NTNU Institutt for fysikk
Contact during the exam:
Professor Ingve Simonsen Telephone: 9 34 17 or 470 76 416
Exam in TFY4240 Electromagnetic Theory Wednesday Dec 10, 2008
09:00–13:00 Allowed help: Alternativ C
Authorized calculator and mathematical formula book This problem set consists of 8=1one page=0.
Problem 1.
R
I O
An infinitely long wire carries a (time-independent) currentI. The wire is bent so as to have a semi-circular detour, of radiusR, around the originO (see figure).
a) Derive an expression for the magnetic field (vector),H, at the originO of the coordinate system.
b) Determine the numeric value of this magnetic field given the currentI = 1A and radius R = 1cm.
Problem 2.
In this problem, we will consider the so-calledattenuation constantfor a plane wave propagat- ing in a good conductor. We aim at, step-by-step, to derive an expression for this constant.
The medium under study is an ohmic conductor of permittivityε, permeability µ and con- ductivityσ. For simplicity these constants are assumed to beindependent of frequency.
a) From the Maxwell’s equations and Ohm’s law, show that the relevant wave equation reads
∇2E−µε∂t2E−µσ∂tE = 0, (1) where E≡E(r, t) and∂t= ∂t∂.
Exam in TFY4240 Electromagnetic Theory, Dec. 10,2008 Page 2 of 8 b) For a wave of angular frequency ω, E(r, t) =E0(r)e−iωt, Eq. (1) can be written in the
from
∇2E0+µ(ω)ω2E0 = 0. (2)
Show this, and identify the function (ω) (different from ε).
A plane wave is incident on the conductor along the inward normal, whose direction is taken to be thez-direction. Then in the conductor the electromagnetic wave can be represented by
E = E0eikz−iωt, (3)
wherekis the wave number.
c) Find an expression for the wave number k in terms of ω and the medium parameters (ε,µ and σ).
d) We write k=k1+ik2, wherek1 andk2 both are real functions. For a good conductor, i.e. forσ/(εω)1, show thatk1 =k2 and determine this common function (again) in terms of ω and the material parameters.
e) Argue why it is reasonable to name the constant δ = 1/k2 the attenuation constant.
Write down the expression for this constant (δ).
Problem 3.
z
x θ
P
zz0
A static electric dipole is located in vacuum at positionr0 = (0,0, z0) (see figure). Its dipole moment can be writtenp=p(sinθ,0,cosθ) whereθ is the angle between p and the positive z-axis. Initially vacuum is filling the whole space (also the regionz≤0).
a) Show that the scalar potential for an individual dipole (without the conducting half- space present) can be written as
V(r) = 1 4πε0
Rˆ ·p
R2 . (4)
What is the meaning of R in this equation? In your proof, you may for simplicity set z0 = 0 andθ= 0. Below, however, this assumption will notbe made.
Exam in TFY4240 Electromagnetic Theory, Dec. 10,2008 Now a perfectly conducting,grounded, half-space is placed atz≤0.
b) Give the boundary conditions that the scalar potential, V(r), satisfies at the interface of the metallic half-space (z = 0). Explain (in words) the essence of the method of images.
c) Use the results from point b) to determine the location and orientation of the image dipole, and make a sketch of the resulting configuration. Moreover, show that the scalar potential for z≥0 can be written as
V(r) = p 4πε0
xsinθ+ (z−z0) cosθ
[x2+y2+ (z−z0)2]3/2 +−xsinθ+ (z+z0) cosθ [x2+y2+ (z+z0)2]3/2
!
. (5)
d) Determine the induced (surface) charge density, σ(x, y) on the surface of the metal.
Express your answer in terms of the spatial coordinates x and y, the dipole height z0, the dipole orientationθ, and the magnitude of the dipole moment|p|=p.
Problem 4.
z O
y
x φ
θ r
v (t)p
Consider a particle of chargeq 6= 0 that is moving with a constant angular frequencyω along a circular path of radiusr0 in the xy-plane (see figure). For instance, this can be achieved by applying a static magnetic fieldH. It is assumed that the particle velocity is non-relativistic (vpc).
An observer point,O, is defined by the spherical coordinates (r, θ, φ) relative to a coordinates system with origin in the centered of the circle (see figure).
a) Write down an expression for the time-dependent particle position, rp(t), and use this to calculate the particle velocity, vp(t), and acceleration, ap(t). What is the direction of the applied (static) magnetic field,H, relativevp(t), for the particle to make circular motion?
We will now study the radiation from this particle. The time-dependent radiated power per solid angle is given by
dP dΩ = 1
4πε0
q2 4πc3
Rˆ ×ap
2
. (6)
Exam in TFY4240 Electromagnetic Theory, Dec. 10,2008 Page 4 of 8 b) Explain what the time-dependent factor ˆR(t) in Eq. (6) means. When calculating
dP/dΩ, what time should be used for this quantity and ap(t)?
c) Under the assumptionr0 rderive an expression for thetime-averagedradiated power per solid angle hdP/dΩi for the particle. Why is this expression independent of the angle φ? Explain why the assumptionr0 r simplifies significantly the calculation.
d) What is the total radiated power, P, from the system (independent of radiation direc- tion)? Compare your result with Larmor’s formula (cf. formula sheet).
e) You have just showed (hopefully) that the particle is radiating, i.e. thatP 6= 0. How- ever, still the particle performs circular motion of constant angular velocity, and there- fore has time independent total energy. Explain how this is possible. Where is the radiated energy coming from?
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