A 1.2 F capacitor with an initial stored energy of 0.43 J is discharged through

Question

A 1.2 F capacitor with an initial stored energy of 0.43 J is discharged through a 2.3 M resistor.

(a) What is the initial charge on the capacitor?

___________C

(b) What is the current through the resistor when the discharge starts?

__________A

(c) Find an expression that gives, as a function of time t, the potential difference VC across the capacitor. (Use the following as necessary: t. Assume S.I. units. Do not include units in your answer.)

Vc=_____________V

(d) Find an expression that gives, as a function of time t, the potential difference VR across the resistor. (Use the following as necessary: t. Assume S.I. units. Do not include units in your answer.)

Vr=_____________V

(e) Find an expression that gives, as a function of time t, the rate at which thermal energy is produced in the resistor. (Use the following as necessary: t. Do not include units in your answer.)

P=______________W

Explanation / Answer

a) C = 1.2 F & E = 0.43 J

=> initial stored energy of capacitor E= 1/2*C*V^2

=> V^2 = 2*E/C = 2*0.43/(1.2*10^-6) =716666.666

=> V =sqrt(716666.666) = 846.56V

initial charge on the capacitor Q= C*V = 1.2*10^-6* 846.56 =1015.872*10^-6 C

b) At the first instant, capacitor acts as a conductor

=> Io =V/R =  846.56/(2.3*10^6) =0.000368 A

c)

For the process of discharging a capacitor C

differential equation is R*dQ/dt + Q/C =0 and Q = C*Vo at t=0

=> Using the boundary condition

=> Q =C*Vo*exp(-t/(R*C)

Vc = Q/C = Vo*exp(-t/(R*C)

d) Vr =Vo - Vc = Vo-Vo*exp(-t/(R*C) = Vo*(1-exp(-t/(R*C) )

e) rate at which thermal energy is produced in the resistor P= I^2*R

and

current through the resistor I =Vo/R*exp(-t/(R*C))

=> P = { Vo/R*exp(-t/(R*C)) } ^2* R

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