Download PHYS4210 Electromagnetic Theory Quiz #1 31 Jan 2011

Name:
PHYS4210 Electromagnetic Theory
Quiz #1
31 Jan 2011
This is a closed book quiz! You have thirty minutes to complete it.
This quiz has three equally weighted parts. (a) Find the gradient of the field Φ(x) = xy 2 z 3 ;
(b) Express ∇ × (∇ × A) as the sum of two terms, using the gradient, divergence, and
Laplacian, but no curls; (c) Let S be the (closed) surface of a cubical box with side length
L, lying in the (+, +, +) quadrant and with one corner at Hthe origin. For the vector field
A(x) = xx̂ + (xy/L)ŷ + 2zẑ, calculate the surface integral S A · dS.
Name:
PHYS4210 Electromagnetic Theory
Quiz #2
7 Feb 2011
This is a closed book quiz! You have thirty minutes to complete it.
This quiz has multiple steps; be sure to work all ofH them. First, show that Gauss’ Law (in
differential form for electric field E(x)) implies that S E·dS = 4πQ for a closed surface S that
contains total charge Q. Use this result to find the electric field E(r) at a (perpendicular)
distance r from the center of a long, cylindrical charge distribution with radius R and
constant charge density ρ0 . (Divide your answer in to the regions r < R and r > R.) Next,
integrate the electric field to find the electric potential Φ(r), fixing Φ(0) = 0. Finally, prove
that Φ satisfies Poisson’s equation for r < R and Laplaces’ equation for r > R.
Name:
PHYS4210 Electromagnetic Theory
Quiz #3
14 Feb 2011
This is a closed book quiz! You have thirty minutes to complete it.
This question has multiple parts, not necessarily equally weighted. If you are stuck on (a),
indicate how you would use (a) to move on to (b) and (c) for partial credit. (a) Find the
electrostatic potential Φ(x) for all points inside a long, thin-walled cylindrical tube of radius
R with thin walls. The potential at the inside surface is Φ(R, φ) = 3 cos(2φ), where φ
measures the angle around the tube’s circumference. There is no charge inside the tube. Do
not assume the tube is a conductor. (b) Find the electric field vector along at the center of
the tube. (c) For a small patch on a large, thin planar surface carrying charge σ per unit
area, use Gauss’ Law to find a relationship between σ and the perpendicular electric field
E⊥ . Use this result to find the surface charge density on the tube as a function of φ.
Name:
PHYS4210 Electromagnetic Theory
Quiz #4
28 Feb 2011
This is a closed book quiz! You have thirty minutes to complete it.
Two large, flat conducting plates, each with area A, are separated by a distance 2d, where
d is much smaller than the sides. A slab with dielectric constant ε and thickness d is flush
against the bottom plate. The bottom plate is grounded, and the top is at potential V0 .
2d d
In terms of A, d, ε, and V0 find (a) the magnitude and direction of the electric field E
everywhere between the plates; (b) the magnitude and direction of the displacement field D
everywhere between the plates; (c) the (free) charge Q on the lower surface of the top plate;
(d) the polarization charge QP on the lower surface of the slab; and (e) the capacitance C
of this system. Remember to continue on, even if you are stuck on one part.
Name:
PHYS4210 Electromagnetic Theory
Quiz #5
7 Mar 2011
This is a closed book quiz! You have thirty minutes to complete it.
This is a “double quiz” with two independent parts. You can work either Part I or Part II.
Or, work both parts. A perfect score on both parts counts as one quiz worth of extra credit.
I. A large, flat conducting plate carries a uniform surface charge density σ. Assuming the
surface lies in the xy plane, find all components of the electrostatic stress tensor at the
surface, and determine the force per unit area on the surface.
II. A spherical shell of radius R carries a surface charge density σ(θ) = σ0 (3 cos2 θ − 1)/2.
Find the total charge Q, all three dipole moment components pi , and the quadrupole moment
component Q33 = Qzz . Explain your reasoning clearly, with equations or otherwise.
Name:
PHYS4210 Electromagnetic Theory
Quiz #6
21 Mar 2011
This is a closed book quiz! You have thirty minutes to complete it.
This quiz has three unrelated parts. Please work all of them for full credit.
(a) If Aµ is a spacetime vector and Bµ is a covector, prove that Aµ Bµ is a scalar.
(b) The four-momentum of a particle is pµ = muµ = mdxµ /dτ , where m is the particle mass
and dτ is the proper time interval. Prove that pµ pµ = m2 c2 .
(c) A four-vector Aµ = (0, Ax , Ay , Az ) in some reference frame. Find the component A00 in
a reference frame moving with velocity +V in the x-direction.
Name:
PHYS4210 Electromagnetic Theory
Quiz #7
28 Mar 2011
This is a closed book quiz! You have thirty minutes to complete it.
This quiz has two unrelated parts. You must work both of them for full credit.
I. A charge q moves with speed V = 3c/5 in the +x-direction, and is located at the origin at
a specific instant in time. At this instant, a stationary charge q is located (a) on the x-axis
at x = a or (b) on the z-axis at z = a, with a > 0. Find both the magnitude and direction
of the force on the stationary charge for cases (a) and (b) in terms of q and a.
II. In class we constructed the field strength tensor
in full contravariant form, namely F µν , in terms of E
and B. Starting from F µν , construct the full covariant
form, namely Fµν . In case you didn’t put it on your
crib sheet, F µν is given on the right.




