preliminary examination: electricity and magnetism...

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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 (a) (b) (c) (d) +2.3 +4.5 +6.8 +1 0 –0.6 –1.5

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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

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 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 Maxwell’s equations in a homogeneous, linear dielectric isgiven by

E(z, t) = E0 cos( 6y + 6 ×101 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?

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 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 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 Ohm’s law with static conductivity σ0,

J = σ 0E .

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

∇2 −1c 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 withcomplex wave number,

k =ωc1+ 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 ×107

(ohm-m)–1,

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

λ = λ0 sinφ , where

λ0 is constant and φ is the usualazimuthal angle. The ring is now set spinning at a constant angular velocity ω 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) =1

4πε0

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

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<<w<<l. Ignoring fringing fields, what is the inductance per unit length?

d

q

d w

l

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 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 = !enev , 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

c2

# 2

#t 2$

% &

'

( ) E =

* p

2

c2E , where

! p

2=

1

4"#0

$

% &

'

( )

4"nee2

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 ε. 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 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 = 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 ρ = 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 σ and radius a is rotating at a constant

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

vector potential given by

~A(r, θ, ϕ) =

13µ0Rσω r sin θ ϕ , r ≤ R , i.e., inside the shell,

13µ0R

4σωsin θ

r2ϕ , 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 ϕ, one has the following relation for its curl:

∇× (A ϕ) =r

r sin θ

∂(A sin θ)∂θ

− θ

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

induced in the ring. Take the solenoid to have n turns per unit length, and radius a, while the

ring has radius b >> a. What is the current in the ring, as a function of dI/dt?

2

6. A previously-charged capacitor, of amount C and charge separation Q, is in a simple opencircuit along with a resistor, R, and an inductor, L. At time t = 0, a switch is closed so thatthis circuit now constitutes a single, closed, series circuit. What is the time dependence ofthe current through the resistor?

3

7. Consider a monochromatic wave moving through vacuum, of frequency ω, and with an electricfield that is the sum of two separate parts, which are presented here in their complex forms:

~E = ~E1 + ~E2 ,

~E1 = E0z ei(kx−ωt) , ~E2 = −E0z e−i(kx+ωt) ,

where E0 is real.Determine the associated, real-valued magnetic field, and the time-averaged Poynting vector forthe entire wave system. Please explain the meaning of your result for the Poynting vector.

8. Consider a circularly-polarized electromagnetic plane wave, propagating in vacuum with fre-quency ω. Write down the complex form for the electric field, and then, before you perform anyaverages over cycles, determine the (real-valued) intensity for the wave, making comments aboutthe time dependence of the result.

9. At a certain time, which we take to be t = 0, we turn on the current, everywhere at once, in aparticular infinitely-long wire, so that the current in this wire may be expressed in the followingway:

I(t) =

0 , t < 0 ,

I0 , t ≥ 0 .

(We take the wire to lie along the z-axis.)At any later time, t > 0, and at any particular measurement location, say a distance s directlyaway from the wire, only some finite portion of the wire can have communicated to this observationpoint the information that there is now a current running in that portion of the wire. For sucha given positive time, t, and distance s, what total length of wire can have communicated thisinformation?

10. An electromagnetic plane wave of frequency ω is traveling in the x-direction through the vacuum.It has amplitude E0, is polarized in the y-direction, and has (time-averaged) intensity I0. Theobserver, S, who made these statements is at rest. However, she sees another observer, S ′, comingpast her, moving in the same direction as the plane wave, at half the speed of light.What are the frequency and intensity of the wave as seen by this other observer, S ′?

4

EM Prelim August 22, 2008 p.1

Preliminary Examination: Electricity and MagnetismDepartment of Physics and Astronomy

University of New MexicoWinter 2008

Instructions:

• the exam consists of 10 problems, 10 points each;

• partial credit will be given if merited;

• personal notes on two sides of 8× 11 page are allowed;

• total time is 3 hours.

1. Two charges +q,−q are separated by a distance d. Define coordinates as in thefigure. Find the potential V (x, y) at any point in the x, y plane.

2. A point charge q is held at a distance h above an infinite conducting sheet.Determine the surface charge density on the sheet.

