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Exercises in Electrodynamics Based on course by Michael Gedalin and Edited By Ben Yellin Department of Physics, Ben-Gurion University, Beer-Sheva 84105, Israel This exercises pool is intended for a undergraduate course in “Electrodynamics 1”. Some of the problems are original, while other were assembled from various sources. In particular I used Gedalin Exercise Pool, Pollack & Stump and Griffiths ====== [Exercise 0010] Del Identity (a) find ∇ × (∇ × ~v ) in general coordinates (b) Show that in cartesian Coordinates it reduce to ∇ × (∇ × ~v ) = ∇(∇ · ~v ) − ∇2~v ====== [Exercise 0011] Del Identity Prove (in general coordinates): ∇(∇ × ~v ) = 0 Taken from Griffiths 1.26 ====== [Exercise 0020] Stokes Theorem ~ x) = ẑ × ~x where ~x = xx̂ + y ŷ + z ẑ Given G(~ (a) sketch field lines H ~ ~l where C = {z = 0; x2 + y 2 = 1} (b) calculate C Gd Taken from Pollack Stump 2.20 ====== [Exercise 0021] Stokes Theorem Check Stokes Theorem for ~v = yẑ over the trangle S (see figure) 2 Taken from Griffiths 1.57 ====== [Exercise 0030] Gauss Theorem Check Gauss Theorem for ~v = r2 sin r̂ + 4r2 cos θθ̂ + r2 tan θϕ̂ over the “ice cream” S (see figure) Taken from Griffiths 1.58 ====== [Exercise 0040] Vector Calculus ~ x) = ~xe−αr Given G(~ ~ (a) Find ∇ · G ~ as function of r (b) Sketch ∇ · G R ~ Where V is a sphere with radii a centered at the origin (c) find V ∇ · G (d) find the the answer to (c) in the limit of a → ∞ Taken from Pollack Stump 2.28 ====== [Exercise 0041] Divergency of 1/r2 Given ~v = rn r̂ (a) Find ∇ · ~v (b) Check the divergernt Theorem for n = −2 on a sphere with radii R that centered at the origin Taken from Griffiths 1.62 ====== [Exercise 0050] Potentials ~ and −∇V . find A ~ and V Given F~ = yz x̂ + zxŷ + xyẑ Show F can be wriiten both as ∇ × A Taken from Griffiths 1.49 ====== [Exercise 0060] 3 Delta Identity 1 Prove δ(kx) = |k| δ(x) Taken from Griffithes Problem 1.15 ====== [Exercise 0070] Charge Disterbution write down the charge density for (a) a sphere of radii R centerd at the origin (b) a thin ring {x2 + y 2 = R2 ; z = 0} (c) infinite cylinder of radii R Taken from Gedalin 3.6 ====== [Exercise 0110] Charge Density from Electric Field ~ r) = Ar̂+B sin θ cos ϕϕ̂ find ρ(~r) Given E(~ r Taken from Griffiths 2.42 ====== [Exercise 0120] Charge Density from Potential Given φ(r) = rq e−αr (a) Find charge density (b) Find total charge Taken from Griffiths 2.42 ====== [Exercise 0140] Electrostatic Energy Part of a spherical shell {r = R; 0 < θ < θ0 ; 0 < ϕ < 2π} charge with a uniform charge density σ. A charge q0 is put on the symmetry axis at equal distances from the center and the surface. What is the energy of interaction of the charge with the surface? Taken from Gedalin 8.2 ====== [Exercise 0145] Electrostatic Force Find the force applied to the charge q0 placed at the top of the cone of the height h and the base radius R. The cone is homogeneously charged with the charge density ρ. Taken from Gedalin 8.3 4 ====== [Exercise 0150] Multipole Expansion - Thin Ring A thin ring of the radius R is homogeneously charged with the linear charge density λ. find the potential on the symmetry axis of the ring far away (a) By explicitly solving Laplace Equation (b) By using multipole expansion Taken from Gedalin 7.6, Pollack Stump 3.30 ====== [Exercise 0151] Multipole Expansion - Thin Disc A thin disc of the radius R is homogeneously charged with the linear charge density σ. find the potential on the symmetry axis of the ring far away (a) By explicitly solving Laplace Equation (b) By using multipole expansion Taken from Griffiths 2.25 ====== [Exercise 0210] 2D cartesian seperation of vars - basic two infinite grounded metal plates lie parallel to the xy plane, one at y = 0 and the other at y = a. at x = 0 the potential is given by V0 (y). find the potential between the plates Taken from Griffiths example 3.3 ====== [Exercise 0220] rectangular pipe An semi infinitely long rectangular metal pipe (a, b) is grounded. the end of the pipe, at x = 0, is maintained at specified constant potential V0 find potential inside the pipe Taken from Griffiths example 3.5 ====== [Exercise 0230] 3D Cartesian separation of vars - poisson Given Charge density ρ = ρ0 sin(kx x) sin(ky y) sin(kz z) find the potential if: (a) the charge density is spread through the all space (b) the charge density is trapped inside a layer (z) < a ====== [Exercise 0240] Cylinder in a constant field 5 Find the potential