Download PHYS-2100 Introduction to Methods of Theoretical Physics Fall 1998 1) 2)

PHYS-2100 Introduction to Methods of Theoretical Physics
Fall 1998
Homework Assignment Due Tuesday, Oct. 20
1) Show explicitly that any function of the form F ( r, t ) = f ( k ⋅ r – ωt ) satisfies the equation
1 ∂2F
2
∇ F – ----2- --------2- = 0
c ∂t
with r = xî + y ĵ + zk̂ and any arbitrary vector k = k 1 î + k 2 ĵ + k 3 k̂ so long as ω = c k = ck .
2) The electric field E = E 0 cos ( kz – ωt )î of a wave is “plane polarized” in the x -direction.
a) Show that this vector can be written as the sum of two vectors E L and E R where
E0
E0
E {L, R} = ------ cos ( kz – ωt )î ± ------ sin ( kz – ωt ) ĵ
2
2
b) Explain why both E L and E R solve the wave equation in free space.
π π 3π π
c) Plot the vectors E L and E R for z = 0 and for t = 0, -------, -------, -------, ----, … . Explain why
4ω 2ω 4ω ω
these are called “left” and “right”-handed circularly polarized waves.
3) This problem desribes a simple waveguide. It is similar to Nettel, Problem 4.13. An electromagnetic wave propagates in a TE mode between a waveguide made of two parallel plates of infinite extent. The plates are made of perfect conductors, and lie parallel to the yz plane, and
separated by a distance ∆x = a . The electric field of the wave is represented by
πx
E = E ( r, t ) ĵ
E ( r, t ) = E 0 sin  ------ exp [ i ( k g z – ωt ) ]
 a
a) Explain why this form satisfies the boundary conditions for the electric field.
b) In what direction does this wave propagate? What is the speed of propagation in terms
of the parameters used to describe E ( r, t ) ? Show that the wavelength is λ g = ( 2π ) ⁄ k g .
c) Show, as we did in class, that to satisfy Maxwell’s Equations in the free space between
1 ∂2E
the plates, E must satisfy the wave equation, that is we need ∇ 2 E – ----2- --------2- = 0 .
c ∂t
λ0 2 1 ⁄ 2
2πc
d) Show that λ g = λ 0 ⁄ 1 –  ------
where λ 0 = --------- .
 2a
ω
e) Find ω c , the cut-off frequency, below which the mode will not propagate. That is, if
ω < ω c , then the wave is exponentially damped in the direction of propagation.
f) Find the cut-off frequency for plates separated by 10cm. Make a plot of the speed of the
wave for ω c ≤ ω ≤ 10ω c .
g) Use Maxwell’s Equations to find the magnetic field B ( r, t ) and show that it also satisfies the appropriate boundary conditions.