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Chapter: 2 -Problem: 59 >> Prove or disprove (with a counterexample) the following Theorem: Suppose a conductor carrying a net charge Q, when placed in an external electric field Ee, experiences a force F; if the external field is now reversed (Ee ??Ee), the force also reverses (F??F). What if we stipulate that the external field is uniform?
Answer Preview: The theorem is false. For example, suppose the conductor is a n…

, Chapter: 7 -Problem: 63 >> 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 field lines are, as it were, “frozen” into the fluid.) (a) Use Ohm’s law, in the form of Eq. 7.2, together with Faraday’s law, to prove that if ? =?and J is finite, then ?B/?t = ? × (v × B).(b) Let S be the surfac
Answer Preview: (a) (b) J= o(E +vxB); J fi…

, Chapter: 2 -Problem: 61 >> What is the minimum-energy configuration for a system of N equal point charges placed on or inside a circle of radius R? Because the charge on a conductor goes to the surface, you might think the N charges would arrange themselves (uniformly) around the circumference. Show (to the contrary) that for N = 12 it is better to place 11 on the circumference and one at the center. How about for N = 11 (i
Answer Preview: Suppose the n point charges are eve…

, Chapter: 7 -Problem: 59 >> 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 ?(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 d?/dt = ?d I/dz. If we stipulate that ?(0) = 0 and I (0) = 0, show that ?(t) = kt, I (z) = ?kz, where k is a constant.(b) Assume for a moment that the
Answer Preview: (a) The charge flowing into d z in time d t is (b) In the quasistatic approximation d d…

, Chapter: 1 -Problem: 6 >> Prove that:[A × (B × C)] + [B × (C × A)] + [C × (A × B)] = 0. Under what conditions does A × (B × C) = (A × B) × C?
Answer Preview: Ax(BXC)+Bx(CxA)+CX (AxB) = B(A C) -C(A B)+C(A B)-A(C B) +A(B C) -B(CA) = 0. So: Ax(BxC…

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, Chapter: 7 -Problem: 54 >> 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 = ?t). An ideal voltmeter (infinite internal resistance) is connected between points P and Q. (a) What is the current in the loop?(b) What does the voltmeter read?
Answer Preview: (a) (b) Inside the shaded region, for a …

, Chapter: 5 -Problem: 24 >> What current density would produce the vector potential, A = k ˆ? (where k is a constant), in cylindrical coordinates?
Answer Preview: 18 k -(sk) 2 = -2…

, Chapter: 11 -Problem: 13 >> A positive charge q is fired head-on at a distant positive charge Q (which is held stationary), with an initial velocity v0. It comes in, decelerates to v = 0, and returns out to infinity. What fraction of its initial energy (1/2mv20) is radiated away? Assume v0 <Answer Preview: The fraction of energy radiated away is equal to the ratio of the work done b…

, Chapter: 11 -Problem: 26 >> An ideal electric dipole is situated at the origin; its dipole moment points in the Z? direction, and is quadratic in time: (a) Use the method of Section 11.1.2 to determine the (exact) electric and magnetic fields, for all r > 0 (there’s also a delta-function term at the origin, but we’re not concerned with that). (b) Calculate the power, P(r, t), passing through a sphere of radius r .(c) Find th
Answer Preview: a) The method of Section 11 1 2 can be used to determine the electric and magnetic fields for an ide…

, Chapter: 2 -Problem: 32 >> Two positive point charges, qA and qB (masses mA and mB) are at rest, held together by a massless string of length a. Now the string is cut, and the particles fly off in opposite directions. How fast is each one going, when they are far apart?
Answer Preview: The speed of each particle can be calculated using the equati…

, Chapter: 7 -Problem: 62 >> A certain transmission line is constructed from two thin metal “ribbons,” of width w, a very small distance h << w apart. The current travels down one strip and back along the other. In each case, it spreads out uniformly over the surface of the ribbon.(a) Find the capacitance per unit length, C.(b) Find the inductance per unit length, L.(c) What is the product LC, numerically? [L and C will, of c
Answer Preview: I = =; W h t 1 (a) Parallel-plate capacitor: E = 0; V = Eh= (b) B = oK: = Bh…

