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UC Berkeley Physics Exam: Physics 110B Spring 2004 (Strovink) Exam 2, Exams of Electromagnetism and Electromagnetic Fields Theory

Central University of South Bihar Electromagnetism and Electromagnetic Fields Theory

The directions and problems for a university physics exam held at the university of california, berkeley during the spring 2004 semester for physics 110b (strovink). The exam covers topics such as relativistic mechanics, electric fields, and potentials. Students are required to solve three problems, which have unequal weight, using only specified materials and without calculators or laptops. The problems involve finding the rapidity and motion of a charged particle in a uniform electric field, the scalar potential seen by an observer due to a moving charged particle, and the leading term in the multipole expansion of the electrostatic potential due to a line charge distribution.

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University of California, Berkeley

Physics 110B Spring 2004 (Strovink)

EXAMINATION 2

Directions: Do all three problems, which have unequal weight. This is a closed-book closed-note

exam except for Griﬃths, Pedrotti, a copy of anything posted on the course web site, and anything

in your own original handwriting (not Xeroxed). Calculators are not needed, but you may use one if

you wish. Laptops and palmtops should be turned oﬀ. Use a bluebook. Do not use scratch paper –

otherwise you risk losing part credit. Show all your work. Cross out rather than erase any work that

you wish the grader to ignore. Justify what you do. Express your answer in terms of the quantities

speciﬁed in the problem. Box or circle your answer.

Problem 1. (35 points)

A positive point charge qof rest mass mis sub-

jected to a uniform constant electric ﬁeld E0

pointing in the ˆxdirection (as observed in the

laboratory). This force causes its (relativistic)

momentum to increase linearly with laboratory

time t.

(a.) (15 points)

At t= 0 the charge is at rest at the origin.

Thereafter, show that sinh η=ω0t, where ηis

the charge’s rapidity and ω0is a constant. De-

termine ω0.

(b.) (20 points)

For t>0 show that the charge moves according

to ω0

cx(t)=1+ω2

0t2−1.

Problem 2. (35 points)

At t= 0 at the origin of a laboratory coordinate

system, a point particle of charge qhas velocity

βc directed along the ˆzaxis. It has been moving

with that constant velocity for a long time.

(a.) (20 points) At t= 0, starting with the

Coulomb potential in the particle’s rest frame,

and using the rules for relativistic transforma-

tion both of coordinates and of em potentials (in

Lorentz gauge), ﬁnd the scalar potential Vseen

by an observer located at Cartesian coordinates

(x, 0,z), where V(∞)≡0. Express Vin terms

of β,x,z, and constants.

(b.) (15 points) As seen at t= 0 by the labora-

tory observer, is −∇Vexpected to point in the

radial direction? Explain.

Problem 3. (30 points)

A thin insulating rod, running from z=−a

to z=+a, carries a static line charge density

(Coul/m)

λ(z)=λ0cos πz

a.

Find the leading term only in the multipole

expansion of the electrostatic potential V(r).

Express your answer in terms of the observer’s

polar angle θand the observer’s distance rto

the origin.

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University of California, Berkeley Physics 110B Spring 2004 (Strovink)

EXAMINATION 2

Directions: Do all three problems, which have unequal weight. This is a closed-book closed-note examexcept for Griffiths, Pedrotti, a copy of anything posted on the course web site, and anything in your own original handwriting (not Xeroxed). Calculators are not needed, but you may use one if you wish. Laptops and palmtops should be turned off. Use a bluebook. Do not use scratch paper – otherwise you risk losing part credit. Show all your work. Cross out rather than erase any work that you wish the grader to ignore. Justify what you do. Express your answer in terms of the quantities specified in the problem. Box or circle your answer.

Problem 1. (35 points) A positive point charge q of rest mass m is sub- jected to a uniformconstant electric field E 0 pointing in the ˆx direction (as observed in the laboratory). This force causes its (relativistic) momentum to increase linearly with laboratory time t.

(a.) (15 points) At t = 0 the charge is at rest at the origin. Thereafter, show that sinh η = ω 0 t, where η is the charge’s rapidity and ω 0 is a constant. De- termine ω 0.

(b.) (20 points) For t > 0 show that the charge moves according to ω 0 c

x(t) =

1 + ω^20 t^2 − 1.

Problem 2. (35 points) At t = 0 at the origin of a laboratory coordinate system, a point particle of charge q has velocity βc directed along the ˆz axis. It has been moving with that constant velocity for a long time.

(a.) (20 points) At t = 0, starting with the Coulomb potential in the particle’s rest frame, and using the rules for relativistic transforma- tion both of coordinates and of em potentials (in Lorentz gauge), find the scalar potential V seen by an observer located at Cartesian coordinates (x, 0 , z), where V (∞) ≡ 0. Express V in terms of β, x, z, and constants.

(b.) (15 points) As seen at t = 0 by the labora- tory observer, is −∇V expected to point in the radial direction? Explain.

Problem 3. (30 points) A thin insulating rod, running from z = −a to z = +a, carries a static line charge density (Coul/m)

λ(z) = λ 0 cos

πz a

Find the leading term only in the multipole expansion of the electrostatic potential V (r). Express your answer in terms of the observer’s polar angle θ and the observer’s distance r to the origin.

UC Berkeley Physics Exam: Physics 110B Spring 2004 (Strovink) Exam 2, Exams of Electromagnetism and Electromagnetic Fields Theory

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