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Preliminary Examination: Electricity and Magnetism Department of Physics and Astronomy

University of New Mexico

Fall 2004

Instructions: • The exam consists two parts: 5 short answers (6 points each) and your pick of 2 out 3 long answer problems (35 points each). • Where possible, show all work, partial credit will be given. • Personal notes on two sides of a 8X11 page are allowed. • Total time: 3 hours

Good luck!

Short Answers: S1. Graphed below are the equipotential contours associated with two point charges. (i) Sketch the electric field lines. Show arrows denoting the direction of the field.

(ii) Which charge distribution could create this potential?

+3q

–q +q

–3q -3q

–q

+3q

+q

�

ψ (x)

(a) (b) (c) (d)

+2.3

+4.5

+6.8

+1

0 –0.6

–1.5

S2. Which field lines could represent a static magnetic field.

S3. A charge q is placed a distance d from a grounded infinite perfectly conducting plane. With what force is it attracted to the plane?

S4. A resistor, capacitor, and inductor are connected in parallel across a battery. At t=0 the switch is closed

Describe the current in the three elements as a function of time for t>0.

S5 A plane wave solution to Maxwell’s equations in a homogeneous, linear dielectric is given by

E(z, t) = E0 cos( 6y + 6 ×10 1 0 t) ˆ x , where t is in seconds, y is in centimeters.

(a) What is the direction of propagation? (b) What is the index of refraction of the medium? (c) What would the wavelength be if this wave traveled in free space?

V R C L

Long Answers: Pick two out of three problems below L1. Two concentric metal spherical shells of radius a and b, respectively, are separated by a weakly conducting material of conductivity σ.

(a) If they are maintained at a potential difference V, what current flows between them? (b) What is the resistance between the shells? (c) Notice that if b>>a the outer radius (b) is irrelevant. How do you account for that? Exploit this observation to determine the current flowing between two metal spheres, each of radius a, immersed deep in the sea held quite far apart, if the potential between them in V. (This arrangement can be used to measure the conductivity of sea water.)

L2. Consider wave incident on an nonmagnetic neutral conductor. Treat the electrons as responding according to Ohm’s law with static conductivity σ0,

�

J = σ 0E .

(a) Using Maxwell’s equations, show that inside the metal the electric field satisfies the following wave equation,

�

∇2 − 1 c 2

∂ 2

∂t 2 ⎛ ⎝ ⎜

⎞ ⎠ ⎟ E = µ0σ 0

∂E ∂t

.

(We neglect here any frequency dependence associated with the conductivity).

(b) Show that plane waves oscillating at frequency ω propagate inside the material with complex wave number,

�

k = ω c 1+ i σ 0

ε0ω .

(c) What is the physical meaning of the real and imaginary parts of

�

k ?

(d) Consider a microwave at 10 GHz reflected from a silver mirror with

�

σ 0 = 6.14 ×10 7

(ohm-m)–1,

�

ε0 = 8.85 ×10 −12 Farad/m. Approximately how many meters will the

microwave penetrate into the mirror (sometimes known as the “skin depth”).

b

a

L3. An insulating circular ring (radius b) lies in the x-y plane, centered at the origin. It carries a linear charge density

�

λ = λ0 sinφ , where

�

λ0 is constant and φ is the usual azimuthal angle. The ring is now set spinning at a constant angular velocity ω about the z axis.

(a) Calculate the total power radiated into the far field (r>>b) as electric dipole radiation?

Hint: Recall the Larmor formula for the instantaneous radiated power,

�

P(t) = 1 4πε0

2 3

˙ ̇ d 2

c 3 , where d is the electric dipole moment.