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, Chapter: 1 -Problem: 44 >> Evaluate the following integrals:(a)  (b)  (c)  d.  Transcribed Image Text: (3x² - 2x - 1) 8(x ? 3) dx.
Answer Preview: a . b . c. ze…

, Chapter: 11 -Problem: 21 >> An electric dipole rotates at constant angular velocity ? in the x y plane. (The charges, ±q, are at  the magnitude of the dipole moment is p = 2qR.)(a) Find the interaction term in the self-torque (analogous to Eq. 11.99). Assume the motion is nonrelativistic (?R ? c).(b) Use the method of Prob. 11.20(a) to obtain the total radiation reaction torque on this system.(c) Check that this result is co
Answer Preview: The total torque is twice the torque on +q; we might as well calculate it at time t = 0. First we need the electric field at +q, due to -q when it was …

, Chapter: 4 -Problem: 38 >> Prove the following uniqueness theorem: A volume V contains a specified free charge distribution, and various pieces of linear dielectric material, with the susceptibility of each one given. If the potential is specified on the boundaries S of V (V = 0 at infinity would be suitable) then the potential throughout V is uniquely determined.
Answer Preview: Given two solutions, V (and E (E3 E2 - E1, D3 = D2 - D1). = == - VV, D = E) and V (E2 = …

, Chapter: 8 -Problem: 15 >> 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 configuration, assuming that w and h are both much less than a (so you can ignore the variation of the fields over the cross section).(b) Now the curr
Answer Preview: (a) The fields are (b) The changing magnetic f…

, Chapter: 11 -Problem: 35 >> Use the result of Prob. 10.34 to determine the power radiated by an ideal electric dipole, p(t), at the origin. Check that your answer is consistent with Eq. 11.22, in the case of sinusoidal time dependence, and with Prob. 11.26, in the case of quadratic time dependence.Data from Prob. 10.34Reference equation 11.22 Transcr
Answer Preview: The fields of a nonstatic ideal dipole (Problem 10 34) contain terms that go like 1…

, Chapter: 9 -Problem: 7 >> Suppose string 2 is embedded in a viscous medium (such as molasses), which imposes a drag force that is proportional to its (transverse) speed:(a) Derive the modified wave equation describing the motion of the string. (b) Solve this equation, assuming the string vibrates at the incident frequency ?. That is, look for solutions of the form ˜ f (z, t) = ei?t ˜F (z). (c) Show that the waves are atten
Answer Preview: a. b. c. d. 0 f …

, Chapter: 7 -Problem: 21 >> Imagine a uniform magnetic field, pointing in the z direction and filling all space (B = B0 z?). 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?
Answer Preview: The answer is indeterminate, until some boundary co…

, Chapter: 1 -Problem: 45 >> Evaluate the following integrals:(a) (b)  (c)  (d)  Transcribed Image Text: ²?(2x + 3) 8 (3x) dx.
Answer Preview: a . b. c…

, Chapter: 8 -Problem: 24 >> A circular disk of radius R and mass M carries n point charges (q), attached at regular intervals around its rim. At time t = 0 the disk lies in the xy plane, with its center at the origin, and is rotating about the z axis with angular velocity ?0, when it is released. The disk is immersed in a (time-independent) external magnetic field B(s, z) = k(?s S? + 2z z?), where k is a constant.(a) Find th
Answer Preview: (a) Initially, the disk will rise like a helicopter. The force on one charge (velocity v = …

, Chapter: 4 -Problem: 31 >> A point charge Q is “nailed down” on a table. Around it, at radius R, is a frictionless circular track on which a dipole p rides, constrained always to point tangent to the circle. Use Eq. 4.5 to show that the electric force on the dipole isNotice that this force is always in the “forward” direction (you can easily confirm this by drawing a diagram showing the forces on the two ends of the dipole)
Answer Preview: Qualitatively, the forces on the negative and positive ends, though equal in magnitude, point in sli…

, Chapter: 5 -Problem: 12 >> Calculate the magnetic field at the center of a uniformly charged spherical shell, of radius R and total charge Q, spinning at constant angular velocity ?.
Answer Preview: Field (at center o…

