Question

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

Answer 1

Given that,

$L=10H$, $R=2\Omega$,

$V=10V$

Now, the maximum energy

$i_0=\frac{V}{2}$ , $i_0=\frac{10}{2}$

$i_0=5A$, $E_0=\frac{1}{2}L{i}_0^2$

$E_0=\frac{1}{2}\times 10\times 5\times 5$

$E_0=125J$

Now, the energy is ¼ of maximum energy

$E=\frac{E_0}{4}$

$E=\frac{125}{4}$

Now, the current is

$E=\frac{1}{2}L{i}^2$

$\frac{125}{4}=\frac{1}{2}\times 10\times i^2$

$i^2=6.25$

$i=2.5A$

Now, the time taken

$i=i_0(1-e^{\frac{-Rt}{L}})$

$2.5=5(1-e^{\frac{-Rt}{L}})$

$e^{\frac{-Rt}{L}}=0.5$

$\frac{-Rt}{L}={\log }_{e}0.5$

$t=\frac{0.693\times 10}{2}$

$t=3.465s$

Hence, the time is $3.465s$