Question

In an oscillating LC circuit, L = 4.15 mH and C = 2.99 ?F. At t = 0 the charge on the capacitor is zero and the current is 1.72 A. (a) What is the maximum charge that will appear on the capacitor? (b) At what earliest time t > 0 is the rate at which energy is stored in the capacitor greatest, and (c) what is that greatest rate?

Answer 1

charge q at time t is

q = Q sin wt

at t= 0

current I = dq/dt

I = d(Qsin wt)/dt

I = Q w Cos Wt

I = Qw

Q = I/W = I * sqrtLC)

Q = 1.72 * sqrt(4.15*10^-3 * 2.99*10^-6)

Q = 1.91 *10^-4 C

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part B:

rate at which Eenrgy is stored is

dU/dt = d/dt (q^2/2c)  

dU/dt = d/dt *(Q^2 W Sin 2Wt)/2C

the rate of change in greatest when sin 2w t =1

2 wt = sin (pi/2)

t = (pi/4) sqrt(LC)

t = (3.14/4) * sqrt(0.00415 * 2.99 *10^-6)

t = 8.744 *10^-5 secs

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part C:

greatest rate of energy is

dU/dt = d/dt*(Q^2 sin^2wt/2c)

dU/dt = Q^2/2C*sqrtLC)

dU/dt = (1.91*10^-4)^2/(2 * 2.99 *10^-6 * sqrt(0.00415* 2.99*10^-6))

dU/dt = 54.76 watts