The current in a discharging `LR` circuit is given by `I = i_0 e^(-t/tau)` where `tau` is the time constant of the circuit. Calculate the rms current for the period `t = 0` to `t = tau`.

Explanation

Mean square current or rms current for the time interval at t = 0 `tau` is given by
`bar(i^(2))=(1)/(tau)int_(0)^(tau)i_(0)^(2)(e^(-t//tau))^(2)dt = (i_(0)^(2))/(tau)int_(0)^(tau)e^(-2t//tau)dt= (i_(0)^(2))/(tau)[(e^(-2t//tau))/(-2//tau)]_(0)^(tau)`
`=- (i_(0)^(2))/(2)(e^(-2)-e^(0))=(i_(0)^(2))/(2)[1-(1)/(e^(2))]=(i_(0)^(2))/(2)[(e^(2)-1)/(e^(2))]`
rms value of current is
`i_("rms")=sqrt(bar(i^(2)))=root(i_(0))((e^(2)-1)/(2e^(2)))`