outside an infinitely long metal pipe of radius R (filled inside with metal) placed at right angle to an otherwise uniform electric field E~0 . find the surface charge induced on the pipe Taken from Griffiths problem 3.24 ====== [Exercise 0250] Cylinder with ”dipole” surface charge A long cylindrical shell of radius R is coaxial with the z axis. the surface charge is given by σ0 for y > 0 and −σ0 for y < 0. find electric potential inside and outside the cylinder Taken from Griffiths problem 3.39 ====== [Exercise 0251] Cylinder with surface charge charge density σ(ϕ) = a sin(5ϕ) is glued over the surface of infinite cylinder of radius R find electric potential inside and outside the cylinder Taken from Griffiths problem 3.25 ====== [Exercise 0260] Ball with charge density A ball of radius R (filled sphere) charge density is given by ρ0 cos θ if r > 0 ρ(r) = 0 if r = 0. (1) find electric potential in all space ====== [Exercise 0270] point charge inside a cavity A conductor is maintened at potential φ0 . inside the conductor there is a spherical cavity of radius R. inside the cavity, at distance d < R there is a point charge. find electric potential inside the cavity ====== [Exercise 0280] ”Images method” (without the method) the lower half of space (z < 0) is a perfect codnuctor. a charge q is held at ~r = (0, 0, d) find electric potential in all space ====== [Exercise 0290] Thin ring, azimotal dependency A thin ring with radius R and charge density λ(ϕ) = λ0 cos(nϕ) find electric potential in all space ====== [Exercise 0300] Spherical Shell, azimotal dependency A spherical shell with radius R is maintened at potential φ(r = R) = φ0 sin θ sin ϕ find electric potential outside the shell ====== [Exercise 0310] 6 Vector potential from current density find the vector potential for the current density r s J~ = j0 cos(nϕ)θ(R − r)ẑ R (2) where n2 6= (s + 2)2 ====== [Exercise 0320] Magnetic field from current density + superposition find the magnetic field for the current density h a r i J~ = j1 ẑ + j2 ϕ̂ θ(b − r)θ(r − a) r a (3) ====== [Exercise 0330] Magnetic force inside a cylinder of radii R1 there is a smaller cylinder of radii R2 . the axis of the small cylinder is in distance l from the axis of the big cylinder (R1 > R2 + l). inside the small cylinder the current is j~2 and inside the big cylinder (beside the volume of the small cylinder) the current is j~1 in the oposite direction. Find the force acting on the small cylinder ====== [Exercise 0340] Quasi-Stationary soltions to maxwell’s equations ~ r) is constant but ρ(~r, t) is not. Suppose J(~ (a) Show that the charge density is a linear function of time ρ(~r, t) = ρ(~r, 0) + ρ̇(~r, 0)t (4) (b) Show that ~ r) µ0 B(~ 4π Z ~ ~0 J(r ) × (~r − r~0 ) |~r − r~0 |3 (5) Taken from Griffiths Problem 7.55 ====== [Exercise 0350] Maxwell Equations Taken from Griffiths Problem 7.42 ====== [Exercise 0360] Cylinder with a slot Taken from Griffiths Problem 7.41 ====== [Exercise 0370] em momentum stored in field A long coaxial cable, of length l, consists of an inner conductor (radius a) and an outer conductor (radius b). It is connected to a battery at one and and a resistor at the other(see figure). The inner conductor carries a uniform charge 7 per unit length lambda, and a steady current I to the right. the outer conductor have oppsite charge and current. What is the em momentum stored in the fields? Taken from Griffiths Example 8.3 ====== [Exercise 0380] Feynman disk paradox Given a long solenoid (R, n, I). Two insulating cylinders are coaxial with the solenoid: (λ, a) and (−λ, b). such that a < R < b (see figure). When the current in the solenoid is gradually reduced, the cylinders begin to rotate. where does the angular momentom come from? Tip: calculate the angular momentum stored in the fields, and compare with the angular momentum due to the turque acting on the cylinders (since an electric field is induced by the coil) Taken from Griffiths Example 8.4 8 ====== [Exercise 0390] Pointing Vector for a solenoid A very long solenoid (n, a, Is ) is coaxial with a circular wire with radius b >> a and resistance R. When the current in the solenoid is gradually decreased, a current Ir is induced in the ring. (a) Calculate Ir , in terms of I˙s (b) Show that the power in the ring (Ir2 ) is due to the solenoid (calculate the flux of energy going out of the solenoid) Taken from Griffiths Problem 8.9 ====== [Exercise 0400] Thomson’s Dipole Soppuse you and an electric charge qe and a magnetic monople