A capacitance C charged to a potential difference V is discharged by connecting its plates across a resistance R . Determine the heat dissipated during one time constant after the connections are established. (This can be evaluated by computing i^2 R , dt or by assessing the reduction in the energy stored within the capacitor.)

Step-by-step solution

The initial energy stored in the capacitor is U_i = 1 2 CV^2 . During the discharging of a capacitor, the voltage across it at any time t is given by V(t) = V e^ -t/RC . After one time constant t = RC , the voltage becomes V(RC) = V e⁻¹ = V e . The energy stored in the capacitor at this time is U_f = 1 2 C ( V e )^2 = 1 2 CV^2 ( 1 e^2 ) . By conservation of energy, the heat dissipated across the resistor is equal to the loss in the electrical potential energy of the capacitor. H = U_i - U_f = 1 2 CV^2 - 1 2 CV^2 (