Due Fri. Feb. 20, 2009, 5PM
Homework Assignment 5
General Comments for HWK 5
The field equations of electrodynamics are an essential part of understanding the world
around us. As you’ll soon see, to understand light, radio waves, phenomena like atomic
and molecular structure, and so on, requires that you have a certain level of competence
with these field equations. However, a huge amount of electrical and electronic
engineering relies on the ‘algebraic’ versions. This approach also provides a useful
background for dealing with the full Maxwell’s Equations in a week or so. This problem
set starts the process of using algebra to deal with the calculus of the basic field
equations.
Problem 1. Inductors and RL circuits (total of 15 points).
a) (5 pts) Design a coil that has a 10 microhenry inductance. Use a cylindrical
solenoid geometry, use 20 gauge copper wire. You will have some flexibility in
the choice of coil dimensions and number of turns. Be sure that the coil length is
consistent with the wire gauge. Try to design something that uses an integer
number of turns. List the coil radius, length, number of turns, and whether the
turns are on one layer or multiple layers.
b) (5 pts) Predict the resistance of your coil. Often, you are interested in making a
coil with the lowest resistance. Do you expect any decrease in coil resistance if
you redesign your inductor to have decreased number of turns, while increasing
the solenoid radius so as to keep the solenoid inductance fixed?
c) (5 pts) Draw a closed loop circuit diagram, with an inductor and resistor in series.
Use Kirchoff’s Law (the sum of the voltage drops around a closed loop is zero) to
write an ordinary differential equation for the current in the loop. Solve this
equation subject to the initial condition that the current in the loop starts at some
initial value, I0.
Problem 2. LC circuits (total of 15 points).
a) (10 pts) Draw a closed loop circuit diagram with an inductor and a capacitor in
series. Use Kirchoff’s Law to write an ordinary differential equation for the
current in the loop. Solve this equation subject to the initial condition that the
current in the loop starts at zero, and that the voltage across the capacitor is some
initial value, V0.
b) (5 pts) Make a sketch that has the current through the inductor and the associated
voltage across the inductor. Make a similar sketch for the current through the
capacitor and the voltage across the capacitor. How much electrical power is
being dissipated as a function of time?
Physics 3320
HW5.1
Spring 2009
Due Fri. Feb. 20, 2009, 5PM
Problem 3. Phasor notation (total of 20 points).
a) (5 pts) Ohm’s Law relates current and voltage drop for resistors by a linear
relationship, V=IR. In comparison, capacitors and inductors relate current and
voltage by first time derivatives of the type we saw in the equations above. For
example, a capacitor connects the voltage drop across the capacitor to the amount
of charge stored by Q=CV. Therefore, a single time derivative of both sides yields
a relationship between the current through a capacitor and the time change of the
voltage drop:
I C
dV
dt
A common technique for electronics is to notice that any time dependent quantity
can be written as a sum of sine and cosine time dependences (Fourier
components!) and therefore that it is useful to think about sinusoidal time
dependent quantities. In particular, if we assume that the current and voltage are
both varying at some angular frequency, , in other words:
I  t   I 0 eit
V  t   V0 eit
then we find that a capacitor is described by a relationship that is linear between
the amplitudes of the two sinusoidally varying quantities, I and V:
I 0  iCV0
or
V0  I 0
1
iC
This result looks a lot like Ohm’s Law, V=IR, and has converted the differential
relationship into an algebraic relationship (under the assumption of sinusoidally
varying current and voltage… but HEY! Anything can be written as a sum of such
things.) We say that a capacitor has a COMPLEX IMPEDANCE,
ZC 
1
iC
Find the complex impedance, ZL of an inductor, L.
Physics 3320
HW5.2
Spring 2009
Due Fri. Feb. 20, 2009, 5PM
b) (5 pts.) Redo your work on the series LC circuit from Problem 2. Again, write
down a differential equation for the current as a function of time. This time,
assume that the solution is of the form:
I  t   I0 eit
Plug this assumed form into the differential equation. Under what assumptions is
this ‘guessed’ solution going to work? In particular, what must the frequency be?
How does this result compare to your solution of the differential equation in
Problem 2?
As a note: You can finish the solution by asking what value of I0 (possibly a
complex number) is required so that the Real part of the assumed form satisfies
the initial conditions. You don’t need to do that here.
c) (10 pts) Draw a closed loop circuit with a resistor, capacitor, and inductor in
series. Again, use Kirchoff to create an equation for the current flow around this
circuit. Using the assumed form from part b), reduce the differential equation to
algebra. What must the ‘frequency’ be if the assumed form is to satisfy the
equation? Argue for a simplified form for the frequency that comes from
assuming that the resistance is ‘small’ and state what ‘small’ means… Small
compared to what? Interpret your result for the ‘frequency’. What type of time
dependence does it predict?
Physics 3320
HW5.3
Spring 2009