A 158 mF capacitor with an initial stored energy of 0198 J i

A 1.58 mF capacitor with an initial stored energy of 0.198 J is discharged through a 1.31 M Ohm resistor. (a) What is the initial charge on the capacitor? (b) What is the current through the resistor when the discharge starts? At time t = 799 s, find (c) the potential difference V_C across the capacitor, (d) the potential difference V_R across the resistor, and (e) the rate at which thermal energy is produced in the resistor. (a) Number Units (b) Number Units (c) Number Units (d) Number Units (e) Number Units

Solution

The initial charge is calculated as follows:

     U = Q²/2C

    Q = sqrt[2UC] = sqrt[2*0.198*1.58x10^-3] = 2.50e-2 C = 25x10^-3 C

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Current is calculated as follows:

     I = Q/RC = 25x10^-3 C / 1.31x10⁶* 1.58x10^-3 = 1.2085e-5

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The potential difference:

     V_C = [Q/C]*e^-t/RC

         = [25x10^-3 /1.58x10^-3 ]*e^-799/1.31x10⁶* 1.58x10^-3   

           = 10.75 V

          = 10.8 V

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The potential difference:

     V_R = [Q/C]*e^-t/RC

         = [25x10^-3 /1.58x10^-3 ]*e^-799/1.31x10⁶* 1.58x10^-3   

           = 10.75 V

          = 10.8 V

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The rate of thermal energy:

    P = [Q²/RC²] * [e^-2t/RC]