# Homework10 Bonus

```BONUS HOMEWORK
Due: June 4, 2015
To be dropped off at my office (If you can not find me in my office, you can
slide your homework below the door)
1. Consider the circuit shown in the figure. The AC source provides a
potential difference given by E = V0 cos ωt
E
r
i2
C
i1
R
(a) Write down equations that would allow you to determine the currents i1 and i2
(b) Solve the equations (Hint: Assume that the currents have the
form i1 = A1 cos(ωt + δ1 ) and i2 = A2 cos(ωt + δ2 ) and solve for
the unknowns A1 , A2 , δ1 and δ2 )
(c) Calculate i1 in the limits ω → 0 and ω → ∞
2. Repeat the previous question if the capacitor in the circuit is replaced
by an inductor of inductance L.
3. A 0.90 m. long FM antenna is oriented parallel to the electric field of
an EM wave. How large must the electric field be to produce a 0.80
mV (rms) voltage between the ends of the antenna? What is the rate
of energy transport per square meter?
1
4. Cosmic microwave background radiation fills all space with an average
energy density of 4 &times; 10−14 J/m3 .
(a) Find the rms value of the electric field associated with this radiation.
(b) How far from a 7.5-kW radio transmitter emitting uniformly in
all directions would you find a comparable value?
5. To detect vehicles at traffic lights, wire loops with dimensions on the
order of 1m are often buried horizontally under roadways. Assume the
self-inductance of such a loop is L = 4.5mH and that it is part of an
LRC with C = 0.15&micro;F and R = 35Ω. The ac voltage has frequency f
and rms voltage Vrms .
(a) The frequency f is chosen to match the resonant frequency f0 of
the circuit. Find f0 and determine what the rms voltage (VR )rms
across the resistor will be when f = f0 .
(b) Assume that f , C, and R never change, but that, when a car is
located above the buried loop, the loop’s self-inductance decreases
by 10% (due to induced eddy currents in the car’s metal parts).
Determine by what factor the voltage (VR )rms decreases in this
situation in comparison to no car above the loop. [Monitoring
(VR )rms detects the presence of a car.]
2
```