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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 × 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 = 0.15µF 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