3. Consider two capacitors of identical construction except one is filled with dielec-tric having K = 2 (capacitor a) and the second is filled with air K ≈ 1 (capacitorb). Initially, switch one is closed and two is open so that capacitor a has aninitial charge q0. Subsequently, switch one is open and switch two is closed, sothat the capacitors are connected in parallel What are the charges on qa, qb onthe capacitors when connected in parallel?

EM Prelim August 22, 2008 p.2

4. An electron (mass m, charge −e) moves between two parallel plates, where theplates have a potential difference V and separation d. Between the plates there isalso magnetic field B (into the page as shown in the figure). The electron startsat rest and follows the trajectory indicated in the sketch– after almost reachingthe distance d in y, the particle will continue to move parallel to the upper platewith constant velocity in the x direction. Find the magnitude of the magneticfield B.

5. Consider the circuit in the figure with values V0 = 12V, R1 = 200kΩ,R2 = 300kΩ,and C = 2µF . The switch is open after having been closed for a very long time.a) What is the voltage on the the capacitor just before the switch is opened?b) After how much time (in seconds) does the voltmeter read 3V?

6. Consider a square loop of wire (with side length d) oriented parallel to a longstraight wire carrying current I. The loop is pulled with constant speed v in adirection perpendicular to the wire.a) What is the induced EMF in the loop?b)What is the direction of the induced current? Indicate on the figure with anarrow.

EM Prelim August 22, 2008 p.3

7. A plane EM wave of angular frequency ω propagates through a material withindex of refraction n1. The wave is normally incident upon the surface interfacewith another material with index of refraction n2. What fraction of the incidentwave energy is reflected from the surface?

8. A straight metal wire of conductivity σ and radius a carries a steady current I.a. Determine the Poynting vector as a function of the distance from the centerof the wire.b. Integrate the normal component of the Poynting vector over the surface ofthe wire and compare to the power loss per unit length due to the resistance ofthe wire.

9. Write Maxwell’s equations for a plane EM wave propagating in a neutral, con-ducting medium having permeability µ, permittivity ε and conductivity σ. Showthat the wave equation for the electric field is given by

~∇2 ~E = µε∂2

∂t2~E + µσ

∂t~E.

10. a) Find the dispersion relation (equation relating the wave number k to theangular frequency ω) for a plane wave of angular frequency ω propagating in aneutral, conducting medium (see previous problem). Note that in this case k isa complex number.b) For σ >> ω, find the skin depth in the medium.

Department of Physics and Astronomy, University of New Mexico

E&M Preliminary Examination

Fall 2012

Instructions:

• The exam consists of 10 problems (10 pts each).

• Partial credit will be given if merited.

• Personal notes on the two sides of an 8.5”x 11” sheet are allowed.

• Total time: 3 hours.

Possibly Useful Formulas

• Relation of spherical polar coordinates, (r, θ, φ), to Cartesian coordinates:

x = r sin θ cosφ, y = r sin θ sinφ, z = r cos θ.

Unit vectors:

r = sin θ cosφ x + sin θ sinφ y + cos θ z;

φ = − sinφ x + cosφ y; θ = φ× r.

• Laplacian in spherical polar coordinates:

∇2 =1r2

∂r

(r2 ∂

∂r

)+

1r2 sin θ

∂θ

(sin θ

∂θ

)+ +

1r2 sin2 θ

∂2

∂φ2.

• Biot-Savart Law for the magnetic field at position ~r due to a steady currentelement I ~d`′ located at position ~r ′:

~B(~r) =µ0

I ~d`′ × (~r − ~r ′)|~r − ~r ′|3 .

1

• Time-averaged power radiated by an oscillating electric dipole:

P =µ0|p|2ω4

12πc.

• Time-averaged power radiated by an oscillating magnetic dipole:

P =µ0|m|2ω4

12πc3.

• Instantaneous power radiated by a non-relativistically moving charge withacceleration a (Larmor formula):

P =q2a2

6πε0c3.

