A capacitor C is charged to an initial potential of 60.0 V, with an initial char

Question

A capacitor C is charged to an initial potential of 60.0 V, with an initial charge of Q_o. It is in a circuit with a switch and an inductor with inductance L = 4.27 times 10^-2 H and resistance R_1. At t=0, the switch is closed, and the curve below shows the potential V across the capacitor as a function of time t. Calculate the energy in the circuit after a time of 11 periods. Note that the curve passes through a grid intersection point. Calculate the time required for 86% of the initial energy to be dissipated.

Explanation / Answer

A.

Information from the given figure:

The sine wave peaks at t0 = 0 sec and t1 = 0.8 ms

This time period corresponds to 11 periods

Time period = T = 0.8 ms/11 = 7.27*10^-5 sec

angular frequency = w = 2*pi/T = 2*pi/(7.27*10^-5) = 86426.21 rad

the peak at time t1 has a height of V1 = 40 V

at t0 = 0 sec, V0 = 60 V

Voltage in a damped oscillating circuit,

V1 = V0*[exp (-R*t/2L)]*cos wt

rearranging the equation:

R = [-2*L*ln (V1/V0)]/t1

R = [-2*5.31*10^-2*ln (40/60)]/(0.0008)

R = 53.83 ohm

R/2L = 53.83/(2*5.31*10^-2) = 506.87 sec^-1

first resonance frequency is given by:

w = sqrt (w0^2 - (R/2L)^2)

w0 = sqrt (w^2 + (R/2L)^2)

w0 = sqrt (86426.21^2 + 506.87^2)

w0 = 86427.70

Now for resonance frequency

w0 = sqrt (1/LC)

C = 1/(w0^2*L) = 1/(86427.7^2*5.31*10^-2)

C = 2.52*10^-9 F = 2.52 nF

Now the initial energy of capacitor is given by:

U = 0.5*C*V^2 = 0.5*2.52*10^-9*60^2

U = 4.54*10^-6 J

Now the capacitor's energy decay according to

Ut = U0*exp (-R*t/L)

After thirteen periods

t13 = 7.27*10^-5*13 = 0.0009451 sec

Ut = 4.54*10^-6*exp (-53.83*0.0009451/(5.31*10^-2))

Ut = 1.74*10^-6 J

B.

in order to 88% of the total energy dissipated, we need 12% of the initial energy remaining

U88 = U0*exp(-R*t88/L)

where

U88 = (1 - 0.88)*U0 = 0.12*U0

0.12 = exp (-R*t88/L)

t88 = -(L/R)*ln 0.12 = -(5.31*10^-2/53.83)*ln 0.12

t88 = 0.00209 sec

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.