ExaminTFY4240ElectromagneticTheory NTNUInstituttforfysikk

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 10, 2012

09:00–13:00 Allowed help: AlternativC

Authorized calculator and mathematical formula book This problem set consists of 4 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 15% of the time on Problem 1;

25% on problem 2; and about 60% 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.

Good luck to all of you!

Exam in TFY4240 Electromagnetic Theory, Dec. 10, 2012 Page 2 of 4 Problem 1.

q V z

λ I

d

A particle of chargeq is traveling in vacuum with velocityv parallel to an infinitely long wire of a constant uniform charge distribution λ > 0 per unit length (see figure). The wire also carries a constant current I directed as shown in the figure. In this problem the gravitational force acting on the particle can be neglected.

a) Discuss and make a drawing of the forces acting on the moving particle.

b) Write down the expressions for the charge density, ρ(r), and current density, J(r), of the wire and use them to obtain the electric (E) and magnetic (H) fields around it.

c) Obtain an expression for the critical velocity vc = vczˆ that the particle must have in order to travel in a straight line parallel to the wire, a distance daway from it.

Problem 2.

A static charge distribution produces a radial electric field E(r) =Ae^−br

r² r,ˆ (1)

whereA >0 and b >0 are constants;r denotes the radial coordinate; and rˆis a unit vector in the radial direction. Assume the medium to be vacuum.

a) (double weight) What is the charge density, ρ(r), that produced the electric field from Eq. (1)? Make a sketch of ρ(r).

b) What is the total electric charge Q?

Exam in TFY4240 Electromagnetic Theory, Dec. 10, 2012 Problem 3.

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x₃

d

ε₁

ε₂(ω)

ε(x₃, ω) Film

Substrate

q k

p

θ₀ θ_s

θ_t

x₁

Consider the scattering geometry depicted above. Here a dielectric film is placed on top of a substrate that fills the region x3 < 0. The dielectric functions of the media above and below the film are ε₁ (a constant) and ε₂(ω), respectively, where ω denotes the angular frequency. The film has thicknessd ≥0 and is made from a spatially local medium that is characterized by a frequency andspatially dependent dielectric function,ε(x3, ω). We remind you that for a spatially local medium, the constitutive relation for the E and D-fields reads D(x, ω) = ε0ε(x3, ω)E(x, ω) (the relation is local in spatial coordinates). Notice that the spatial dependence of the dielectric function of the film only entersviathe vertical coordinate x₃. All media are assumed to be non-magnetic, i.e. the relative permeability is µ= 1 for all media. In the first part of the problem we will neglect the existence of the film, and it will only be considered in the last part.

Onto this geometry, ap polarized plane electromagnetic wave is incident in the direction of the wave vectork(see figure). A coordinate system is chosen so that the projection ofkonto thex₁x₂ plane is directed along xˆ₁.

a) State the general boundary conditions that the normal and tangential components of the electromagnetic field have to satisfy at a general surface (assuming no sources at the surface).

First we assume zero film thickness, i.e. d= 0, so that the film does not exist.

b) Why can the magnetic field component of the electromagnetic wave above the substrate be written in the form [where kk ∝xˆ1 and xk = (x1, x2,0)]

H^>(x, t) =H^>(x|ω) exp (−iωt), (2a)

Exam in TFY4240 Electromagnetic Theory, Dec. 10, 2012 Page 4 of 4 where

H^>(x|ω) =H₀xˆ₂h

exp ik_k·x_k−iα₁(k_k)x₃

+r(q_k|k_k) exp

iq_k·x_k+ iα₁(q_k)x₃i . (2b) Explain the symbolsk_k andq_kand defineα1(k_k) so that theH^>(x, t) satisfies the wave equation for the magnetic field. What is the physical meaning of r(q_k|k_k) and use this to identify what part of Eq. (2b) corresponds to the reflected and incident wave.

In the substrate, the magnetic field takes the form H^<(x|ω) =H0xˆ2 t(p_k|k_k) exp

ip_k·xk−iα2(pk)x3

(3) wheret(p_k|k_k) is the transmission amplitude.

c) (double weight) How isα₂(p_k) entering Eq. (3) defined? Derive expressions for the elec- tric fields above and below the substrate, denotedE^>(x|ω) andE^<(x|ω), respectively.

d) (double weight) Use the boundary conditions on the electromagnetic field to obtain a relation between k_k, q_k and p_k. Obtain expressions for the r(q_k|k_k) and t(p_k|k_k) expressed in terms of α_i,ε_i and µ_i (i= 1,2).

e) The intensity of an electromagnetic wave crossing a plane parallel to the x1x2 plane is I = |hSi ·xˆ₃|. What is the meaning of the symbols used to define the intensity, and give the SI-unit used forI. Calculate the intensity of the reflected (Ir) and transmitted (It) light for our scattering geometry. Express your answers in terms of r,t,αi,εi, and µ_i.

The reflection (R) and transmission (T) coefficients (not to be confused with amplitudes r andt) are defined as

R= I_r

I₀, T = I_t

I₀, (4)

whereI₀ is the intensity of the incident light.

f ) Derive the expressions for Rand T. Also here express your answers in terms ofr,t,αi, ε_i, and µ_i. Under the assumption that there is no absorption of energy in the media involved, i.e. Imεi = 0, demonstrate that the conservation of energy conditionR+T = 1 is satisfied.

We will now assume a non-zero film thicknessd >0.

g) For this case, obtain a mathematical expression that determines the direction of the transmitted light. The expression should be valid for a general form ofε(x₃, ω).

h) In your own words, describe a method that can be used to determining the intensity of the transmitted light (and not only its direction) when d6= 0.

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