Download Exercises in Electrodynamics

Exercises in Electrodynamics
Based on course by Michael Gedalin and Edited By Ben Yellin and Daniel Hurowitz
Department of Physics, Ben-Gurion University, Beer-Sheva 84105, Israel
This exercise pool is intended for an undergraduate course in “Electrodynamics 1”. Some of
the problems are original, while others were assembled from various sources. In particular I used
Gedalins Exercise Pool, Pollack & Stump and Griffiths
====== [Exercise 0010]
Del Identity
(a) Find ∇ × (∇ × ~v ) in general coordinates.
(b) Show that in cartesian Coordinates it reduces 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 the 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 triangle S (see figure)
2
Taken from Griffiths 1.57
====== [Exercise 0030]
Gauss Theorem
Check Gauss’s theorem for ~v = r2 sin θr̂ + 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 radius a centered at the origin.
(c) Find V ∇ · G,
(d) Find the the answer to (c) in the limit a → ∞
Taken from Pollack Stump 2.28
====== [Exercise 0041]
Divergence of 1/r2
Given ~v = rn r̂
(a) Find ∇ · ~v .
(b) Check the divergernce theorem for n = −2 on a sphere with radius R that is centered at the origin.
Taken from Griffiths 1.62
====== [Exercise 0050]
Potentials
Given F~ = yz x̂ + zxŷ + xyẑ, show that F~ can be written both as
~
(a) ∇ × A
(b) −∇V .
3
~ and V
Find A
Taken from Griffiths 1.49
====== [Exercise 0060]
Delta Identity
1
Prove that δ(kx) = |k|
δ(x).
Taken from Griffiths Problem 1.15
====== [Exercise 0070]
Charge Distribution
Write down the charge density for
(a) A sphere of radius R centered at the origin.
(b) A thin ring {x2 + y 2 = R2 ; z = 0}.
(c) An infinite cylinder of radius R.
Taken from Gedalin 3.6
====== [Exercise 0080]
Greens Function in 1D
Derive Greens function for the one dimensional Laplace equation.
Taken from Gedalin, 4.2
====== [Exercise 0081]
Greens Function in 2D
Derive Greens function for the two dimensional Laplace equation.
Taken from Gedalin, 4.1
====== [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 the charge density.
(b) Find the total charge.
Taken from Griffiths 2.42
4
====== [Exercise 0140]
Electrostatic Energy
Part of a spherical shell {r = R; 0 < θ < θ0 ; 0 < ϕ < 2π} is charged with uniform charge density σ. A charge q0 is
placed on the axis of symmetry 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 a charge q0 placed at the top of a cone of height h and base radius R. The cone is
homogeneously charged with charge density ρ.
Taken from Gedalin 8.3
====== [Exercise 0150]
Multipole Expansion - Thin Ring
A thin ring of radius R is homogeneously charged with linear charge density λ. Find the potential on the symmetry
axis of the ring far away from the ring.
(a) By explicitly solving Laplace equation.
(b) By using the multipole expansion.
Taken from Gedalin 7.6, Pollack Stump 3.30
====== [Exercise 0151]
Multipole Expansion - Thin Disc
A thin disc of radius R is homogeneously charged with linear charge density σ. Find the potential on the symmetry
axis of the disc far away
(a) By explicitly solving Laplace Equation.
(b) By using the multipole expansion.
Taken from Griffiths 2.25
====== [Exercise 0152]
Multipole expansion - Dipoles and Quadrupoles
A linear quadrupole consists of three charges: q, −2q, and q, on the z axis. The positive charges are at z = ±d and
the negative charge is at the origin.
(a) Show that this system is the same as two dipoles, with dipole moments +qdk̂ and −qdk̂, centered at z = +d/2
and z = −d/2, respectively.
(b) Calculate the potential V (r, θ) in spherical coordinates for r d.
(c) Sketch the electric field lines in the xz plane.
5
Taken from Pollack and Stump, 3.38
====== [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
A semi-infinite rectangular metal pipe with sides (a, b) is placed along the x axis and 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 0221]
Rectangular pipe
A rectangular metal pipe of length L with sides (a, b) is placed along the x axis and centered at the origin. The
potential on the sides V (x, y = ±a/2, z = ±b/2) = 0. The potential at x = −L/2 is V1 , the potential at x = L/2 is
V2 .
Find potential inside the pipe.
====== [Exercise 0222]
Surface charge density
Consider a plane x − y charged with the surface charge density σ(x, y) = σ0 sin(ax + by). Find the potential in all
space.
====== [Exercise 0223]
Charged half-space
The half-space z ≤ 0 is charged with the charge density ρ = ρ0 cos(~k · ~r), where ~k is not parallel to ẑ. Find φ.
Taken from Gedalin 9.4
====== [Exercise 0225]
Infinite Wires
Find the electric field of an infinite parallel array of infinitely long charged lines aligned parallel to the y-axis and
charged with opposite charges:
ρ=
∞
X
(−1)n λδ(x − na)δ(z)
n=−∞
Taken from Gedalin
====== [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:
(1)
6
(a) The charge density is spread through the entire space.
(b) The charge density is trapped inside a layer |z| < a.
====== [Exercise 0240]
Cylinder in a constant field
Find the potential outside an infinitely long metal pipe of radius R (filled inside with metal) placed at a 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 the 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 an infinitely long cylinder of radius R.
Find the electric potential inside and outside the cylinder.
Taken from Griffiths problem 3.25
====== [Exercise 0260]
Ball with charge density
The charge density of a sphere of radius R is given by
ρ0 cos θ if r > 0
ρ(r) =
0
if r = 0.
(2)
Find the electric potential in all space.
====== [Exercise 0261]
Conducting sphere in uniform electric field
~ 0 . Find the potential.
