Fig. 6.7, a distance r apart, (a) using Eq. 6.2, and (b) using Eq. 6.3.
> You are asked to referee a grant application, which proposes to determine whether the magnetization of iron is due to “Ampère” dipoles (current loops) or “Gilbert” dipoles (separated magnetic monopoles). The experiment will involve a cylinder of iron (ra
> Calculate the volume integral of the function T z2 over the tetrahedron with corners at (0,0,0), (1,0,0), (0,1,0), and (0,0,1).
> A magnetic dipole m is imbedded at the center of a sphere (radius R) of linear magnetic material (permeability μ). Show that the magnetic field inside the sphere (0 < r ≤ R) is / What is the field outside the sphere?
> At the interface between one linear magnetic material and another, the magnetic field lines bend (Fig. 6.32). Show that tan θ2/ tan θ1 μ2/μ1, assuming there is no free current at the
> Compare Eqs. 2.15, 4.9, and 6.11. Notice that if ρ, P, and M are uniform, the same integral is involved in all three: Therefore, if you happen to know the electric field of a uniformly charged object, you can immediately write d
> Notice the following parallel: Thus, the transcription turns an electrostatic problem into an analogous magnetostatic one. Use this, together with your knowledge of the electrostatic results, to rederive (a) the magnetic field inside a uni
> Imagine two charged magnetic dipoles (charge q, dipole moment m), constrained to move on the z axis (same as Problem 6.23(a), but without gravity). Electrically they repel, but magnetically (if both m’s point in the z direction) they attract. (a) Find th
> A familiar toy consists of donut-shaped permanent magnets (magnetization parallel to the axis), which slide frictionlessly on a vertical rod (Fig. 6.31). Treat the magnets as dipoles, with mass md and dipole moment m. (a) If you put two back-to-back magn
> In Prob. 6.4, you calculated the force on a dipole by “brute force.” Here’s a more elegant approach. First write B(r) as a Taylor expansion about the center of the loop: where r0 is the position of th
> (a) Show that the energy of a magnetic dipole in a magnetic field B is [Assume that the magnitude of the dipole moment is fixed, and all you have to do is move it into place and rotate it into its final orient
> How would you go about demagnetizing a permanent magnet (such as the wrench we have been discussing, at point c in the hysteresis loop)? That is, how could you restore it to its original state, with M = 0 at I = 0?
> Starting from the Lorentz force law, in the form of Eq. 5.16, show that the torque on any steady current distribution (not just a square loop) in a uniform field B is m × B.
