(a) Find the capacitance C , the resistance R , and the initial charge Q 0 on th

Question

(a) Find the capacitance C, the resistance R, and the initial charge Q0 on the capacitor. [Hint: You will need to solve three equations simultaneously for the three unknowns. You can find both the initial current and the time constant from the graph.]

R=22.446 ?

Q0= 2.4386e-4C

A charged capacitor is discharged through a resistor. The current I(t) through this resistor, determined by measuring the voltage VR(t) = I(t)R with an oscilloscope, is shown in the graph. The total energy dissipated in the resistor is 2.60 10-4 J. Find the capacitance C, the resistance R, and the initial charge Q0 on the capacitor. [Hint: You will need to solve three equations simultaneously for the three unknowns. You can find both the initial current and the time constant from the graph.] At what time is the stored energy in the capacitor 4.60 10-5 J?

Explanation / Answer

The current is defined by i0*(e^-(t/?)) where ? is the time constant = RC (time constant is the time for the current to dissipate to 0.368 of it max value)

Reading from the graph the value of t when I = 36.8mA is approximately 12m/s

So 0.012 = R*C so C = 0.012/22.446 = 5.346x10^-4F

The total energy stored is the energy dissipated

U = 1/2*C*V^2

= 1/2*5.346x10^-5*(V)^2 =2.6x10^-4

v= sqrt[(2.6x10^-4)/0.5 * 5.346x10^-4 ]

v=0.9859V

R=v/i

=0.9859/0.368= 2.67

b) The energy is when the voltage is sqrt(.565)*V0 = 0.751V0

So when e^-(t/?) = 0.751

so -t/0.012 = ln(0.751)

t = -0.012*ln(.751)

= 0.003435s = 3.43ms

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