Here is a list of some old exam questions I have given over the past few years in Electromagnetics. These are intended solely to provide a flavor for the kind of questions I ask, and the format I follow.  

 

Instructions: Work through the following problem neatly and professionally, and without collaboration of any kind. Please put your name on every page of your work. A perfect score on this exam is 25 points. You may not consult any notes or text. You may not use a calculator. Please be very clear in the presentation of your work.

 

 

Lossy Plane Waves (12 points).

Starting from Maxwell's equations, derive the wave equation for a lossy material:


 

Solve the equation for plane wave type solutions.

 

Wave Equation Derivation (9 points):

Starting with Maxwell's equations and the simple material relations derive the scalar wave equation for an arbitrary component of the magnetic field, H.

 


What is c equal to? What does it mean physically?

Hint: Your written comments, and explanations will be as important to your grade as the mathematics of the derivation. Please be verbose, and show me not only that you know the math, but also that you understand what is behind the math.

 Div, Grad, Curl and all that (6 points):

(a)    determine the gradient of:

 


(b)   determine the divergence of:

 


(c)    determine the curl of

 


Reflection off a sheet of glass (5 points):

An electric field has a linear polarization parallel to a glass surface, and it is incident on the glass surface at an angle qi. The ratio of the reflected electric field component to the incident electric field is:

 


 

The ratio of the transmitted electric field component to the incident electric field is:

 


 

Snell's law relates qt to qi:

 

 


 

(d)   Tell me what each quantity is in each of these three equations. What are the units, what conventions follow, what are typical numerical values. Use pictures if necessary.

 

 

 

(e)    Show that power is conserved in this situation.

 

 

 

(c) What is the angle of minimum reflection?

 

 

 

Plane Waves (9 points):

A plane wave of 1 GHz is travelling in free space, and encounters a shiny metal surface at a 45° angle of incidence. A sheet of glass (h= 250, note the ratio of 250/377 is about 2/3) is located as shown in the drawing, 18.5 feet away and at 45° relative to the metal sheet.

 

Write down the expressions for the E field, the H field and the Poynting vector everywhere. You will have to introduce a coordinate system, and assume a polarization that makes sense.

 

Calculate any Standing Wave Ratios (SWRs) and discuss any nodes present.

 

In class we described plane waves as being infinite in extent. In this problem, that assumption is challenged significantly. Explain why. In writing this problem I am visualizing a plane wave with square aperture of about 6 feet by 6 feet. Discuss this difference in some detail both qualitatively and rigorously. What principle allows us to consider plane waves of finite aperture?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


 


Transmission Lines (12 points)

A 50 Ohm transmission line of semi-infinite length is connected to a 75 Ohm transmission line of length L. The other end of the 75 Ohm line is connected to a second semi-infinite 50 Ohm line.

 

What are the voltage reflection and voltage transmission coefficients at points 1 and 2?

What are the power reflection and voltage transmission coefficients at points 1 and 2?

 

Suppose an input sinusoidal wave of frequency w is moving in A from left to right. What is the total voltage in C?

 

In other words, what is the overall, frequency dependent transmission coefficient for the structure? (Why would it be frequency dependent?)


Hint: you will need the sum:

 

 

For ½x½<1.

 

Relate your answer to the generic feedback equation you learned about in electronic circuits.