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Preliminary Examination: Electricity and MagnetismDepartment of Physics and Astronomy

University of New Mexico

Fall 2004

Instructions: The exam consists two parts: 5 short answers (6 points each) and your pickof 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

(x)

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

+2.3

+4.5

+6.8

+1

00.6

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 conductingplane. With what force is it attracted to the plane?

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

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

S5 A plane wave solution to Maxwells equations in a homogeneous, linear dielectric isgiven by

E(z, t) = E0 cos( 6y + 6 101 0 t) 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?

VR C L

Long Answers: Pick two out of three problems belowL1. Two concentric metal spherical shells of radius a and b, respectively, are separatedby a weakly conducting material of conductivity .

(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 betweenthem 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 asresponding according to Ohms law with static conductivity 0,

J = 0E .

(a) Using Maxwells equations, show that inside the metal the electric field satisfies thefollowing wave equation,

2 1c 2

2

t 2

E = 0 0

Et

.

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

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

k = c1+ i 0

0.

(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

0 = 6.14 107

(ohm-m)1,

0 = 8.85 1012 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. Itcarries a linear charge density

= 0 sin , where

0 is constant and is the usualazimuthal angle. The ring is now set spinning at a constant angular velocity about the zaxis.

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

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

P(t) = 140

23

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

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

r0

= mv /qB(r0) 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(r0,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 . 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 Maxwells 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 = !enev , where ne is the density of electrons and with instantaneous velocity v

driven by the electric field. Use Maxwells equations to show that these waves satisfy the equation

!2 "1

c2

# 2

#t 2$ % &

' ( ) E =

* p2

c2E , where

! p2

=1

4"#0

$ % &

' ( ) 4"nee

2

m is the square of the plasma frequency.

Ignore any electron damping.

R L C

V0cos!t

I(t)

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

Er

=1! " /"

0

1+ " /"0

E0.

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

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

r

-q

+q

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 = pz is situated at the origin. What is the force, caused bythe dipole, on each of two separate point charges, of amount q. The first is located at a distancea from the origin along the x-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 z-axis, i.e. so that the chargehas the Cartesian coordinates (0, 0, a)?

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

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

4. A uniformly charged shell of surface charge density and radius a is rotating at a constant

angular velocity ~, and we take the z-axis along ~. At an arbitrary location, ~r, it has a magnetic

vector potential given by

~A(r, , ) =

130R r sin , r R , i.e., inside the shell,130R

4sin r2

, r R , i.e, outside the shell.

Show that the magnetic field inside the rotating shell is uniform, and along the z-direction. Also

determine the magnetic field outside the shell. Can you describe that field in simple language?

Is the field continuous at the boundary of the shell? Explain physically your answer.Note that for a vector of the form ~A = A , one has the following relation for its curl:

(A ) = rr sin

(A sin )

r

(Ar)r

.

5. A very long solenoid carries a current I. Coaxial with the solenoid is a large, circular ring of

wire, with resistance R. When the current in the solenoid is gradually decreased, a current is

indu