• Fresnel formulas for the amplitude reflection coefficient of a plane wave in-cident at a planar interface between two dielectrics:

r⊥ =n cos θ − n′ cos θ′

n cos θ + n′ cos θ′; r‖ =

n′ cos θ − n cos θ′

n′ cos θ + n cos θ′,

where ⊥, ‖ refer, respectively, to polarizations perpendicular and parallel tothe plane of incidence. The angles of incidence and refraction are θ andθ′, and n, n′ are the refractive indices of the medium of incidence and themedium of transmission, respectively.

2

1. In the Cartesian coordinate system (x, y, z), the electrostatic potentialhas the form V = a|z|, where a is a constant. Does the potentialobey the Laplace equation? Derive the specific charge distributionthat produces such a potential.

2. A uniformly charged ring of radius R, charge Q, and centered at theorigin in the xy plane rotates uniformly at an angular velocity ω aboutits axis. Determine the electric and magnetic fields at the center of thering.

3. A point charge q of mass m is released from rest a distance d from aninfinitely extended, uniformly charged plane of surface charge densityσ. Take q and σ to have the same sign. Either using the work-energytheorem or otherwise, write down an expression for the acceleration ofthe charge as a function of its speed without making any non-relativisticapproximations. By integrating this expression, obtain the speed of thecharge as a function of time. After how long will the charge achieve aspeed equal to 0.8c? Neglect any radiation from the accelerating chargefor this problem. Hint: You may find useful the indefinite-integralidentity, ∫ dx

(1− x2)3/2=

x

(1− x2)1/2.

4. Consider a charge q of mass m orbiting on a circle under the action ofa uniform static magnetic field ~B normal to the plane of the circularorbit. Using the radiative power loss formula for circular orbits,

P =q2a2γ4

6πε0c3,

where a is the acceleration and γ the relativistic Lorentz factor of theorbiting charge, show that the charge loses energy at a rate proportionalto γ2 as its speed approaches c.

5. A solid sphere of radius a is uniformly polarized with a permanentpolarization (density) ~P = zP along the z axis. Take the sphere to becentered at the origin. What are the bound volume and surface chargedensities in the sphere as a function of the position coordinates (r, θ, φ)inside and on the sphere? Show that the potentials

V <(r, θ) =P

3ε0

r cos θ, V >(r, θ) =Pa3

3ε0r2cos θ

3

correctly solve the electrostatic problem inside and outside the sphere,respectively, i.e., they both solve the Laplace equation and satisfy thetwo required boundary conditions at the spherical surface. Using V <,calculate the value of the electric field everywhere inside the sphere.

6. A uniform, infinitely extended current plane of surface current density~K = Kx is located in the xy plane. A small circular loop of radius acarrying current I and located above the plane is free to rotate abouta diameter that is held fixed and parallel to the x axis.

y

x

K

z

What is the equilibrium orientation of the loop relative to the cur-rent plane? At what frequency will the loop perform small oscillationsabout the equilibrium orientation if it is rotated slightly away fromthat orientation? Express your answer in terms of µ0, I, a, and Im, themoment of inertia of the loop about a diameter.

7. A plane electromagnetic (EM) wave is incident on a large planar metal-lic foil of area A at angle θ from the normal. The foil is slightly black-ened so only a fraction R of the EM energy is reflected and the rest isabsorbed.

4

θ

A

What are the magnitude and direction of the radiation force generatedby the wave in terms of its (time-averaged) intensity I, R, A, and θ?

8. A long skinny bar magnet of magnetization ~M parallel to its axis ap-proaches a highly permeable material with a plane surface. Take themagnetization of the magnet to be perpendicular to the material sur-face. With what force will the magnet attach to the surface of thematerial, if the cross-sectional area of the magnet is A?

Make any approximations that are reasonable to arrive at your answers.You may find useful the facts that the magnetic field inside a solenoidof n turns per unit length and current I is µ0nI parallel to the axis ofthe solenoid and that the force on a uniformly magnetized bar may bewritten, in perfect analogy with electrostatics, as the effective magnetic

5

charge of amount AM times the external magnetic field ~Bext to whichthe bar is exposed.

9. An unpolarized monochromatic electromagnetic plane wave of angularfrequency ω is incident from vacuum on the plane surface of an idealplasma. The refractive index of the plasma may be exressed as

n(ω) =

√1− ω2

P

ω2,

where ωP is the plasma frequency which we assume to be a constantand smaller than ω.

