A coil having an inductance `L` and a resistance `R` is connected to a battery of emf `E` . Find the time taken for the magnetic energy stored in the circuit to change from one fourth of the steady-state value.

Explanation

`i=i_(0)(1-e^(-(Rt)/(L)))` , `i_(0)=E//R`
`U=(1)/(2)Li^(2)=(1)/(2)Li_(0)^(2)(1-e^(-(Rt)/L))^(2)`
`=U_(0)(1-e^(-(Rt)/(L)))^(2)`
`U_(0)(=(1)/(2)Li_(0)^(2))` : steady-state value of magnetic energy stored
`t=t_(1)` , `U_(1)=(U_(0))/(4)`
`(U_(0))/(4)=U_(0)(1-e^(-(Rt_(2))/L))^(2) implies 1-e^(-(Rt)/(L))=(1)/(2)d`
`.e^((-Rt_(1))/(L))=2 implies (Rt_(1))/L1n e=1n2 implies t_(1)=(L)/(R)1n2`
`t=t_(2)` , `U_(2)=(U_(0))/(2)`
`(U_(0))/(2)=U_(0)(1-e^((-Rt_(1))/(L)))^(2) implies 1-e^((-Rt_(1))/(L))=(1)(sqrt(2))`
`e^((-Rt_(1))/(L))=1-(1)/(sqrt(2))=(sqrt(2)-1)/(sqrt(2))`
`e^((-Rt_(1))/(L))=(sqrt(2))/(sqrt(2-1))=(sqrt(2)(sqrt(2)+1))/((sqrt(2-1))(sqrt(2+1)))(2+sqrt(2))`
`(Rt_(2))(L)1ne=1n(2+sqrt(2))`
`t_(2)=(L)/(R)1n(2+sqrt(2))`
`t_(2)-t_(1)=(L)/(R)[1n(2+sqrt(2))-1n(2)]`
`=(L)/(R)1n((2+sqrt(2)/(2))=(L)/(R)1n((sqrt(2)+1)/(sqrt(2)))`