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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 14, 2010
09:00–13:00 Allowed help: AlternativC
Authorized calculator and mathematical formula book This problem set consists of 5 pages.
This exam consists of two problems each containing several sub-problems. Each of the sub- problems will be given approximately equal weight during grading, except point 2e that will be given double weight. For your information, I estimate that you will spend about twice the amount of time on the 2nd problem relative the 1st.
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. 14, 2010 Problem 1.
Consider a rectangle of horizontal and vertical diagonalsaand b, respectively, and where we have placed (static) chargesqi (i= 1, . . . ,4) at each corner as shown in the above figure.
A coordinate system is placed with its origin at the center of this rectangle so that the rectangle is in the xz-plane. Relative to this coordinate system, the distance vector to the observation pointP will be denotedr, and the distance vector to chargei isri.
a) Write down the general expression for the (total) scalar potential V(r) for the four charges that is valid at anypointr (6=ri).
b) Describe, in your own words, what is meant by a multi-pole expansion for the scalar potential. In particular, point out the essential assumption that must be satisfied for the first few terms of this expansion to represent a good approximation to the potential.
We will from now onwards assume that the dimensions of the rectangle are small compared to the distance, r, to the observation point P; that is a/r 1 and b/r 1 with |r| = r.
Approximations to the scalar potential will now be studied for various charge configurations when the observation point is far away from all the charges. With an approximation, we mean the dominating (“the largest”) non-zero term contributing to the potential.
c) Write down an approximative expression for the potential V(r) (valid for large r) for the four charges under the assumption that q1+q2+q3+q46= 0.
Exam in TFY4240 Electromagnetic Theory, Dec. 14, 2010 Page 3 of 5 d) Now set q2 =q4 = 0, q1 =−q3 =q and b=` >0. Show in this case that the potential
for the system can be written as
V(r) = 1 4πε0
p·ˆr
r2 , (1)
and give an expression for p. What is the quantity p called? Here ˆr denotes a unit vector in the radial direction. Make a sketch the angular distribution of the of this potential (for given r).
Assume now that all charges are non-zero (qi 6= 0 for all i); b =` > 0 (with `/r 1); and thata/b=a/`1.
e) We first consider the case where q1 =q4 =q and q2 =q3 =−q. Make a sketch of the charge distribution indicating the positions of the positive and negative charges. What is the potential V(r) in this case? [Hint: Make use of the physics of the problem, and do not try to calculate this directly Of course, such a more lengthy calculation will work though]. Make a sketch of the of the angular distribution of the potential (for a given distance r).
f ) Assume now instead thatq1=q3 =qandq2 =q4=−q. Also for this case make a sketch of the charge distribution indicating the positions of the positive and negative charges.
Why will in this case (the angular distribution of) the potential be more complicated than in the previous sub-problem?
g) Find an approximative expression for the potential V(r) in the above case (i.e. when q1=q3 =q and q2 =q4=−q) and show that it satisfies
V(r)∝ 1
r3. (2)
Make a sketch of the angular distribution of the potential in this case (assuming a constant distance r). What is this angular pattern called?
Exam in TFY4240 Electromagnetic Theory, Dec. 14, 2010 Problem 2.
We consider a thin, straight, conducing wire of length` that is oriented along the z-axis as shown in the figure above. The wire carries the time-varying current
I(t) =I0exp (−iωt), (3)
everywhere along its length `. Here I0 and ω are both positive constants. The system is a simple model for an antenna.
a) Charge will only build up at the endpoints of the wire. Explain why this is so. Find an expression for the time-dependent charge, Q(t), building up at one endpoint.
b) Use your expression forQ(t) to determine the dipole momentp(t) of the simple antenna.
c) Argue why the current density can be written as
J(r, t) = ˆzI(t)δ(x)δ(y)θ(`/2)− |z|), (4) where δ(·) is the Dirac delta-function, θ(·) is the (Heaviside) step-function, and ˆz is a unit vector in the positive z-direction.
We will now study the electromagnetic field radiated from the antenna. To this end, we will start by calculate the potentials. In the Lorentz-gauge the vector potential satisfies the equation
∇2A(r, t)− 1
c2∂t2A(r, t) =−µ0J(r, t), (5) wherec= 1/√
ε0µ0is the speed of light in vacuum. Hereε0andµ0 are the electric permitivity and magnetic permeability of free space , respectively.
Exam in TFY4240 Electromagnetic Theory, Dec. 14, 2010 Page 5 of 5 d) By using the Green’s function for the wave-equation
g(r, t|r0, t0) =− δ
t−t0− |r−r0| c
4π|r−r0| , (6)
show that a solution to Eq. (5) is
A(r, t) = µ0 4π
Z
d3r0 J(r0, tr)
|r−r0|. (7) What is the meaning ofr and r0? Moreover, what is the expression fortr, and what is the physical interpretation of this quantity?
e) Calculate the vector potential A(r, t) under the assumption that r `and show that it can be written as (k=ω/c)
A(r, t)≈ˆzµ0 4π
2I0 cosθsin
k`
2 cosθ
exp (ikr−iωt)
kr . (8)
It will now be assumed that the wavelength (λ= 2π/k) and distance to the observation point (r) are so thatkr 1.
f ) Derive an expression for the magnetic field H(r, t) (under the condition kr1).
[Answer: H≈ µ1
0 ik×A.]
g) Calculate the corresponding electric field E(r, t) (under the same assumption as in the previous sub-problem). [Hint: Use Amperes law].
h) Obtain an expression for the time-averaged Poyntings vector, hSit (= 12E×H∗), in terms of the known quantities of the problem.
The radiation pattern of the antenna is defined as dP
dΩ =|hSit|r2, (9)
whereP =R
dΩdP/dΩ =R
hSit·dA is the total power radiated by the antenna.
i) Calculate dP/dΩ and show that dP
dΩ ∝sin2 k`
2 cosθ
tan2θ. (10)
j) Assume that the antenna is small compared to the wavelength, i.e. k`1, and obtain the expression fordP/dΩ in this limit.
k) Argue why your expression fordP/dΩ in the limit k`1 is reasonable. Make a sketch of the radiation pattern in this case. What is this pattern called?
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