qm . The Fields are: 1 qe r̂ 4π0 r2 ~ = µ0 qm r̂ B 4π r2 ~ = E Find the total angular momentum stored in the fields if the two charges are separated by a distance d Tip: Z ∞ r 1 = 2 + d2 − 2rd cos θ r d(1 − cos θ) 0 (6) (7) (8) Taken from Griffiths Problem 8.12 ====== [Exercise 0410] Conservation for dialectrics Taken from Griffiths Problem 8.15 ====== [Exercise 0420] reflection and transmission Suppose the xy plane forms the boundary between two linear media (1 , µ1 and 2 , µ2 ). A plane wave of frequency ω, traveling in the z direction and polarized in the x direction, approaches the interface from the left. (see section 9.3.2 in Griffiths) (a) Calculate The exact transmission and reflection coefficients (notice that µ1 6= µ2 ). (b) Show that R + T = 1 Taken from Griffiths Problem 9.13 ====== [Exercise 0430] TEM Waves Show that a infinite coaxial transmission line does admit TEM waves. In order to do so, consider two infintly long hollow cylinders (radii a, b), the cylinders are coaxial with the z axis. show that it is possible to find a solution with Ez = 0 and Bz = 0. 9 (Note: it is not possible to do so in a case of hollow cylinder, see section 9.5.1 for a proof) Taken from Griffiths Section 9.5.3 ====== [Exercise 0440] Spherical wave Suppose sinθ 1 ~ E(r, θ, φ, t) = A cos(kr − ωt) − sin(kr − ωt) φ̂ r kr (9) with ωk = −c. (This is, incidentally, the simplest possible spherical wave. For notational convenience, let (kr − ωt) = u in your calculations.) ~ obeys all four of Maxwell’s equations, in vacuum, and find the associated magnetic field. (a) Show that E ~ over a full cycle to get the intensity vector I. ~ (Does it point in the (b) Calculate the Poynting vector. Average S 2 expected direction? Does it fall off like r , as it should?) (c) Integrate I~ · d~a over a spherical surface to determine the total power radiated. Taken from Griffiths Problem 9.33 ====== [Exercise 0450] Reflection and Transmition Light of (angular) frequency ω passes from medium 1 to medium 2 at z = 0 and from medium 2 to medium 3 at z = d. Show that the transmission coefficient for normal incidence is given by 1 n2 ωd (n2 − n22 )(n23 − n22 ) 2 T −1 = sin (10) (n1 + n3 )2 + 1 4n1 n3 n22 c (assume µ1 = µ2 = µ3 = µ0 ) Taken from Griffiths Problem 9.34 ====== [Exercise 0460] Evanescent Waves Taken from Griffiths Problem 9.37 ====== [Exercise 0470] Rsonant Cavity Consider the resonant cavity produced by closing off the two ends of a rectangular wave guide, at z = 0 and at z = d making a perfectly conducting empty box. Show that the resonant frequencies for both TE and TM modes are given by s 2 m 2 n 2 l ωl,m,n = cπ + + (11) d a b for integers l, m, and n. Find the associated electric and magnetic fields. Taken from Griffiths Problem 9.38 10 ====== [Exercise 0480] Retarded Potential - Semi Ring A piece of wire bent into a loop, as shown in the figure, carries a current that increases linearly with time: I(t) = kt (12) Calculate the retarded vector potential A at the center. Find the electric field at the center. Why does this (neutral) wire produce an electric field? (Why can’t you determine the magnetic field from this expression for A?) Taken from Griffiths Problem 10.10 ====== [Exercise 0490] Retarded Potential - Ring Suppose you take a plastic ring of radius a and glue charge on it, so that the line charge density is λ = λ0 |sin(θ/2)|. Then you spin the loop about its axis at an angular velocity ω. Find the (exact) scalar and vector potentials at the center of the ring. Taken from Griffiths Problem 10.21 ====== [Exercise 0500] Moving Charge - “Horizon” consider a particle in hyperbolic motion along the x axis: p w(t) ~ = b2 + (ct)2 (13) (In Special Relativity this is the trajectory of a particle subject to a constant force F = me /b.) (a) Sketch the graph of ω versus t. At four or five representative points on the curve, draw the trajectory of a light signal emitted by the particle at that pointboth in the plus x direction and in the minus x direction. What region on your graph corresponds to points and times (x, t) from which the particle cannot be seen? At what time does someone at point x first see the particle? (Prior to this the potential at x is evidently zero.) Is it possible for a particle, once seen, to disappear from view? (b) Determine the Lienard-Wiechert potentials for a charge in hyperbolic motion. Assume the point r is on the x axis and to the right of the charge. Taken from Griffiths Problem 10.15+10.16 ====== [Exercise 0510] Moving