A conducting sphere is place in a uniform electric field E
====== [Exercise 0262]
Spherical separation of variables
One half of a spherical surface of the radius R has the potential φ1 and the other half has the potential φ2 , while
φ1 6= φ2 . Find the potential outside the sphere if there are no charges there.
====== [Exercise 0270]
point charge inside a cavity
A conductor is maintained 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 the electric potential inside the cavity.
====== [Exercise 0280]
7
”Images method” (without the method)
The lower half-space (z < 0) is a perfect codnuctor. A charge q is held at ~r = (0, 0, d)
Find the electric potential in all space.
====== [Exercise 0290]
Thin ring, azimuthal dependency
A thin ring of radius R is charged with charge density λ(ϕ) = λ0 cos(nϕ).
Find the electric potential in all space.
====== [Exercise 0300]
Spherical Shell, azimuthal dependency
A spherical shell of radius R is maintained at potential φ(r = R) = φ0 sin θ sin ϕ.
Find the electric potential outside the shell.
====== [Exercise 0310]
Vector potential from current density
Find the vector potential for the current density
r s
J~ = j0
cos(nϕ)θ(R − r)ẑ
R
where n2 6= (s + 2)2 .
(3)
====== [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
(4)
====== [Exercise 0325]
Magnetic field from current density - spherical symmetry
A sphere of radius R has surface current density J~ = J0 (θ)φ̂. Calculate the magnetic field in all space.
====== [Exercise 0330]
Magnetic force
Inside a cylinder of radius R1 there is a smaller cylinder of radius R2 . The axis of the small cylinder is at distance l
from the axis of the large 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 solutions 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
(5)
(b) Show that
~ r) µ0
B(~
4π
Z ~ ~0
J(r ) × (~r − r~0 )
|~r − r~0 |3
(6)
8
Taken from Griffiths Problem 7.55
====== [Exercise 0350]
Maxwell Equations
~ = 0, and any net charge resides on the surface (just as it
In a perfect conductor, the conductivity is infinite, so E
does for an imperfect conductor in electrostatics).
~
• Show that the magnetic field is constant (∂ B/∂t
= 0), inside a perfect conductor.
• Show that the magnetic flux through a perfectly conducting loop is constant.
~ inside is in fact zero
A superconductor is a perfect conductor with the additional property that the (constant ) B
(This is known as the Meissner effect).
• Show that the current in a superconductor is confined to the surface.
• Superconductivity is lost above a certain critical temperature TC , which varies from one material to another.
Suppose you had a sphere (radius a) above its critical temperature, and you held it in a uniform magnetic field
B0 ẑ while cooling it below TC . Find the induced surface current density as a function of the polar angle θ.
Taken from Griffiths Problem 7.42
====== [Exercise 0355]
infinite cylinder, quasistationary aprox.
An infinitely long thin cylindrical shell (inner radius a, thickness d << a) made of a conductor with the conductivity
σ is placed in a magnetic field Re(B0 eiωt ) parallel to the cylinder axis. Find the magnetic field inside the cylinder.
Taken from Gedalin
====== [Exercise 0360]
Cylinder with a slot
~ for a circuit can acutally be calculated is the following: Imagine an
A rare case in which the electrostatic field E
infinitely long cylindrical sheet, of uniform resistivity and radius a. A slot (corresponding to the battery) is maintained
~ φ̂.
at ±V0 /2 at φ = ±π and a steady current flows over the surface K||
1. Find the potential on the cylinder V (a, φ).
2. Find the potential in all space.
3. Find the surface charge density on the cylinder.
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
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?
9
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
====== [Exercise 0390]
Pointing Vector for a solenoid
10
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
Suppose you have 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
=
r2 + d2 − 2rd cos θ
d(1 − cos θ)
0
(7)
(8)
(9)
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.
(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
11
====== [Exercise 0435]
Resonant cavity
Taken from Griffiths 9.38
====== [Exercise 0436]
Transmission coefficient in a wave guide
Taken from Pollack and Stump 14.8
====== [Exercise 0437]
Radiation pressure
Taken from Pollack and Stump 13.11
====== [Exercise 0440]
Spherical wave
Suppose
~ θ, φ, t) = A sinθ cos(kr − ωt) − 1 sin(kr − ωt) φ̂
E(r,
r
kr
(10)
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
expected direction? Does it fall off like r2 , 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
(n21 − n22 )(n23 − n22 )
n2 ωd
1
2
2
−1
(n1 + n3 ) +
sin
(11)
T =
4n1 n3
n22
c
(assume µ1 = µ2 = µ3 = µ0 ) Taken from Griffiths Problem 9.34
====== [Exercise 0460]
Evanescent Waves
12
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
+
+
(12)
d
a
b
for integers l, m, and n. Find the associated electric and magnetic fields.
Taken from Griffiths Problem 9.38
====== [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
(13)
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 0481]
Retarded potential - electron on a ring
Taken from Griffiths 10.13
====== [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
13
====== [Exercise 0500]
Moving Charge - “Horizon”
consider a particle in hyperbolic motion along the x axis:
p
w(t)
~
= b2 + (ct)2
(14)
(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
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
w(t)
~
= b2 + (ct)2
(15)
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 0521]
Linear antenna radiation
Taken from Gedalin 9.23
====== [Exercise 0522]
14
Oscillating charge
~
A charge q oscillates according to X(t)
= a sin(Ωt)ẑ Calculate the angular distribution of the radiated power.
====== [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
(16)
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
(17)
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
====== [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
15
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
(18)
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
====== [Exercise 0640]
motion under constant force
Show that
µ
K Kµ =
1−
v2
c2
cos2 (θ)
1−
v2
c2
F2
(19)
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 ?
16
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