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Problem Set 12 Solutions

# Problem Set 12 Solutions - MASSACHUSETTS INSTITUTE OF...

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MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Fall 2007 Problem Set 12 Solutions Problem 1: LC Circuit (a) Initially, the capacitor in a series LC circuit is charged. A switch is closed, allowing the capacitor to discharge, and after time T the energy stored in the capacitor is one- quarter its initial value. Determine L if C and T are known. If the energy is at ¼ of it’s initial value then the voltage has fallen in half. That is, ( ) ( ) 2 0 0 2 1 9 cos 2 3 3 V T V T V T T L T C LC π π ω ω ω π = = = = = = (b) A capacitor in a series LC circuit has an initial charge and is being discharged. Find, in terms of L and C , the flux through each of the N turns in the coil at time t , when the charge on the capacitor is Q ( t ). 0 Q The flux depends on the current and inductance: LI N = Φ . So we just need to write the current in terms of the charge. By conservation of energy 2 2 2 1 0 2 2 2 Q C LI Q C + = . So, ( ) ( ) 2 2 0 Q Q t LI L N N LC Φ = = (c) An LC circuit consists of a 90.0-mH inductor and a 1.000- F μ capacitor. If the maximum instantaneous current is 0.100 A, what is the greatest potential difference across the capacitor? By conservation of energy, 2 2 1 1 max max 2 2 LI CV = , so ( ) ( ) ( ) max max 90 mH 0.1 A 30 V 1.0 μ F L V I C = = = Problem 2: Solenoid Consider a long solenoid of length D and radius a with N turns, placed along the z-axis (it is centered at z=0). It is hooked in series with a capacitor C , a resistor R , and a power supply that is driving the circuit on resonance. At time t = 0 the current flowing through the solenoid is a maximum, I 0 . Problem Set 12 Solutions p. 1 of 7

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