(a) What fraction of the power of the plane wave would be reflectedback at normal incidence?

(b) For what range of values of the angle of incidence would the planewave be fully reflected?

(c) For what angle of incidence would the reflected wave be perfectlyplane polarized?

10. Consider a monochromatic TEM mode of a planar metallic waveguideof plate separation w that is filled with a dieletric material of refractiveindex n. If the rms value of the electric field of the mode is E0 and itsangular frequency is ω, then write down complete expressions for theelectric and magnetic fields, including their magnitudes and directions,inside the guide. What is the speed of propagation of the TEM mode?Calculate the time-averaged Poynting vector and energy density of theguided mode.

6

1

Department of Physics and Astronomy, University of New Mexico

Electricity and Magnetism Preliminary Examination

Fall 2014

Instructions:

• You should attempt all 10 problems (10 points each).

• Partial credit will be given if merited.

• NO cheat sheets are allowed.

• Total time: 3 hours.

2

Possibly Useful Formulas

• Divergence of a vector ~A = Aρρ+Aφφ+Az z in cylindrical coordinates:

~∇ · ~A =1

ρ

∂ρ(ρAρ) +

1

ρ

∂Aφ∂φ

+∂Az∂z

.

• Laplacian in cylindrical coordinates:

∇2 =1

ρ

∂ρ

(ρ∂

∂ρ

)+

1

ρ2∂2

∂φ2+

∂2

∂z2.

• Vector identity:

~∇× (~∇× ~A) = ~∇(~∇. ~A)−∇2 ~A .

• Maxwell’s equations:

~∇. ~E =ρ

ε0,

~∇. ~B = 0 ,

~∇× ~E = −∂~B

∂t,

~∇× ~B = µ0

(~J + ε

∂ ~E

∂t

).

• Biot-Savart law for the magnetic field at position ~r due to a steady current element Id~l′ at position ~r′:

~B(r′) =µ0

Id~l′ × (~r − ~r′)|~r − ~r′|3

.

• Time-averaged power radiated by an oscillating electric dipole:

P =µ0|p|2ω4

12πc.

1

2

3

4

5

Preliminary Examination: Electricity and Magnetism Department of Physics and Astronomy

University of New Mexico Spring 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. Consider a ring of radius R with a charge q uniformly distributed on the circumference. Show that the electric field a distance z from the origin on the axis of the ring is,

E(z) =q

4!"0

z

z2

+ R2( )

3 / 2ˆ z .

Show that the limit z>>R (keeping first nonvashing term) is what you expect. Problem 2. Now let the ring rotate at angular speed ω. What is its magnetic moment? Problem 3. A dielectric medium of permittivity ε fills half of a parallel plate capacitor. Each plate has area A. The distance between the plates is d. What is its capacitance? (Ignore fringing fields) Problem 4. A model of an atom consists of a point charge e is imbedded inside a uniform cloud of charge -e distributed throughout a spherical volume of radius a. An external electric field is applied which polarizes the atom, displacing the cloud (without distorting it) a distance b from equilibrium. Calculate b.

z

R

ω

ε

•e -e

•e

-e b E

d

Problem 5. A square ring of iron, with very large magnetic permeability µ, is wrapped with a coil of wire with n1 turns per unit length on one side and another coil with n2 turns per unit length on the opposite side. What is the mutual inductance between the coils? Problem 6. Consider a circuit with resistor and capacitor, driven by an ac-voltage source V0cos(ωt). In steady state, show the voltage across the resistor as a function of time is:

Comment on the limits

! << 1/RC, ! >> 1/RC . Problem 7. The electric field associated with a plane wave traveling in a nonmagnetic dielectric medium is given by,

E(r, t) = (10Volts/cm) cos(2cm!1

z ! 30ns!1

t) ˆ x + cos(2cm!1

z ! 30ns!1

t) ˆ y ( ) (i) To what part of the electromagnetic spectrum does this wave correspond? (ii) In what direction is the wave propagating? (iii) What is the polarization of the field (linear, circular, elliptical)? (iv) What is the dielectric constant of the medium? (v) What is the intensity of the radiation? Problem 8. A electromagnetic wave travels through a neutral plasma generating a current density J. Use Maxwell’s equations to show that the electric field satisfies the wave equation,

!2 "1

c2

# 2

#t 2$

% &

'

( ) E = µ

0

#J#t

.