, Chapter: 2 -Problem: 14 >> Find the electric field inside a sphere that carries a charge density proportional o the distance from the origin, ? = kr, for some constant k. (Fig. 2.25). Find the electric field in the three regions: (i) r < a, (ii) a < r < b, (iii) r > b. Plot |E| as a function of r , for the case b = 2a. Transcribed Image Text:
Answer Preview: Gaussian surface f E-da …

, Chapter: 2 -Problem: 46 >> If the electric field in some region is given (in spherical coordinates) by the expression for some constant k, what is the charge density? Transcribed Image Text: k E(r) = [3f+ 2 sin 0 cos 0 sin þê + sin0 cos 4 ]
Answer Preview: The charge density is the amount of charge per unit volume in a given region. It is usually expresse…

, Chapter: 11 -Problem: 18 >> A point charge q, of mass m, is attached to a spring of constant k. At time t = 0 it is given a kick, so its initial energy is Now it oscillates, gradually radiating away this energy.(a) Confirm that the total energy radiated is equal to U0. Assume the radiation damping is small, so you can write the equation of motion as (b) Suppose now we have two such oscillators, and we start them off with ide
Answer Preview: (a) To confirm that the total energy radiated is equal to U0, we can use the formula for the energy of an oscillator: E = (1/2)kx^2 + (1/2)mv^2 where …

, Chapter: 1 -Problem: 46 >> (a) Show that(b) Let ?(x) be the step functionShow that d?/dx = ?(x). Transcribed Image Text: d - x (8(x)) = -8(x). dx
Answer Preview: a. b. f(x) [x8(x)] dx = x (x)8(x) | The first …

, Chapter: 10 -Problem: 29 >> We are now in a position to treat the example in Sect. 8.2.1 quantitatively. Suppose q1 is at x1 = ?vt and q2 is at y = ?vt (Fig. 8.3, with t < 0). Find the electric and magnetic forces on q1 and q2. Is Newton’s third law obeyed?Figure 8.3       Transcribed Image Text: Fe ya 92 V2 - B? -Fm B? m 91 F
Answer Preview: The electric force on q1 is given by F1 = q1E = q1(E2 - …

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, Chapter: 11 -Problem: 8 >> A parallel-plate capacitor C, with plate separation d, is given an initial charge (±)Q0. It is then connected to a resistor R, and discharges, Q(t) = Q0e?t/RC. (a) What fraction of its initial energy (Q20 /2C) does it radiate away? (b) If C = 1 pF, R = 1000 , and d = 0.1 mm, what is the actual number? In electronics we don’t ordinarily worry about radiative losses; does that seem reasonable, in th
Answer Preview: (a) (b) SO The power radiated (Eq. 11 60) is -p. In this …

, Chapter: 2 -Problem: 40 >> (a) A point charge q is inside a cavity in an uncharged conductor (Fig. 2.45). Is the force on q necessarily zero?(b) Is the force between a point charge and a nearby uncharged conductor always attractive? Transcribed Image Text: Gaussian surface + + + + q• E #0 Conductor + X X + + E=0+ ++
Answer Preview: (a) No. For example, if it is very close to th…

, Chapter: 3 -Problem: 35 >> A solid sphere, radius R, is centered at the origin. The “northern” hemisphere carries a uniform charge density ?0, and the “southern” hemisphere a uniform charge density ??0. Find the approximate field E(r, ?) for points far from the sphere (r >> R).
Answer Preview: The total charge is zero, so the dominant te…

, Chapter: 10 -Problem: 34 >> Find the (Lorenz gauge) potentials and fields of a time-dependent ideal electric dipole p(t) at the origin. (It is stationary, but its magnitude and/or direction are changing with time.) Don’t bother with the contact term.
Answer Preview: The retarded potentials are t t and hence - We want a source localized at the origin, so we expand in powers of r', keeping terms up to first order: 2 …

