A solenoid has an inductance of `10H` and a resistance of `2Omega`. It is connected to a `10V` battery. How long will it take for the magnetic energy to reach `1//4` of its maximum value?

Explanation

`U=1/2Li^2,` i.e. `Upropi^2`
`U` will reach `1/4th` of its maximum value when current is reached half of its maximum value. In `L-R` circuit, equation of current growth is written as
`i=i_0(1-e^(-t//tau)L)`
Here, `i_0=`Maximum value of current
`tau_L=`Time constant`=L//R`
`tau_L=(10 H)/(2 Omega)5s`
Therefore, `i=i_0//2=i_0(1-e^(-t//5))`
or `1/2=1-e^(-t//5)` or `e^(-t//5)=1/2`
or `-t//5=ln(1/2)` or `t//5=ln(2)=0.693`
`:. t=(5)(0.693)` or `-t=3.465s`