Problem 9. According to classical physics, an electron orbiting the nucleus will decay due to electromagnetic radiation. For a hydrogen atom, modeled as an electron in a circular orbit initially with radius a0 (Bohr radius), what is the instantaneous rate of decay of energy at time t=0? Problem 10. An infinite line charge, with charge λ per unit length in the “lab frame” moves at a speed β=v/c=.8, where c is the speed of light. Using relativistic length contraction, find the electric and magnetic fields (magnitude and direction) in the lab frame and rest frame of the rod. Show that these satisfy the Lorentz transformation of the electromagnetic field.

! E ||

= E||, ! E " = # E" +

r

$ % cB( )! B ||

= B||, c ! B " = # cB" &

r

$ % E( )

where || and ⊥ denote the vector components parallel and perpendicular to the direction of relative motion

between reference frames,

r

! =v

c

, and

! = 1/ 1" # 2 .

VR(t) = Re V0 e

! i"t R

R + i /"C#

$

%

&

= V0"RC( )

2

1+ "RC( )2 cos"t !

"RC1+ "RC( )

2 sin"t#

$ '

%

& (

R

C

n1 n2

µ

Preliminary Examination: Electricity and MagnetismDepartment of Physics and Astronomy

University of New MexicoSpring, 2007

Instructions: ??????

Problem 1: A thin ring, of radius a, has a non-uniform, linear charge density on it, of amount λ =

λ0 sin φ. Please find the net charge of the ring, and its dipole moment. Along the direction normal

to the ring, which we call z, at very large distances the electrostatic potential is approximately

proportional to 1/zn. What is the value of n?

Problem 2: Find the capacitance of a pair of concentric, spherical metal shells, which have radii

a and b, with b > a.

Problem 3: A particular parallel plate capacitor has an area A and a distance h separating the

plates. It has been charged so that one plate has a charge Q, and the other the negative of that,

but has recently been disconnected from the charging battery. Half of the area is filled, over the

entire separation distance, with a dielectric material with dielectric constant ε. You now attempt

to pull this material out from between the two plates. What is the minimum force you will have

to exert in order to do this?

Problem 4: An originally-uncharged, metal sphere of radius a is placed in an otherwise uniform

electric field, ~E = E0z, which induces a charge distribution on the sphere. As a result of this the

total electrostatic potential exterior to the sphere may be written as

V (r, θ) = −E0

(r − a3

r2

)cos θ .

What is the induced charge distribution? What is the potential’s dependence on r interior to the

sphere?

Problem 5: A rectangular loop of wire hangs vertically, and supports a mass m that hangs

downward from it, under the influence of gravity. The upper end of the loop finds itself in a

region where there is a uniform magnetic field ~B, which points into the page. For what current,

I, in the loop, would the mass be suspended in mid-air? What direction must that current have?

Problem 6: What current density, ~J , would produce the magnetic vector potential, ~A = k φ, in

cylindrical coordinates, where k is a constant?

Problem 7: A square loop of wire, of side length a, lies on a table, a distance s away from a very

long, straight wire which carries a current, I = I0 sin ωt. The square loop has a total resistance

R. What current flows in it?

Problem 8: Using Maxwell’s Equations with sources, show that the total charge inside any fixed,

finite volume is conserved, i.e., constant in time.

Problem 9: The magnetic field associated with a plane wave travelling in a nonmagnetic, di-

electric medium is given by

~B(~r, t) = B0cos[ω(2x/c− t)]z + sin[ω(2x/c− t)]y .

a. What is the direction of propagation of the wave?

b. What is the direction of the polarization of the wave, and what is its nature?

c. What is the dielectric constant of the material through which it is travelling?

d. What is the intensity of the radiation?

Problem 10: A particular observer, O, measures fields in a small region in space, where he finds

approximately uniform electric and magnetic fields as follows, where A is a constant:

~E = Ax , ~B = 3Ay .

a. Another observer, O ′, passes by at a velocity ~v = αz. What must the value of α be in order

that this observer measures no electric field?

b. Could there be yet another observer, O ′′, moving at a different velocity such that she would

see no magnetic field? If so, what would her velocity be?

