ExaminTFY4240ElectromagneticTheory NTNUInstituttforfysikk

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NTNU Institutt for fysikk

Contact during the exam:

Professor Ingve Simonsen Telephone: 9 34 17 or 470 76 416

Exam in TFY4240 Electromagnetic Theory Dec 20, 2013

09:00–13:00 Allowed help: AlternativC

Authorized calculator and mathematical formula book This problem set consists of 6 pages.

This exam consists of three problems each containing several sub-problems. Each of the sub-problems will be given approximately equal weight during grading. However, some sub- problems may be given double weight, but only if so is indicated explicitly.

For your information, it is estimated that you will spend about 10% (or less) of the available time on problem 1; 50% on problem 2; and about 40% on problem 3.

I will be available for questions related to the problems themselves (though not the answers!).

The first round (of two), I plan to do around 10am, and the other one, about two hours later.

The problems are given in English only. Should you have any language problems related to the exam set, do not hesitate to ask. For your answers, you are free to use either English or Norwegian.

Notethat some formulas are given after the last problem!

Good luck to all of you!

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Exam in TFY4240 Electromagnetic Theory, Dec. 20, 2013 Problem 1.

z O

x

θ r

y

φ r_||

r =a_||

r =b||

Figure 1: Schematics for problem 1

A very long non-magnetic cylindrical conductor with inner radiusaand outer radiusbcarries a constant currentI. The current density in the conductor is uniform. A coordinate system is defined so that the cylindrical axis coincides with the z-axis, and the direction of the current is along zˆas depicted in Fig. 1.

a) Obtain an expression for the current density, J(r), of the system (wherer is a general position vector).

b) Obtain the magnetic field,H(r), as function of the distance from the z-axis, r_k =|r_k| (r_k =xxˆ+yy), for the regions: (i)ˆ r_k < a; (ii) a < r_k < b; and (iii) r_k > b. Make a sketch of the amplitude of the magnetic field. Remember to indicate the direction of the magnetic field.

c) Repeat the previous sub-problem, but now for the electric field. ¹ Problem 2.

A dielectric sphere of radius a and dielectric constant ε1 is placed in a dielectric liquid of infinite extent and dielectric constant ε₂ (see Fig. 2). Originally a uniform static electric field, E₀, is present in the liquid, i.e., before the sphere is placed in the liquid. We choose a coordinate system with the origin located at the center of the sphere and the positive z-axis directed along the original electric fieldE₀ =E₀zˆ(E₀ >0), as illustrated in Fig. 2. As usual,

ˆ

z denotes the unit vector in thez-direction.

The purpose of this problem is to find the scalar potential and the total electric field inside and outside the dielectric sphere so we can determine how it reacts to the external field.

a) Argue why the scalar potential, V(r), for the system can be written in the form V(r) =

∞

X

`=0

A_`r^`+ B`

r^`+1

P_`(cosθ). (1)

1This means: Obtain the electric field, E(r), as function of rk = |rk| for the regions: (i) rk < a; (ii) a < rk< b; and (iii)rk> b.

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Exam in TFY4240 Electromagnetic Theory, Dec. 20, 2013

ε₁ ε₂

(r,θ,φ)

E

θ

0

r=a

z x

Explain why we have both positive and negative powers of r, and what the functions P_`(cosθ) are. Note that you are not asked to derive this result, only motivate it!

b) Write down the four boundary conditions satisfied by the scalar potential at (i) r= 0;

(ii) r→ ∞; and (iii)r =a. [GivenP1(x) =x]

c) (double weight) Use these boundary conditions to show that the scalar potential has the form

V(r) =

(−k₁E₀rcosθ, r < a

−h

1−k₂ ^a_r3i

E₀rcosθ, r > a, (2) where k₁ and k₂ are constants. Obtain expressions for the constants k₁ and k₂, and express your answer in terms of ε₁ andε₂.

d) From the expression for the potentialV(r), Eq. (2), obtain the total electric field,E(r), inside and outside the sphere. Express one of the two terms for the electric field outside the sphere in terms of

3 (E0·r)ˆ rˆ r³ −E0

r³

, (3)

where rˆdenotes a unit vector in the direction ifr.

e) What is the physical interpretation of the expression for the electric field (or potential) outside the sphere (r > a)? Obtain en expressions for the induced dipole moment of the dielectric sphere, p, and its polarizability, α, defined via p=αE0. Determine the polarization of the sphere and show that it satisfies

P ∝k₂E₀. (4)

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f ) Calculate the charge density, σ(θ), induced on the surface of the sphere. Make a sketch of σ(θ). Is the result as expected?

g) If the sphere is rotated at an angular frequencyωabout the direction of the electric field E0, will a magnetic field be produced? Give reasons for your answer! (Only answering yes or no will give no credit.)