F µν = 
0 −Ex −Ey −Ez
Ex
0
−Bz By 


Ey Bz
0
−Bx 
Ez −By Bx
0

Name:
PHYS4210 Electromagnetic Theory
Quiz #8
4 Apr 2011
This is a closed book quiz! You have thirty minutes to complete it.
A long, cylindrical resistor with radius a
and length b carries a steady current I.
a
There is a potential difference V across the
length of the resistor. The electric field in
I
the resistor is a constant. Answer the folb
lowing in terms of V , I, a, and b.
(a) What is the magnitude of the electric field?
(b) Draw the direction of the electric field on the diagram.
(c) What is the magnitude of the magnetic field on the cylindrical surface of the resistor?
(d) Draw the direction of the magnetic field on the cylindrical surface of the resistor.
(e) What is the magnitude of S = (c/4π)E × B on the cylindrical surface of the resistor?
(f ) Draw the direction of S = (c/4π)E × B on the cylindrical surface of the resistor.
(g) By integrating the energy flux vector over the surface of the resistor, find the rate at
which energy appears in the resistor volume, i.e. the power P dissipated by the resistor.
Name:
PHYS4210 Electromagnetic Theory
Quiz #9
11 Apr 2011
This is a closed book quiz! You have thirty minutes to complete it.
A magnetic dipole m1 rests at the origin. A second magnetic dipole m2 sits on the x-axis
at x = a. Both dipoles point in the +z direction. Find (a) the magnetic field B due to m1
at the position of m2 , (b) the torque on m2 from B, and (c) the force on m2 from B. You
can ignore the variation of B over the size of m2 for part (b).
Name:
PHYS4210 Electromagnetic Theory
Quiz #10
18 Apr 2011
This is a closed book quiz! You have thirty minutes to complete it.
A uniform magnetic field B = B0 x̂ fills all space in a vacuum.
A long, paramagnetic cylinder with radius R and permittivity µ is then inserted into the field, with its axis along the
z-axis. Find the resulting magnetic field B(r, θ) inside the
cylinder, in terms of B0 , R, and µ. Show that you get the
correct answer for µ = 1, that is, susceptibility χ = 0. It
is useful to know that a solution to Laplaces’ Equation in
cylindrical coordinates takes the general form
∞ h
X
i
an rn + bn r−n cos(nθ)
n=1
Also find the magnetic energy per unit length in the cylinder.
B0
y
x
R
Name:
PHYS4210 Electromagnetic Theory
Quiz #11
25 Apr 2011
This is a closed book quiz! You have thirty minutes to complete it.
An electromagnetic plane wave propagates in the z-direction in vacuum. The electric field is
E(t, x) = E0 ŷ cos(ωt − kz)
What is the wavelength λ? Find the magnetic field B(t, x). Show that it satisfies at
least one of the time-dependent Maxwell’s Equations. Find the Poynting vector S(t, x).
Consider a square loop centered at the origin, in the xy-plane, each side with length λ/2.
Find the induced EMF E(t) around the loop from B(t, x). Also find the average rate
dŪ /dt at which energy is carried through the loop, by the wave. Repeat the calculations
of E(t) and dŪ /dt for a loop of the same dimensions lying at the origin in the yz-plane.
Name:
PHYS4210 Electromagnetic Theory
Quiz #12
2 May 2011
This is a closed book quiz! You have thirty minutes to complete it.
A plane wave of visible light in vacuum is normally incident on the planar surface of a
thick slab of non-conducting material. The intensity (that is, the power per unit area) of
the reflected light is 1/9 that of the incident light. What is the index of refraction of the
material? You may assume that the magnetic permittivity µ = 1 for the material.
Name:
PHYS4210 Electromagnetic Theory
Quiz #13
9 May 2011
This is a closed book quiz! You have thirty minutes to complete it.
Write the best choice in the space next to the question.
1. A single charged particle undergoes simple harmonic motion with angular frequency ω. The average radiated power is proportional to
A. ω 0
B. ω 1
C. ω 2
D. ω 3
E. ω 4
2. An electric dipole is formed from two stationary charges ±q separated by a fixed
distance b. For large distances r b from the dipole, the electric field falls like
A. 1/r
B. 1/r2
C. 1/r3
D. 1/r4
E. 1/r5
3. A long, thin, cylindrical rod of length d and radius R, with d R, sits in an
external, uniform magnetic field B whose direction is aligned with the axis of the rod.
If the rod is made of a homogenous material with magnetic permittivity µ, what is the
magnitude of the magnetic field inside the rod?
A. B
B. µB
C. B/µ
D. (R/d)µB
E. (R/d)B/µ
4. Given a four-vector potential Aµ = (0, −Kct, 0, 0) where K is a constant and t is
time, what are the electric and magnetic fields?
A. E = B = 0
B. E = B = K x̂
C. E = 0 and B = K x̂
D. E = K x̂ and B = 0
E. E = K x̂ and B = K ŷ