MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Fall 2007 Problem Set 12 Solutions Problem 1: LCCircuit (a) Initially, the capacitor in a series LCcircuit is charged. A switch is closed, allowing the capacitor to discharge, and after time Tthe energy stored in the capacitor is one-quarter its initial value. Determine Lif C andT are known. If the energy is at ¼ of it’s initial value then the voltage has fallen in half. That is, ()()200219cos233VTV TVTTLTCLCππωωωπ==⇒=⇒==⇒=(b) A capacitor in a series LCcircuit has an initial charge and is being discharged. Find, in terms of Land C, the flux through each of the Nturns in the coil at time t, when the charge on the capacitor is Q(t). 0QThe flux depends on the current and inductance: LIN=Φ. So we just need to write the current in terms of the charge. By conservation of energy 22210222QCLIQC+=. So, ( )()220QQ tLILNNLC−Φ ==(c) An LCcircuit consists of a 90.0-mH inductor and a 1.000- Fμcapacitor. If the maximum instantaneous current is 0.100 A, what is the greatest potential difference across the capacitor? By conservation of energy, 2211maxmax22LICV=, so ()()()maxmax90 mH0.1 A30 V1.0 μFLVIC===Problem 2: Solenoid Consider a long solenoid of length Dand radius awith Nturns, 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, I0. Problem Set 12 Solutions p. 1 of 7
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