Problem 3.

z

y x

θ r

I(t)

φ

( ,θ,φ) r

A small circular loop of wire has radiusaand carries a time-dependent current

I(t) =I0exp (−iωt), (5)

as shown in Fig. 3. Here ω denotes the angular frequency that is assumed to be a positive constant. It is also assumed that the wire-loop is placed in vacuum in the the xy-plane. In this problem, we aim to calculate the radiation pattern of the system.

To this end, we will start by obtaining some useful results that will be used later in the problem. In the Lorentz gauge, a general expression for the vector potential is

A(r, t) = µ0

4π Z

d³r⁰ J(r⁰, tr)

|r−r⁰|, (6)

where t_r=t− |r−r⁰|/c and the remaining quantities you should know the meaning of. For anarbitraryobservation point,r, the calculation of the vector potentialA(r, t) is a non-trivial task. However, in the far field (the radiation zone), this calculation simplifies significantly.

We recall that in the far field the following conditions apply kr 1 (where k = ω/c) and rr⁰.

a) Assume a general time-harmonic current density of the form J(r, t) = J(r|ω)e^−iωt. Under the assumption of being in the far field, show that the vector potential (6) can be written in the form

kr→∞lim A(r, t) = µ0

4π

e^ikr−iωt

r f(θ, φ), (7a)

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Exam in TFY4240 Electromagnetic Theory, Dec. 20, 2013 where an angular factor is defined by

f(θ, φ) = Z

d³r⁰ J(r⁰|ω)e^−ikrˆ·r⁰. (7b) In passing we note that since one assumes krkr⁰ 1, the exponential contained in f(θ, φ), may be expanded and one may keep only the leading term.

b) Demonstrate that the mathematical expression for the current density for the geometry in Fig. 3 in spherical coordinates reads

J(r, t) = I(t)

a δ(r−a)δ(θ−π/2)φ,ˆ (8)

where φˆ is the unit vector in direction φ, and δ(·) denotes a Dirac delta function.

Explain in particular why a factor 1/ais needed in Eq. (8).

For the system in Fig. 3, the leading term off(θ, φ) in thefar field can be shown to be (like we did in the lectures)

f(θ, φ) =−ikm₀×r,ˆ (9a)

with

m₀= 1 2

Z

d³rr×J(r|ω). (9b)

c) Based on Eq. (9), show explicitly that

A(r, t)∝sinθexp (ikr−iωt)

r φ.ˆ (10)

What is the physical significance (meaning) of m(t) =m0e^−iωt?

d) (double weight) What is meant by radiation fields? Argue why it is sufficient to use Eq. (10) for the calculation of the radiation fields. Use this equation to calculate the electric and magnetic fields, E(r, t) and H(r, t), respectively, in thefar field [or radiation zone] (kr 1).

The time averaged total power radiated from the system is defined as P =

Z

dA · hSi= Z

dΩdP

dΩ, (11)

where dA is a small surface element (pointing outward), and hSi= 1

2E×H^∗ (12)

denotes the time-averaged Poynting vector. Moreover, dP/dΩ is the averaged power radiated per unit solid angle, also known as the differential power, and it defines theradiation pattern of the system.

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e) Obtain an expression for hSi in thefar field and show that it satisfies hSi ∝ sin²θ

r² r.ˆ (13)

Use it to calculate the radiation pattern, dP/dΩ. Make a sketch of the radiation pattern of the wire-loop. For what directions are the radiation the strongest and the weakest?

f ) Calculate the total averaged power, P, radiated by the loop.

g) If the amplitude of the current in the wire is doubled, say, how much smaller must the radius of the loop be in order for the loop to radiate the same amount of power. Argue from the definition of the magnetic dipole moment, that you answer is reasonable.

Formulas

Some formulas that you may, or may not, need. The meaning of the symbols you should know.

p(t) = Z

d³rrρ(r, t) m(t) = 1

2 Z

d³rr×J(r, t) Z 1

−1

dx Pm(x)Pn(x) = 2 2n+ 1δmn

V(r) = 1 4πε₀

p·rˆ r² A(r) = µ₀

4π

m×rˆ r² 1

|r−r⁰| = 1

√

r²+r⁰²−2rr⁰cosθ⁰ = 1 r

∞

X

n=0

Pn(cosθ⁰) r⁰

r n

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