2

Preliminary Examination: Electricity and Magnetism Department of Physics and Astronomy

University of New Mexico Spring 2008

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. Consider two point charges +q and -2q located on the y-axis as shown below. Sketch the electric field lines and equipotential lines in the x-y plane. Problem 2. The positive terminal of a battery (ground taken at infinity) is attached to a perfectly conducting sphere of radius R, bringing it to potential V. How much work does the battery do in bringing the sphere to the same potential? Problem 3. Two spherical cavities, of radii a and b, are hollowed out from the interior of a solid (neutral) conducting sphere of radius R. At the center of each cavity a point charge is placed: qa and qb.

(Next page)

y

x

-2q

+q

+V

R

qa

qb

b

a R

(a) Find the surface charge densities at radii a, b, and R: σa, σb, and σR. (b) What is the electric field outside the conductor? (c) What is the force on qa and qb? (d) Which of these answers will change if a third charge, qc, is brought near the conductor? Problem 4. A parallel plate capacitor is filled with two equal thickness layers of linear dielectrics, of permittivity

!1

= 2!0 and

!2

= 1.5!0, respectively (ε0 is the permittivity of

free space). A surface charge density ±σ is placed on the two plates (a) What is the electric field inside each of the dielectrics (ignore fringing fields). (b) What is the surface charge density at the interface between the two dielectrics? Problem 5. A bar magnetic with magnetic dipole m is place on the axis, a distance d from an infinitely long wire carrying current I. With what force is the magnetic dipole attracted to the wire for the two orientations shown? (i) m perpendicular to I and r. (ii) m parallel to I, perpendicular to r. Problem 6. A charge/area σ is distributed on the surface of a very long cylinder of radius R. The cylinder is spun about its axis so that it’s instantaneous angular velocity is ω(t). Find the electric field as a function of position and time.

ε1 ε2

I

N N

I r r

ω(t)

σ

R

Problem 7. Consider an RLC circuit with

R /L << 1/ LC . A battery charges the capacitor to voltage V0. At time t=0, the switch is open with the battery and closed with the resistor and inductor in series. Sketch the energy stored in the capacitor and inductor as a function of time, denoting any relevant time dependencies on your graph. Problem 8. A monochromatic electromagnetic wave with complex amplitude,

E(r, t) = E(r)e!i"t , travels through a neutral plasma generating a current density

J(r, t) = !i"Ne

2

mE(r)e

!i"t , with e, m the electron’s charge and mass, and N the electron

density. Use Maxwell’s equations to show that the electric field satisfies the following wave equation,

!2+" 2

c2

#

$ %

&

' ( E(r) =

" p

2

c2E(r),

where

! p = Ne2/m"

0 is the plasma frequency.

Problem 9. Based on the wave equation in a plasma given Problem 8, consider a plane wave with electric field

E(r, t) = E0ei(kz!"t )

ex.

(a) Derive the dispersion relation

!(k). (b) What is the magnetic field associated with this wave? (c) What is the intensity of the wave?

Problem 10. A charge q oscillating sinusoidally with frequency ω on line segment of length d, radiates electromagnetic radiation, observed very far away r>>d. (a) Under what condition is the dominant contribution electric-dipole radiation? (b) The dipole radiation per solid angle is not isotropic; it varies as

sin2! . Explain why.

(c) What is the total electric dipole power radiated into all directions, time averaged over a cycle of oscillation.

r>>d

d θ

q

C L

R

V0

Preliminary Examination: Electricity and Magnetism Department of Physics and Astronomy

University of New Mexico Spring 2009

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. Consider two point charges +q and -q fixed on the y-axis, separated by a distance a. A charge Q is placed on the x-axis a distance r from the bisector.

What is the direction and magnitude of the force on Q in the limit a<<r (lowest nonvanishing term in a/r). Problem 2. A charge of magnitude Q is uniformly distributed throughout a sphere of radius R. What is the electric field everywhere (both inside and outside the sphere, magnitude and direction)? Problem 3. A battery of voltage V is used to charge of a coaxial capacitor of length L, inner radius a, outer radius b. How much energy is stored in the capacitor once it is fully charged (ignore fringing fields) in terms of the parameters given (a, b, V)?

