An inductance L and a resistance R are connected in series with a battery of emf epsilon. Find the maximum rate at which the energy is stored in the magnetic field.

Explanation

The energy stored in the magnetic field at tiem t is ` U = (1)/(2) Li^2 = (1)/(2) Li_0^2 (1-e^9-t/tau)^2`
The rate at which the energy is stored is `P = (dU)/(dt) = Li_0^2(1-e^(-t/tau) (-e ^(-t /tau)) (-(1)/(tau))` `= (Li_0^2)/(tau) (e^(-t/tau -e^(-2t/tau)`
This rate will be maximum when (dP)/(dt) = 0 or, (Li_0^2)/(tau) (-(1)/(tau)e^(-t/tau)+(2)/(tau)e^(-2t/tau)) = 0` or, `e^(-1=t/tau) = (1)/(2)` putting in (i), `P_max = (Li_0^2)/(tau)((1)/(2) - (1)/(4))` =`(Lepsilon^2)/(4R^2(L/R) = (epsilon^2)/(4R)