Discharging a Capacitor in an RC Circuit Consider a capacitor of capacitance C that is being discharged through a resistor of resistance R as shown in figure (c). (a) A capacitor in series with a resistor, (b) When the switch is thrown to position a, the capacitor begins to charge up. switch, and battery. + E % E 96 www %0 (c) When the switch is thrown to position b, the capacitor discharges. R www @ 8 22 96 (a) After how many time constants is the charge on the capacitor one-fourth its initial value? SOLUTION Conceptualize Study the figure and imagine throwing the switch to position b as shown in figure (c). Upon doing so, the capacitor begins to discharge. Categorize We categorize the example as one involving a discharging capacitor and use the appropriate equations. Analyze (Use the following as necessary: t, r, C, and R.) Substitute q(t) = into g(t)- Qe-t/RC lo VC Take the logarithm of both sides of the equation and solve for t -In(4) CH t RC (b) The energy stored in the capacitor decreases with time as the capacitor discharges. After how many time constants is this stored energy one-fourth its initial value? SOLUTION (Use the following as necessary: t, T, C, and R.) Q Use UE 2C (1) u(t) and g(t) 4 2C Substitute U(t)= -In(4) t= = 9 2C 2C l VC, - -4(2) 2C Q-2U/RC to express the energy stored in the capacitor at any time t: into equation (1): Take the logarithm of both sides of the equation and solve for t: 20 RC Finalize Notice that because the energy depends on the square of the charge, the energy in the capacitor drops more rapidly than the charge on the capacitor. EXERCISE Suppose the same type of series circuit as in the example has R = 9.50 x 100, C= 5.00 F, and an initial voltage across the capacitor of 5.00 V. How many electrons are on the negative plate of the capacitor after 14.235 seconds? (Enter the number of electrons that constitute the net negative charge on the plate.) Hint electrons