Columbia UniversityDepartment of Physics

QUALIFYING EXAMINATION

Monday, January 13, 20143:10PM to 5:10PMClassical Physics

Section 2. Electricity, Magnetism & Electrodynamics

Two hours are permitted for the completion of this section of the examination. Choose4 problems out of the 5 included in this section. (You will not earn extra credit by doing anadditional problem). Apportion your time carefully.

Use separate answer booklet(s) for each question. Clearly mark on the answer booklet(s)which question you are answering (e.g., Section 2 (Electricity etc.), Question 2, etc.).

Do NOT write your name on your answer booklets. Instead, clearly indicate your ExamLetter Code.

You may refer to the single handwritten note sheet on 812” × 11” paper (double-sided) you

have prepared on Classical Physics. The note sheet cannot leave the exam room once theexam has begun. This note sheet must be handed in at the end of today’s exam. Pleaseinclude your Exam Letter Code on your note sheet. No other extraneous papers or booksare permitted.

Simple calculators are permitted. However, the use of calculators for storing and/or recov-ering formulae or constants is NOT permitted.

Questions should be directed to the proctor.

Good Luck!

Section 2 Page 1 of 6

1. One end of a conducting rod rotates with angular velocity ω in a circle of radius r makingcontact with a horizontal, conducting ring of the same radius. The other end of the rodis fixed. Stationary conducting wires connect the fixed end of the rod (A) and a fixedpoint on the ring (C) to either end of a resistance R. A uniform vertical magnetic field~B passes through the ring.

(a) Find the current I flowing through the resistor.

(b) What is sign of the current if positive I corresponds to flow in the direction of thearrow in the figure?

(c) What torque must be applied to the rod to maintain its rotation at the constantangular rate ω?

Section 2 Page 2 of 6

2. Consider a very thin rubber sheet of radius a containing a uniform surface charge densityσ.

(a) Calculate an exact expression for the electric potential V (r, θ = 0) at an arbitraryposition r above the center of the disk.

(b) Calculate the electric potential V (r, θ, φ) at all points in space for r >∼ a

X Y

Z

V (r, θ = 0)part a

V (r, θ, φ)part b

aσ

Section 2 Page 3 of 6

3. A rectangular copper strip 1.5cm wide and 0.10cm thick carries a current of 5.0 A. A 1.2T magnetic field is applied perpendicular to the strip. Find the resulting Hall voltage.The molar mass of copper is 63.5 g, and the density of copper is 8.95 g/cm3. Assumeeach copper atom contributes one free electron to the body of the material.

Section 2 Page 4 of 6

4. Consider an infinitesimally thin charged disk of radius R and uniform surface chargedensity σ that is in the xy plane centered on the origin.

(a) Calculate the potential along the z axis from Coulomb’s law.

(b) Calculate the monopole moment, dipole moment and quadrapole moment of thedisk. These are defined as

Q =

∫d3r ρ(r)

x~P =

∫d3r ρ(r)~r

Qij =

∫d3r ρ(r)

[3rirj − r2δij

]

Section 2 Page 5 of 6

5. Consider a conductor in which the free current density and the electric field are relatedby the linear relationship

~jfree = ~g × ~E = g(z × ~E)

where ~g is a given real constant vector directed along the z-axis (~g = gz with g > 0).

The conductor’s polarization and magnetization vectors are both zero (~P = 0, ~M = 0).

Consider a monochromatic plane wave traveling through such a conductor in the +zdirection (propagation parallel to the vector ~g:)

[~E(z, t)~B(z, t)

]= Real Part

{[~E0

~B0

]ei(kz−ωt)

}where ω is the given (real) angular frequency of the wave. ~E0, ~B0 and k are constants tobe determined.

(a) Find the possible values for the index of refraction (N = kc/ω) for this wave.Express your answers in terms of g and ω.

(b) Show that there is a cut-off frequency ωc such that for ω > ωc all the waves whichhave definite N propagate without attenuation. Also show, for ω < ωc, that onlyone of these waves propagates and that the other wave does not propagate but justundergoes attenuation. Find the value for ωc, and briefly explain your reasoning.

(c) For each possible value of N , find the corresponding electric-field polarization ~E0.

