Examination paper for TFY4240 Electromagnetic theory

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Department of Physics

Examination paper for

TFY4240 Electromagnetic theory

Academic contact during examination: Associate Professor John Ove Fjærestad Phone: 97 94 00 36

Examination date: 16 December 2015 Examination time (from-to): 9-13

Permitted examination support material: C

Approved calculator

Rottmann: Matematisk Formelsamling (or an equivalent book of mathematical formulas)

Other information:

This exam consists of three problems, each containing several subproblems. The sub- problems will be given approximately equal weight during grading. However, some sub- problems may be given double weight, but only if this is indicated explicitly. In many cases it is possible to solve later subproblems even if earlier subproblems were not solved. Some formulas can be found on the pages following the problems.

Language: English

Number of pages (including front page and attachments): 10

Checked by:

____________________________

Date Signature

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Exam in TFY4240 Electromagnetic Theory, Dec. 16, 2015 Problem 1.

a) Briefly describe the ”method of images” and the type of problems it can be used to solve.

A point chargeq is held at a fixed position outside a grounded conducting sphere of radius R. The distance between the point charge and the center of the sphere is a. We take the z axis to pass through both the center of the sphere (wherez= 0) and the point charge (where z=a). See Fig. 1 for an illustration of the geometry.

𝑧

𝑞 𝒓

𝑎 𝜃

𝑅

Figure 1: A point charge q outside a grounded conducting sphere of radiusR.

b) (Double weight) We wish to find the potential V at an arbitrary point r outside the sphere. Show that the problem can be solved by introducing an image charge q⁰ where

q⁰ =−R

aq, (1)

which is positioned on the z axis atz=bwhere b= R²

a . (2)

Give an expression for the potential V(r) outside the sphere.

c) Find the induced surface charge density σ (which depends on the angle θ, see figure) and the total induced charge Qon the surface of the sphere.

d) Suppose next that the conducting sphere is not grounded, but is instead held at a fixed potential V₀ 6= 0 (with respect to infinity, where V → 0). Again we wish to find the potential everywhere outside the sphere. Solve this problem by introducing one more image charge. What is the charge and position of this second image charge?

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𝐼 𝐼

𝑥 𝑦

𝑧 𝑑

1 2

Figure 2: Two infinite straight wires a distance dapart.

Consider two infinitely long, straight, electrically neutral, parallel wires labeled 1 and 2 as shown in Fig. 2. The wires lie in the xy plane, are oriented along they axis, and each wire carries the same currentI in the positivey direction. The wires havex-coordinate±d/2, so the distance between the wires isd.

a) Show that the magnetic field produced by each wire has magnitudeµ0I/2πr wherer is the distance to the wire. Use this to find the force, per unit length, on wire 1. Is the force attractive or repulsive?

b) For a general problem in electrodynamics, briefly explain the meaning of the 3 terms in the equation

F = I _↔

T ·da− d dt

Z S

c² d³r. (3)

c) (Double weight) By an appropriate application of (3), give an alternative calculation of the force per unit length on wire 1. [Hint: In your calculation, give a key role to the plane x= 0 of points equidistant from both wires.]

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Exam in TFY4240 Electromagnetic Theory, Dec. 16, 2015 Problem 3.

a) Show that in the Lorenz gauge, the electromagnetic potentials V and A satisfy the inhomogeneous wave equations

∇²V − 1 c²

∂²V

∂t² = −ρ/₀, (4)

∇²A− 1 c²

∂²A

∂t² = −µ₀J. (5)

State the definition of the Lorenz gauge.

𝑣 𝑡 𝒛 𝑧 𝒓

𝑞 𝑹 𝜃

Figure 3: A particle of charge q moving with constant velocityv along the z axis.

In the following we will consider a derivation of the fields produced by a point charge q moving withconstant velocity, by directly solving the inhomogeneous wave equations for the potentials for this special case. Taking the particle to be moving along the z axis with velocityv=vzˆwith v >0 (see Fig. 3), the source densities for the point charge are

ρ(r, t) = qδ(x)δ(y)δ(z−vt), (6)

J(r, t) = ρ(r, t)v (7)

(we take the particle to be at the origin att= 0). SinceJ ∝z, the only nonzero componentˆ ofAwill be A_z. Furthermore, because of the uniform motion, the z andtdependence of the potentials must be through the combinationz−vt≡ξ.

b) Show that in this case (4) can be rewritten as the differential equation

∂²V

∂x² +∂²V

∂y² + (1−β²)∂²V

∂ξ² =−q

₀δ(x)δ(y)δ(ξ) (8)

where β=v/c.

c) By making another change of variables from ξ to z⁰ = γξ, where γ = 1/p 1−β², rewrite (8) as a differential equation in the variables x,y, and z⁰. Based on the form of this differential equation, and using your knowledge of the solution of a mathematically related but physically simpler problem, show that V is given by

V(x, y, z, t) = γq 4π₀p

x²+y²+γ²(z−vt)². (9)

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ities between the problems for V and Az.]

e) Show that the electric field is given by E(r, t) = γq

4π₀

xxˆ+yyˆ+ (z−vt) ˆz

[x²+y²+γ²(z−vt)²]^3/2. (10) f ) Show that (10) can be rewritten as

E(r, t) = q 4π0

Rˆ R²

1−β²

(1−β²sin²θ)^3/2, (11) where R is the vector pointing from the positionvtzˆof the particle to the observation point r=xxˆ+yyˆ+zz, andˆ θ is the angle betweenR and v (see Fig. 3).

g) Briefly discuss how the magnitude of E varies with direction θ for an ultrarelativistic particle (β ≈1), especially comparing the forward/backward directions (θ≈0, π) with the transverse directions (θ≈π/2). Do the same thing for a very nonrelativistic particle (β≈0).

h) Consider an (imagined) sphere of radius R centered on the particle at time t. What is the energy per unit time flowing through the surface of this sphere at timet?

i) Give a brief definition of radiation for a general problem in electrodynamics. For the special problem considered earlier, namely that of a particle moving with constant velocity, does the particle radiate? Justify your answer.

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Exam in TFY4240 Electromagnetic Theory, Dec. 16, 2015

Formulas

Some formulas that you may or may not need (you should know the meaning of the symbols and possible limitations of validity):

σ=−₀ ∂V

∂n

_outside−∂V

∂n _inside

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F =I`×B (13)

Tij =0

EiEj−1 2δijE²

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µ0

BiBj−1 2δijB²

(14) δ(ax) = 1

|a|δ(x) (15)

∇²V =−ρ/₀ (16)

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