A coil having an inductance `L` and a resistance `R` is connected to a battery of `emf epsilon`. Find the time elapsed before (a) the current reaches half its maximum value, (b) the power dissipated in heat reaches half its maximum value and © the magnetic field energy stored in the circuit reaches half its maximum value.

Explanation

Maximum Current `=(E/R)`
`In steady state magnetic field energy stored
`(E^2)/(2R^(2))L`
The fourth of steady energy `=(1/8)L(E^2)/R^2)`
`one half steady energy =(1/4)L(E^2)/(R^2) (1-e^(-(t_1)R//L)^2`
`implies1-e^(tR//L)=(1/2)`
`implies t_(1)(R/L)=In2`
`Again (1/4)L((E^2)/(R^2))=(1/2)L((E^2)/(R^2))(1-e^(-(t_2)R//L)^2`
`implies e^(-(t_2)R//L)=(sqrt(2)-1)/(sqrt(2))=(2-(sqrt(2)))/2`
`implies t_1=(tau)In((1)/(2-(sqrt(2)))+In2`
`so, t_2-t_1=(tau)InI1)/(2-(sqrt(2)))` .