(c) What is the polarization of this radiated field on the z-axis and on y-axis? (d) What power is radiated as magnetic dipole radiation?

x

z

y

ω

r

Preliminary Examination: Electricity and Magnetism Department of Physics and Astronomy

University of New Mexico Fall 2006

Instructions: • The exam consists of 10 problems, 10 points each; • Partial credit will be given if merited; • Personal notes on two sides of an 8 × 11 page are allowed; • Total time is 3 hours. Problem 1: A charge q is uniformly distributed on the surface of a sphere of radius R. What is the potential energy stored in this charge configuration? Problem 2: A charge q is placed a distance d above an infinite, grounded, conducting plane. Find the induced surface charge density as a function of coordinates on the plane. Problem 3: Hall effect: A uniform magnetic field B0 in the z-direction is applied to a semiconducting material carrying current in the y-direction. In steady state, a voltage develops across d and the charge velocity vy does not vary (electric and magnetic forces balance).

- What is the Hall voltage Vh in terms of vy, B0 and d? - Can this measurement determine the polarity of the charge carriers (n-type vs. p-type semiconductor)? Explain.

Problem 4: Consider a transmission line consists of two parallel conducting strips of width w, length l, separated by distance d

Problem 5: Betatron: An electron with speed v, undergoing cyclotron motion in a magnetic field B(r) at the cyclotron radius

�

r 0

= mv /qB(r 0 ) can be accelerated by ramping B-field in time.

- Since magnetic fields do no work, what is increasing the kinetic energy of the electron? - Show that if the field at r0 is half of the average across the orbit,

�

B(r 0 ,t) =

1

2

B(r,t) ! da" #r

o

2 ,

then the radius of the orbit is constant in time. Assume nonrelativistic speeds. Problem 6: Consider a parallel RLC circuit driven by an ac voltage source at frequency ω. In steady state, what is the current drawn from the source as a function of time (Hint: this is easiest if you use complex impedance). Problem 7: Starting with Maxwell’s equations, show that the electric and magnetic fields are derivable from scalar and vector potentials,

�

E = !A

!t " #$ ,

�

B = !"A .

How can we change the vector and scalar potentials without changing the electric and magnetic fields? Problem 8: A transverse electromagnetic wave travels inside a neutral plasma, inducing a current density

�

J = !en e v , where ne is the density of electrons and with instantaneous velocity v

driven by the electric field. Use Maxwell’s equations to show that these waves satisfy the equation

�

!2 " 1

c 2

# 2

#t 2 $ % &

' ( ) E =

* p 2

c 2 E , where

�

! p 2

= 1

4"# 0

$ % &

' ( ) 4"nee

2

m is the square of the “plasma frequency”.

Ignore any electron damping.

R L C

�

V 0 cos!t

�

I(t)

Problem 9: Consider a plane wave of amplitude E0, normally incident on a dielectric with permittivity ε. Use the boundary conditions on E and B at the interface to show the amplitude of the transmitted wave is

�

E r

= 1! " /"

0

1+ " /" 0

E 0 .

Problem 10: A +q is set in circular orbit above a charge –q as shown with angular velocity ω.

What is instantaneous the rate at which the charge loses energy by electromagnetic radiation?

r

ρ

-q

+q

ω

Preliminary Examination: Electricity and Magnetism

Department of Physics and Astronomy

University of New Mexico

Fall, 2007

Instructions:

• The exam consists of 10 problems, 10 points each. • Partial credit will be given if merited. • Personal notes on two sides of an 8× 11 page are allowed. • Total time is 3 hours.

1. An ideal electric dipole of moment ~p = pẑ is situated at the origin. What is the force, caused by the dipole, on each of two separate point charges, of amount q. The first is located at a distance a from the origin along the x̂-axis, i.e., so that the charge has the Cartesian coordinates (a, 0, 0), and the other is also at a distance a from the origin, but along the ẑ-axis, i.e. so that the charge has the Cartesian coordinates (0, 0, a)?

2. Please find the capacitance per unit length of two coaxial, hollow, metal, cylindrical tubes, of radius a and b > a.

3. A hollow sphere carries charge density ρ = c/r2 in the region a ≤ r ≤ b. Find the electric field in each of the three regions: within the hollow of the sphere, i.e., for r ≤ a; within the interior of the sphere, i.e., for a ≤ r ≤ b; and exterior to the sphere, i.e., for b ≤ r. Provide the result in terms of the total charge, q, of the shell. Provide a plot of the magnitude of the electric field as a function of the distance r from the center of the system.

4. A uniformly charged shell of surface