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 19 cos 23 3 V T VT V T T L TC LC ππ ωω 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 22 2 1 0 2 QC L IQC += . So, () 2 2 0 QQ t LI L NN L C 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, 11 max max LI CV = , so max max 90 mH 0.1 A 30 V 1.0 μ F L VI 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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Problem 2: Solenoid continued (a) At this moment (t = 0), calculate the magnetic field B 0 everywhere inside the solenoid. Show your work. To calculate the magnetic field all we need is the current. At t = 0, I = I 0 . Using Ampere’s law (you should draw a picture) we find 00 NI BD NI B D μ =⇒ = . (b) Calculate the self inductance L of the solenoid. Show your work. 22 2 0 NI a N N BA a L DI μπ π Φ Φ= = ⇒ = = D (c) Write an equation for the time dependence of the magnetic field The circuit is being driven is resonance, so the drive frequency 1 LC ω = where C is given and L is from (b). The magnetic field is a max at t = 0 (because that is when the current is largest) so () 0 cos B Bt = , where B 0 is from (a). Note that it is cosine and not sine because we are told the current is largest at t = 0. (d) Write an equation for the time dependent voltage across the power supply Since we are in resonance, the impedance is just R and we are in phase with the current, which as we said above is a cosine: 0 cos S VI R t = Problem 3: Filters When describing a filter’s behavior, one typically gives the “corner frequency” or “3 dB point,” the frequency at which the power is cut in half (the voltage is cut by 2 , which corresponds to 3 dB).
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This homework help was uploaded on 04/07/2008 for the course 8 8.02 taught by Professor Hudson during the Fall '07 term at MIT.

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Problem Set 12 Solutions - MASSACHUSETTS INSTITUTE OF...

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