In a discharging RC circuit, how many time constants must elapse for the charge on a capacitor to decrease to 0.1 % of its maximum value?

Step-by-step solution

The charge on a discharging capacitor at any time t is given by the equation: q = q₀ e^ -t/ where q₀ is the maximum initial charge and is the time constant of the circuit. Given that the charge decreases to 0.1 % of its maximum value: q = 0.1 % of q₀ = 0.1 100 q₀ = 10⁻³ q₀ Substituting this into the discharging equation: 10⁻³ q₀ = q₀ e^ -t/ e^ -t/ = 10⁻³ Taking the natural logarithm on both sides: -t = (10⁻³) -t = -3 (10) Using the value (10) 2.303 : t = 3 2.303 t 6.9 Thus, 6.9 time constants must elapse.