Section 2 Page 6 of 6

N. Christ November 30, 2013

Quals E&M Problem

1. One end of a conducting rod rotates with angularvelocity ω in a circle of radius r making contactwith a horizontal, conducting ring of the same ra-dius. The other end of the rod is fixed. Stationaryconducting wires connect the fixed end of the rod(A) and a fixed point on the ring (C) to eitherend of a resistance R. A uniform vertical mag-netic field ~B passes through the ring.

(a) Find the current I flowing through the resistor. [10 points]

(b) What is sign of the current if positive I corresponds to flow in thedirection of the arrow in the figure? [2 points]

(c) What torque must be applied to the rod to maintain it rotationat the constant angular rate ω? [8 points]

1

Suggested Solution

1. (a) Apply the integrated version of Faraday’s law to the closed loopL made up of the segment of the circle connecting the movingend of the rod and the point C, moving counter clockwise (whenviewed from above) from that end to C, the segment from C to Acontaining the resistor R and then the rod itself:

∮

Ld~r · ~E = −1

c

d

dtΦB

where ΦB(t) is the flux of magnetic field passing through thatloop. Evaluating the left and right hand sides of this equation:

IR =1

c

ωr2

2B or I =

r2Bω

2cR

(b) As found in (a), I is positive.

(c) Consider a segment of the rod of length dx a distance x from thepivot. Because of the flowing current this segment will contain acharge dQ moving at velocity v where I = dQ/(dx/v) = vdQ/dx.The torque τ exerted by the magnetic field on the rod carryingthe current I is then given by:

τ =∫ r

0xv

cB

dQ

dxdx =

IB

c

∫ r

0xdx =

IBr2

2c=

r4B2ω

4c2R

pointed downward. Thus, −~τ must be applied to the rod to main-tain its rotational motion.

2

annie

Text Box

Sec 2 - 2

Quals Problem #2

January 2014

Problem

A rectangular copper strip 1.5 cm wide and 0.10 cm thick carries a current of 5.0 A. A 1.2 T magnetic fieldis applied perpendicular to the strip. Find the resulting Hall voltage. The molar mass of copper is 63.5 g,and the density of copper is 8.95 g/cm3. Assume each copper atom contributes on free electron to the bodyof the material.

1

Suggested Solution to Quals Problem #2

January 2014

Solution

One mole of copper has the following volume:

V =m

ρ=

63.5 g

8.95 g/cm3 = 7.09 cm3 (1)

Therefore, the number density of free electrons in copper is:

n =NA

V=

6.02 × 1023 electrons

7.09 cm3(2)

which can be straightforwardly converted to 8.48 × 1028 electrons/m3. The Hall voltage is:

VH =IB

nqt=

(5.0 A)(1.2 T)

(8.48 × 1028 m−3)(1.60 × 10−19 C)(0.10 × 10−2 m)= 0.44 µV (3)

2

Electromagnetism Quals Problem

Fall 2013

Robert Mawhinney

Consider an infinitesimally thin charged disk of radius R and uniformsurface charge density σ that is in the xy plane centered on the origin.

1. Calculate the potential along the z axis from Coulomb’s law.

2. Calculate the monopole moment, dipole moment and quadrapole mo-ment of the disk. These are defined as

Q =

∫d3r ρ(r) (1)

~P =

∫d3r ρ(r)~r (2)

Qij =

∫d3r ρ(r)

[3rirj − r2δij

](3)

1

Solution

1.

φ(x) =

∫d2x

σ√z2 + x2 + y2

(4)

= σ

∫ R

0

2πρ dρ1√

z2 + ρ2(5)

= 2πσ√z2 + ρ2|R0 (6)

= 2πσ(√

z2 +R2 − |z|)

(7)

2. The monopole moment is just the total charge, which is σπR2. Thedipole moment vanishes by symmetry. The quadrapole moment is zero,unless i = j. We need ∫

d2x σ x2 =

∫d2x σ y2 (8)

by symmetry. Thus we have∫d2x σ x2 =

1

2

∫d2x σ (x2 + y2) (9)

= πσ

∫ R

0

dρ ρρ2 (10)

=πσ

4ρ4|R0 (11)

=πσ

4R4 (12)

(13)

This gives

Qij =πσR4

4

3− 2 0 00 3− 2 00 0 0− 2

=πσR4

4

1 0 00 1 00 0 −2

(14)

2

annie

Text Box

Sec 2 - 5