For a discharging LR circuit, the current is described by i = i₀ e^ -t/ , where represents the circuit's time constant. Determine the rms current over the interval from t = 0 to t = .

Step-by-step solution

The root mean square (rms) current over the time interval from t = 0 to t = is given by: I_ rms = 1 ₀^ i^2 dt Given i = i₀ e^ -t/ , we square the current: i^2 = i₀^2 e^ -2t/ Now, integrate i^2 from t = 0 to t = : ₀^ i^2 dt = ₀^ i₀^2 e^ -2t/ dt ₀^ i^2 dt = i₀^2 [ e^ -2t/ -2/ ]₀^ ₀^ i^2 dt = - i₀^2 2 (e⁻² - e⁰) = i₀^2 2 (1 - e⁻²) Substitute this back into the rms formula: I_ rms = 1 i₀^2 2 (1 - e⁻²) I_ rms = i₀^2 2 ( 1 - 1 e^2 ) I_ rms = i₀^2 2 ( e^2 - 1 e^2 ) I_ rms = i₀ e e^2 - 1 2