+q

x Q

R

y

a r

Q

V a b

–q

Problem 4. A current density J flows uniformly along the y-direction in a slab (extending to infinity in the y and z directions) of thickness d along x. A cross section of the slab in the x-y plane is sketched below. What is the direction and magnitude of the magnetic field as function of x (do this for all x, positive and negative, inside the slab and out). Problem 5. A charge q moves in uniform electric field

E = Eˆ x and uniform magnetic field

B = Bˆ z . The charge starts at the origin at rest. Show that the velocity of the charge obeys the following equation of motion,

d2vdt 2

= −q2B2

m2 v +q2

m2 E × B( ).

Sketch the trajectory of the charge for

E = 0 and

E ≠ 0 . Problem 6. A charge/area σ is uniformly distributed and fixed on the surface of a very long cylinder of radius R and length L and zero mass. The cylinder is spun slowly about its axis from rest to a final angular velocity is ω. How much energy is stored in resulting magnetic field (ignore fringing fields)? Considering the fact that the cylinder is massless, where did this energy come from and how would you calculate the work done to create it?

ω

σ

R

x

J

d

y

Problem 7. Consider an RLC circuit with

R /L << 1/ LC . An ac-voltage drives the circuit

V (t) = V0 cos ωt( ) . In steady state, find the time averaged power dissipated in the resistor. Sketch a plot of this power as a function of ω. Comment on its form.

Problem 8. A monochromatic electromagnetic wave travels in a material with dielectric permittivity ε1 and magnetic permeability µ1. It comes to an interface with a second material at normal incidence, with dielectric permittivity ε2 and magnetic permeability µ2. Use the boundary conditions dictated by Maxwell’s equations to show that the ratio of the transmitted to incident electric field amplitude is,

Etrans

Einc

=2Z2

Z1 + Z2,

where

Zi = µi /ε i is the wave impedance in the material. Problem 9. Consider a electromagnetic field traveling in a nonmagnetic, nonconducting dielectric, with bound charge described by polarization density (electric dipole per unit volume) P(r,t). Use Maxwell’s equations to show that the electric field satisfies the wave equation with source,

∇2 − µ0ε0∂ 2

∂t 2⎛ ⎝ ⎜

⎞ ⎠ ⎟ E(r,t) = µ0

∂ 2P(r, t)∂t 2

.

Problem 10. Consider now monochromatic wave solutions to Problem 9. Take an Ansatz for the fields as plane waves,

E(r, t) = E0ei(kz−ωt ),

P(r, t) = P0ei(kz−ωt ).

Suppose the medium “nonlinear”, so that

P0 = ε0χ(3)E0

2E0 .

Show that the phase velocity of the wave is,

vphase =cn(I)

, where the intensity-dependent

index of refraction is

n(I) = 1+ χ (3) E02 .

L R

C V(t)

ε2, µ2 ε1, µ1 Einc

kinc Binc

Etrans

ktrans Btrans

Preliminary Examination: Electricity and MagnetismDepartment of Physics and Astronomy

University of New MexicoWinter 2016

Instructions:

• The exam consists of 10 problems, 10 points each.

• Partial credit will be given if merited.

• Total time is 3 hours.

Useful formulas and relations:

• Relation of spherical polar coordinates, (r, θ, φ), to Cartesian coordinates:

x = r sin θ cosφ, y = r sin θ sinφ, z = r cos θ. (1)

• Laplacian in spherical polar coordinates:

∇2 =1

r2∂

∂r

(r2∂

∂r

)+

1

r2 sin θ

∂θ

(sin θ

∂θ

)+

1

r2 sin2 θ

∂2

∂φ2. (2)

• Maxwell’s equations:

∇ · ~E =ρ

ε0,

∇ · ~B = 0,

∇× ~E = −∂~B

∂t,

∇× ~B = µ0

(~J + ε0

∂ ~E

∂t

).

• Biot-Savart Law for the magnetic field created by a steady current element Id~l:

d ~B (~r) =µ04π

Id~l × ~r|~r|3

.