, Chapter: 8 -Problem: 14 >> An infinitely long cylindrical tube, of radius a, moves at constant speed v along its axis. It carries a net charge per unit length ?, uniformly distributed over its surface. Surrounding it, at radius b, is another cylinder, moving with the same velocity but carrying the opposite charge (??). Find: (a) The energy per unit length stored in the fields. (b) The momentum per unit length in the fields.
Answer Preview: The fields are zero for s < a an…

, Chapter: 9 -Problem: 11 >> Consider a particle of charge q and mass m, free to move in the xy plane in response to an electromagnetic wave propagating in the z direction (Eq. 9.48—might as well set ? = 0). (a) Ignoring the magnetic force, find the velocity of the particle, as a function of time. (Assume the average velocity is zero.) (b) Now calculate the resulting magnetic force on the particle. (c) Show that the (time) av
Answer Preview: a. b. The magnetic force is c. The (time) average force is d. Adding i…

, Chapter: 10 -Problem: 21 >> For a point charge moving at constant velocity, calculate the flux integral ? E · da (using Eq. 10.75), over the surface of a sphere centered at the present location of the charge.Reference equation 10.75 Transcribed Image Text: E(r, t): = 9 1-v²/c² R 3/2 47 €0 (1-v² sin² 0/c²) ³/² R²°
Answer Preview: By Gauss's law (in integral form) the answer has to be q/0. 1 q(1-v/c) R 47 (1 2 sin 0)3/2 R…

, Chapter: 5 -Problem: 51 >> Consider a plane loop of wire that carries a steady current I; we want to calculate the magnetic field at a point in the plane. We might as well take that point to be the origin (it could be inside or outside the loop). The shape of the wire is given, in polar coordinates, by a specified function r (?) (Fig. 5.62).(a) Show that the magnitude of the field is26(b) Test this formula by calculating th
Answer Preview: (a) (b) (c) (d) so B …

, Chapter: 7 -Problem: 20 >> 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. Transcribed Image Text: B (in) (a) B (in) (b) changing magnetic field FIGURE 7.21 B (c)
Answer Preview: B/t is nonzero along the left and right edges of the shaded rectangle: The (inward) flux through t…

, Chapter: 2 -Problem: 60 >> Prove or disprove (with a counterexample) the following Theorem: Suppose a conductor carrying a net charge Q, when placed in an external electric field Ee, experiences a force F; if the external field is now reversed (Ee ??Ee), the force also reverses (F??F). What if we stipulate that the external field is uniform?
Answer Preview: The initial configuration consists of a point charge q at the center, …

, Chapter: 1 -Problem: 18 >> Calculate the curls of the vector functions in Prob. 1.15.Data from problem 1.15    Transcribed Image Text: (a) va = x² + 3xz² ? - 2xz 2. (b) V, =xyx+2yz ý+3zxê. (c) Vc = y²x + (2xy + z²) ? + 2yz î.
Answer Preview: a. b. c . V XVa = …

, Chapter: 8 -Problem: 20 >> Consider an ideal stationary magnetic dipole m in a static electric field E. Show that the fields carry momentumSo far, this is valid for any localized static configuration. For a current confined to an infinitesimal neighborhood of the origin we can approximate V(r) ? V(0) ? E(0) · r. Treat the dipole as a current loop, and use Eqs. 5.82 and 1.108.
Answer Preview: (E B) dr of v VB x da + 00 S P = 0 So = 0 =-0 For a current loop, J dr- [(VV) x Bdr = co = 1/2 LV …

, Chapter: 10 -Problem: 7 >> A time-dependent point charge q(t) at the origin, ?(r, t) = q(t)?3(r), is fed by a current J(r, t) = ?(1/4?)(q/r 2) ˆr, where q ? dq/dt.(a) Check that charge is conserved, by confirming that the continuity equation is obeyed.(b) Find the scalar and vector potentials in the Coulomb gauge. If you get stuck, try working on (c) first.(c) Find the fields, and check that they satisfy all of Maxwell’s eq
Answer Preview: (a) The charge density for the point charge is given by (r, t) = q(t)3(r). The current density is gi…

, Chapter: 2 -Problem: 12 >> Use Gauss’s law to find the electric field inside a uniformly charged solid sphere (charge density ?).
Answer Preview: R Gaussian surface f E …