Charge - “Horizon”2 11 One particle, of charge q1 , is held at rest at the origin. Another particle, of charge q2 , approaches along the x axis, in hyperbolic motion: p (14) w(t) ~ = b2 + (ct)2 it reaches the closest point, b, at time t = 0, and then returns out to infinity. (a) What is the force F2 on q2 (due to q1 ) at time t? R∞ (b) What total impulse I2 = −∞ F2 dt is delivered by q1 to q2 ? (c) What is the force F1 on q1 (due to q2 ) at time t? R∞ (d) What total impulse I1 = −∞ F1 dt is delivered by q2 to q1 ? Taken from Griffiths Problem 10.24 ====== [Exercise 0520] Radiation from Rotating Ring An insulating circular ring (radius b) lies in the xy plane, centered at the origin. It carries a linear charge density λ = λ0 sin θ, where λ0 is constant and θ is the usual azimuthal angle. The ring is now set spinning at a constant angular velocity ω about the z axis. Calculate the power radiated. Taken from Griffiths Problem 11.9 ====== [Exercise 0530] infinite plane Suppose the (electrically neutral) y z plane carries a time-dependent but uniform surface current K(t)ẑ. (a) Find the electric and magnetic fields at a height x above the plane if (i) a constant current is turned on at t = 0 (K(t) = K0 ẑ for t > 0, 0 otherwise). (ii) a linearly increasing current is turned on at t = 0: (K(t) = αtẑ for t > 0, 0 otherwise). (b) Show that the retarded vector potential can be written in the form Z ∞ µ0 c x ~ A(x, t) = ẑ K t − − u du 2 c 0 (15) and from this determine E and B. (c) Show that the total power radiated per unit area of surface is ~ t) = µ0 c [K(t)]2 A(x, 2 (16) Explain what you mean by ”radiation,” in this case, given that the source is not localized. Taken from Griffiths Problem 11.24 ====== [Exercise 0540] hyperbolic motion p Problem 11.31 (a) Does a particle in hyperbolic motion w(t) ~ = b2 + (ct)2 radiate? (Use the exact formula (Eq. 11.75) from Griffiths to calculate the power radiated.) Taken from Griffiths Problem 11.31 12 ====== [Exercise 0550] Radiation with Image Charge When a charged particle approaches (or leaves) a conducting surface, radiation is emitted, associated with the changing electric dipole moment of the charge and its image. If the particle has mass m and charge q, find the total radiated power, as a function of its height z above the plane. Taken from Griffiths Problem 11.25 ====== [Exercise 0560] Decelerated Electron Suppose an electron decelerated at a constant rate a from some initial velocity v0 down to zero. What fraction of its initial kinetic energy is lost to radiation? (The rest is absorbed by whatever mechanism keeps the acceleration constant.) Assume v0 << c so that the Larmor formula can be used. Taken from Griffiths Problem 11.13 ====== [Exercise 0600] C.O.M Frame Suppose you have a collection of particles, all moving in the x direction, with energies Ei and momenta pi , Find the velocity of the center of momentum frame, in which the total momentum is zero. Taken from Griffiths Problem 12.30 ====== [Exercise 0610] Compton Scattering A photon of energy E0 ”bounces” off an electron, initially at rest. Find the energy E of the outgoing photon, as a function of the scattering angle θ Taken from Griffiths example 12.9 ====== [Exercise 0620] Newton’s Second Law In classical mechanics Newton’s law can be written in the more familiar form F = ma. The relativistic equation, F = dp/dt, cannot be so simply expressed. Show, rather, that ~v · ~a F~ = γm ~a~v 2 (17) c − v2 where a = dv/dt is the ordinary acceleration. Taken from Griffiths Problem 12.36 ====== [Exercise 0630] motion under constant force A particle of mass m is subject to a constant force F . If it starts from rest at the origin, at time t = 0, find its position (x), as a function of time. Taken from Griffiths Example 12.10 13 ====== [Exercise 0640] motion under constant force Show that µ K Kµ = 1− v2 c2 cos2 (θ) 1− v2 c2 F2 (18) Where K µ is the Minkowski 4-force Taken from Griffiths Problem 12.39 ====== [Exercise 0650] Invariants (a) Show that E · B is relativistically invariant. (b) Show that E 2 − c2 B 2 is relativistically invariant. (c) Suppose that in one inertial system B = 0 but E 6= 0 (at some point P ). Is it possible to find another system in which the electric field is zero at P ? Taken from Griffiths Problem 12.46 ====== [Exercise 0660] Invariants Compute the tensor invariants F µν Fµν , Gµν Gµν and F µν Gµν in terms of E and B. Compare with exercise 0560 (Griffiths Prob. 12.46) Taken from Griffiths Problem 12.50