• Time-averaged power radiated by an oscillating electric dipole:

P =µ0 |p|2 ω4

12πc.

• Time-averaged power radiated by an oscillating magnetic dipole:

P =µ0 |m|2 ω4

12πc3.

EM Prelim Version of December 20, 2015 2/5

1. (10 points) Consider three point charges +q, +q, and −2q fixed on the y−axis as shown in thefigure. Another charge Q is placed on the x−axis a distance r from the bisector

+q

+q

Q2a x

y

r

-2q

What is the direction and the magnitude of the force on Q in the limit a r (lowest nonvan-ishing term in a/r)?

2. (10 points) Consider three different charges with values: +2q, +q, and −q placed in threedifferent configurations

-q+q+2q +q+2q-q +2q-q+q(a) (b) (c)

Which of the three configurations has the minimum energy?

3. (10 points) Three conducting spheres with radius a, b, and c (a < b < c) are connected asfollows: the inner and outer spheres are connected to the ground, while the middle one isconnected to a potential source V . Find the electric potential in the two regions: a ≤ r ≤ b,and b ≤ r ≤ c. Also, calculate the charge on each of the spheres.

ab

cground

ground

groundV

EM Prelim Version of December 20, 2015 3/5

4. (10 points) A charge of magnitude Q is uniformly distributed throughout a sphere of radiusR. Find an expression for the electric potential φ (r) everywhere (both inside and outside thesphere)? Sketch a graph showing φ (r) as a function of r.

QR

5. (10 points) Consider two capacitors of identical construction except one is filled with a dielec-tric having dielectric constant K = 2 (capacitor A) and the second is filled with air, K = 1(capacitor B). Initially, the switch S is open and capacitor A has a charge q0. At some time,S is closed, so that the capacitors are connected. What are the final charges qA and qB on thecapacitors?

S

Κ=2 Κ=1A B

+q0

-q0

6. (10 points) A uniformly charged ring of radius R and charge Q is rotated at a uniform angularvelocity ω about its axis. Calculate the magnitude and direction of the magnetic field due tothe rotating disk at a distance d from it along its axis.

ω

d

EM Prelim Version of December 20, 2015 4/5

7. (10 points) A very long solenoid carries a current I (t) = I0 (1− αt), where I0 and α areconstants. Coaxial with the solenoid is a large, circular ring of wire, with resistance R. As thecurrent in the solenoid changes, a current is induced in the ring. Take the solenoid to have nturns per unit length, and radius a, while the ring has radius b a. What is the current inthe ring?

2a

2b

8. (10 points) An electron (mass m, charge e) moves between the plates of a parallel plate ca-pacitor. The voltage across the capacitor is V , and the plate separation is d (as shown in thefigure). The electron starts from rest at the cathode and for a constant value of B it follows acertain periodic trajectory, almost reaching the anode before returning to the cathode.

batteryd

B

xy

V

(a) Verify that the electron velocity is determined by the following equations

vx (t) =V

dB

[1− cos

eB

mt

]vy (t) =

V

dBsin

eB

mt

(b) Find the trajectory x(t), y(t) and draw it.

EM Prelim Version of December 20, 2015 5/5

9. (10 points) Consider a plane electromagnetic wave of amplitude EI propagating in vacuum thatis normally incident on a nonmagnetic, nonconducting dielectric material with permittivity ε1.Use the boundary conditions on the electric ~E and magnetic ~B fields at the interface to findthe amplitude of the reflected and transmitted waves (denoted by ER and ET respectively).Find ε1 such that the transmitted amplitude is half of the reflected one.

ε0, μ0 ε1, μ0

EI

kI

BI

ET

kT

BT

ER

kR

BR

10. (10 points) The magnetic field associated with a plane wave traveling in a nonmagnetic dielec-tric medium is given by

~B (~r, t) =(9 · 10−8 T

) [cos(20µm−1z − 3fs−1t

)x− sin

(20µm−1z − 3fs−1t

)y]

(a) To what part of the electromagnetic spectrum does this wave correspond?

(b) In what direction is the wave propagating?

(c) What is the polarization of the field (linear, circular, elliptical)?

(d) What is the dielectric constant of the medium?

(e) What is the intensity of the radiation?