ELECTROMAGNETISM STUDY INFO, PROBLEMS, SOLUTIONS 

 

1.  ECE 107 WEBSITE:  http://cem01.ucsd.edu/~vitaliy/courses/ece107/ 

 

 

 

 

 

 

 

Problem 1

You are given a parallel plate transmission line made of perfect electrically conducting plates of

width with the distance between the plates of 1w cm= 0.25d cm= . The space between the plates

is filled with vacuum. The transmission line is short-circuited at 0z = . It is excited by a source of

frequency . 1.8GHzf =

a) Find parameters characterizing the transmission line, i.e. 0', ', ', ', , , , pR L G C Z uβ λ .

b) The voltage at is 1 3z c= − m 41( ) 5 jV z . e Vπ=

− Give expressions describing the voltage and current in the transmission line in

the form of phasors and in the time domain representation.

− Find the reflection coefficient Γ at the point AA′ .

− Find the reflection coefficient Γ at the point BB′ .

AB

Fig. 1

A′B′ z1z 0z =

Problem 2

Consider an infinitesimally thin ring with inner radius and outer radius that resides in the 1r 2rx y− plane and is centered at (Fig. 2(a)). The ring has a surface charge density 0z =

0ss r

ρρ = ,

where 2 2r x y= + . a) Find the electric field ( 0, 0, )x y z= =E on the axis. z Now suppose the ring is rotating counterclockwise (as viewed from above) about its center with a constant angular velocity 0ω (Fig. 2(b)). b) Find the surface current distribution sJ on the ring. (Hint: s sρ=J v , where is the linear

velocity). v

c) Find the magnetic field ( 0, 0, )x y z= =H on the axis generated by the surface current density

z

sJ .

sρ

1r

2r

sρ

1r

2r 0ω

(a) (b)

Fig. 2 Here are some integrals you might find useful.

( )

( )

( )

212 2 2 2 22 2

2 2

2 22 2

3/ 2 2 2 22 22

12 2

3/ 2 2 22 22 2

2 2

2 2

2 2

1 1 1tan ln( )2

1 1ln( )2

tan 1

1 ln( )

x xdx C dx x a x a x a x Ca x a aa x

x xdx a x C dx Ca xa a xa xx xdx x a C xa x a dx C

a xa xdx x a x Ca x x

x dx a x Ca x

−

−

⎛ ⎞= +⎜ ⎟ = + − + + ++ ⎝ ⎠ += + + = ++

++⎛ ⎞= − +⎜ ⎟+ ⎝ ⎠ = − +

++= + + +

+

= + ++

∫ ∫

∫ ∫∫

∫∫

∫ ( )2

2 23/ 2 2 22 2

ln( ) xdx x a x Ca xa x

= + + − +++

∫

Problem 3

Consider an interface between two dielectric half-spaces with material parameters 1 0,ε μ and 2 0,ε μ (Fig. 3(a)). A perpendicularly (TE) polarized plane wave with amplitude 0E is incident

on this interface from medium 1 under an angle θ with respect to the normal to the interface.

a) Consider the case where 0θ ≠ (oblique incidence) and 1 2ε ε≠ (different half-spaces)

− Write expressions for the incident, reflected, and transmitted electric and magnetic fields.

− Write expressions for the incident, reflected, and transmitted power densities (Poynting vectors) including the magnitude and direction.

− What are the surface currents and charges

b) Consider the case where 0θ = (normal incidence) and 1 2ε ε≠ (different half-spaces).

− Give conditions for which the amplitude of the transmitted electric field is greater than that of the incident electric field. Give conditions for which the amplitude of the transmitted magnetic field is greater than that of the incident magnetic field.

− For these cases, what are the ratios between the transmitted and incident power densities (is it greater or smaller than unity)? What are the relations between the incident, reflected and transmitted power densities?

c) Consider the case where 0θ = (normal incidence), 1 2ε ε= (identical half-spaces), but there is an infinitesimally thin sheet inserted on the interface between the half-spaces (Fig. 3(b)). Under an applied electric field, there is a surface current on the sheet that is given by ˆ ˆs sσ× = ×z J z E , where sσ is a given constant (of units 1−Ω ).

− Find the reflection and transmission coefficients for this problem. Hint: Note that the total field in the left half-space ( 0z < ) is given by the sum of the incident and reflected field (i.e. ; 0i r z= + <E E E ), whereas the total field in the right half-space ( ) is given only by the transmitted field (i.e. 0z > ; 0t z= >E E ).

x

(a) (b)

Fig. 3

1 0,ε μ 2 0,ε μ

iθtθz

1 0,ε μ 1 0,ε μ

sJ

x

z