Homework Set 5, Physics 3320, Spring 2012
Due Wednesday, Feb 29, 2012 (start of class)
1. RLC circuit. This parallel RLC circuit is driven
I
by a AC voltage V  V0 cos(t) .
A) What is the total complex impedance that the
V
C
R
voltage source sees? Write the answer in the
j
format Z  | Z | e ; i.e. give the magnitude and
phase angle of the impedance. (Hint: It is simplest to begin by computing the
magnitude and phase of 1/Z.)
B) Using complex analysis (the phasor method) solve for the "true" current Itrue
through the voltage source: Itrue = Re[ I ].
C) Make a sketch of the magnitude of the current I vs. frequency. At what frequency
is the current at an extremum?
2. LR filter. The following LR circuit is driven by an AC voltage source
Vin  V0 cos(t) . It can be regarded as a filter which changes a Vin into a Vout.
A) Using complex analysis (the phasor method) solve for the "true" current Itrue in
the circuit: Itrue = Re[ I ].
Vin
R
Vout
B) Solve for the complex ratio Vout / Vin (give magnitude and phase of this ratio) as
a function of frequency. Check that your answer makes sense in the limits 0 and

C) Make a sketch of the magnitude of Vout / Vin vs. frequency. How would describe
this filter? Is it a high-pass filter? Low-pass? Band-pass?
L
Page 2 of 3
3. KE vs Magnetic energy. We have shown that the energy stored in an inductor is
UM 
1 2
L I which is the energy stored in the magnetic field. This ignores the
2
kinetic energy of the conduction electrons, due to their drift velocity.
A) Derive an expression for the ratio of kinetic to magnetic energies
1
2
KE
in an
L I2
inductor. Assume the inductor is a long, single-layer solenoid of radius R, made of
wire of radius r . Assume also that there is one conduction electron per atom, and
the volume per atom is d3 (d is roughly the distance between atoms). If you simplify
the ratio as much as possible, you will find that it depends only on R, r, d, and
fundamental constants such as the mass and charge of the electron. All other factors
such as current I, number or turns N, etc. cancel out. Hint: it may help to notice that
the length of the solenoid is given by  S  N  2 r and the length of the wire in the
solenoid is given by  W  N  2  R .
wire diameter = 2r
R
ls
B) Compute the value of the ratio, using d = 21010 m, r = 0.5 mm, and R = 1 cm.
Are we justified in ignoring the KE of the electrons when computed the energy in a
solenoid?
Page 3 of 3
4. Charging capacitor. A capacitor with circular plates of radius R separated by
distance d (d<<R) is being charged by a steady current I. The plates are sufficiently
close that fringe effects can be ignored.
A) Compute the magnitude of the B-field between the plates at all distances r from
the center of the plates (both r < R and r > R). Sketch the magnitude of this B-field
vs. R.

B) Compute the Poynting vector S (magnitude and direction) on the rim of the
capacitor, between the plates, at r = R. (The "rim" is the ribbon of area at r=R
between the plates. See the diagram.)
 d U 
 is
 d t 
C) Show that the rate at which the capacitor's stored energy is increasing 

equal to the rate at which field energy is entering through the rim
rim
d
I
I
R
rim

 S da