, Chapter: 3 -Problem: 41 >> Buckminsterfullerine is a molecule of 60 carbon atoms arranged like the stitching on a soccer-ball. It may be approximated as a conducting spherical shell of radius R = 3.5Å. A nearby electron would be attracted, according to Prob. 3.9, so it is not surprising that the ion C? 60 exists. (Imagine that the electron— on average—smears itself out uniformly over the surface.) But how about a second ele
Answer Preview: 1 1 + -40% (-a + (a - b)) - The second term is identical to Problem 3 9, and …

, Chapter: 7 -Problem: 9 >> An infinite number of different surfaces can be fit to a given boundary line, and yet, in defining the magnetic flux through a loop, ?=? B. da, I never specified the particular surface to be used. Justify this apparent oversight.
Answer Preview: This oversight is justified because the magnetic flux is determined by the strength and direction o…

, Chapter: 2 -Problem: 58 >> (a) Consider an equilateral triangle, inscribed in a circle of radius a, with a point charge q at each vertex. The electric field is zero (obviously) at the center, but (surprisingly) there are three other points inside the triangle where the field is zero. Where are they? [Answer: r = 0.285 a—you’ll probably need a computer to get it.](b) For a regular n-sided polygon there are n points (in addit
Answer Preview: (a) One such point is on the x axis (see diagram) at x = r. Here the field …

, Chapter: 1 -Problem: 36 >> (a) Show that(b) Show that Transcribed Image Text: [ƒ(V x A) · da = [[A × (Vƒ)] · da + f ƒA. dl. $ . P
Answer Preview: a. Use the pro…

, Chapter: 7 -Problem: 51 >> An infinite wire carrying a constant current I in the z? 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). Transcribed Image Text: N I FIGURE 7.54
Answer Preview: In the quasistatic approximation the magnetic field of the wire is B = (ol/27…

, Chapter: 2 -Problem: 51 >> Find the potential on the rim of a uniformly charged disk (radius R, charge density ?).
Answer Preview: Let us/R. Then V = - …

, Chapter: 4 -Problem: 34 >> The space between the plates of a parallel-plate capacitor is filled with dielectric material whose dielectric constant varies linearly from 1 at the bottom plate (x = 0) to 2 at the top plate (x = d). The capacitor is connected to a battery of voltage V. Find all the bound charge, and check that the total is zero.
Answer Preview: X Say the high voltage is connected to the bottom plate, so the electric field points in …

, Chapter: 10 -Problem: 25 >> Figure 2.35 summarizes the laws of electrostatics in a “triangle diagram” relating the source (?), the field (E), and the potential (V). Figure 5.48 does the same for magnetostatics, where the source is J, the field is B, and the potential is A. Construct the analogous diagram for electrodynamics, with sources ? and J (constrained by the continuity equation), fields E and B, and potentials V and A
Answer Preview: POTENTIALS V. A V A= 1 av c t v=dT; A = dr Vz…

, Chapter: 2 -Problem: 33 >> Two positive point charges, qA and qB (masses mA and mB) are at rest, held together by a massless string of length a. Now the string is cut, and the particles fly off in opposite directions. How fast is each one going, when they are far apart?
Answer Preview: When the string is cut, each particle will fly off …

, Chapter: 8 -Problem: 9 >> Two concentric spherical shells carry uniformly distributed charges +Q (at radius a) and ?Q (at radius b > a). They are immersed in a uniform magnetic field B = B0 z? (a) Find the angular momentum of the fields (with respect to the center). (b) Now the magnetic field is gradually turned off. Find the torque on each sphere, and the resulting angular momentum of the system.
Answer Preview: a. b. Between the shells, E = Q …

, Chapter: 12 -Problem: 27 >> A cop pulls you over and asks what speed you were going. “Well, officer, I cannot tell a lie: the speedometer read 4 × 108 m/s.” He gives you a ticket, because the speed limit on this highway is 2.5 × 108 m/s. In court, your lawyer (who, luckily, has studied physics) points out that a car’s speedometer measures proper velocity, whereas the speed limit is ordinary velocity. Guilty, or innocent?
Answer Preview: Guilty. A speedometer measures the proper velocity of a car, which is the velocity of the …

