Question

Consider a Sinusoidally Driven LC Electrical Circuit, which Contains an Electric Potential Oscillator, E E, cos(or), an Inductor, L, and a Capacitor, C. Note that an Oscillating Charge,g).Forms on the Capacitor Plates, as well as an Oscillating Current, I(). throughout the Circuit, which is Associated with the Driven Frequency, ω , as Shown. 1. 1(6) gt) E(r) Recall that the Electric Potential Over an Inductor is Given by E , and the dl dr Electric Potential Over a Capacitor is E, =-111 , where the Current is l (1) , and the Relation of the Current in the Circuit to the Charge on the Capacitor Plates, q(a). is ) Recall that the Kirchhoff Law of Electric Potential Balance around the Circuit is E. (r)+E, (r)=E(r). a) Given Kirchhoff's Equation, and the Current/Charge Relationship, Derive the Second Order Differential Equation for the Electrical Circuit, in Terms of the Charge on the Capacitor as a Function of Time, q(t), in Terms of the Given Circuit Parameters, L,C.o,E. b) Determine the Particular Driven Solution which is Directly Related to the Sinusoidal Drive (that is, Neglect the Homogeneous Solution), for the Charge as a Function of Time,g) , in Terms of the Given Parameters, and Determine the Current as a Function of dr Time, 1(r), And Determine a Driving Frequency, ω, Choice which Leads To a Large Amplitude Resonance Charge ), And Current, 10). Response. o) For the Specife Numerical Choice of Parameters E, = 10V, ω-9rad/s,に10H, C=ImF (note that the system of units relate the Volt, V,the Henry, H, the Farad, F, and the Ampere, A, for the Current, which is a Coulomb Per Second, and note that m is milli), Determine the Current Solution, (), and Evaluate the Result for the Current at the Time r-O.1s, or I(0.1). d) Finally, For L Arbitrarily Small, L→0, Determine the Charge, q(,), And Current, 1(), Results, And Compare the Time Evolution of the Charge, 4t), To the Current, I(), And the Potential, E(t), In Terms Of the Phase Differences, , Between Them, Which Can Be Made Clear By Plotting the Results For One Period Of Oscillation, And Describe Physically Why the Result Occurs, And Do the Same In the Opposite Domain Of Arbitrarily Large L, L-o

Answer 1

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