, Chapter: 3 -Problem: 28 >> A circular ring in the xy plane (radius R, centered at the origin) carries a uniform line charge ?. Find the first three terms (n = 0, 1, 2) in the multipole expansion for V(r, ?).
Answer Preview: K N r R y = For a line charge, p(r') dr' (r') dl', which in this case becomes …

, Chapter: 10 -Problem: 30 >> A uniformly charged rod (length L, charge density ?) slides out the x axis at constant speed v. At time t = 0 the back end passes the origin (so its position as a function of time is x = vt, while the front end is at x = vt + L). Find the retarded scalar potential at the origin, as a function of time, for t > 0. [First determine the retarded time t1 for the back end, the retarded time t2 for the f
Answer Preview: c(t-t) = vt, t = t(1+v/c), t = x = vt = vt 1+v/c If L vt, V = V = x2 = vt + L V (0, t) = X 40 …

, Chapter: 3 -Problem: 58 >> Find the charge density ?(?) on the surface of a sphere (radius R) that produces the same electric field, for points exterior to the sphere, as a charge q at the point a < R on the z axis.
Answer Preview: The charge density on the surface of a sphere that produces the …

, Chapter: 6 -Problem: 22 >> First write B(r) as a Taylor expansion about the center of the loop: where r0 is the position of the dipole and ?0 denotes differentiation with respect to r0. Put this into the Lorentz force law (Eq. 5.16) to obtainOr, numbering the Cartesian coordinates from 1 to 3:where i j k is the Levi-Civita symbol (+1 if i j k = 123, 231, or 312; ?1 if i j k = 132, 213, or 321; 0 otherwise), in terms of whic
Answer Preview: F = I dl x B=I = 1 (f) (because fdl=0). Now = 1 Ifa dlx [(r. Vo)Bo] - I 1 (fa) = dl x Bo + I = I tij…

, Chapter: 3 -Problem: 57 >> A stationary electric dipole p = p ˆz is situated at the origin. A positive point charge q (mass m) executes circular motion (radius s) at constant speed in the field of the dipole. Characterize the plane of the orbit. Find the speed, angular momentum and total energy of the charge.
Answer Preview: The plane of the orbit is in the x y - plane . The speed of the charge is given by v = s , where is …

, Chapter: 7 -Problem: 4 >> Suppose the conductivity of the material separating the cylinders in Ex. 7.2 is not uniform; specifically, ?(s) = k/s, for some constant k. Find the resistance between the cylinders.  Transcribed Image Text: E L FIGURE 7.2
Answer Preview: V I = J(s) 2 sL J(s) …

, Chapter: 5 -Problem: 39 >> Analyze the motion of a particle (charge q, mass m) in the magnetic field of a long straight wire carrying a steady current I. (a) Is its kinetic energy conserved? (b) Find the force on the particle, in cylindrical coordinates, with I along the z axis. (c) Obtain the equations of motion. (d) Suppose ?z is constant. Describe the motion
Answer Preview: (a) No, the kinetic energy of the particle is not conserved. The particle accelerates due to the Lor…

, Chapter: 2 -Problem: 2 >> Find the electric field (magnitude and direction) a distance z above the midpoint between equal and opposite charges (±q), a distance d apart (same as Example 2.1, except that the charge at x = +d/2 is ?q). Example 2.1: Transcribed Image Text: ZA 9 d/2 P N (a) d/2 9 x E? 9 d/2 E E? N (b) 2 d/29 x
Answer Preview: This time the verti…

, Chapter: 8 -Problem: 5 >> Imagine two parallel infinite sheets, carrying uniform surface charge +? (on the sheet at z = d) and ?? (at z = 0). They are moving in the y direction at constant speed v  (a) What is the electromagnetic momentum in a region of area A? (b) Now suppose the top sheet moves slowly down (speed u) until it reaches the bottom sheet, so the fields disappear. By calculating the total force on the charge (
Answer Preview: (a) (b) (i) There is a magnetic force, due to the (average) magnetic field at the upp…

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