Question

A solenoid of inductance $L$ and resistance $R$ is connected to a battery. The time taken for the magnetic energy to reach 1/4 of its maximum value is:

• A. $\frac{L}{R}ln\left(1\right)$
• B. $\frac{L}{R}ln\left(2\right)$
• C. $\frac{L}{R}ln\left(3\right)$
• D. $\frac{L}{R}ln\left(4\right)$

Answer 1

The correct option is B $\frac{L}{R}ln\left(2\right)$

Given : A solenoid of inductance LL and resistance RR is connected to a battery. The time taken for the magnetic energy to reach 1/4 of its maximum value is:

Solution :

Initial magnetic energy=$\frac{1}{2}L{i}^2$

Final magnetic energy =$\frac{1}{4}$ of initial magnetic energy

=$\frac{1}{4}$$\left(\frac{1}{2}L{i}^2\right)$

For final energy to become $\frac{1}{4}$ of initial magnetic energy final current have to become $\frac{1}{2} of initial current .

We have a relation $i=i_o(1-e^{\frac{-Rt}{L}})$

t= time , L=inductance , R= resistance

Substituting the value of current here we get $\frac{i_o}{2}=i_o(1-e^{\frac{-Rt}{L}})$

$\frac{1}{2}=(1-e^{\frac{-Rt}{L}})$

$\frac{-1}{2}=(-e^{\frac{-Rt}{L}})$ (taking $ln$ both sides)

$ln\left(\frac{1}{2}\right)=\frac{-Rt}{L}$

$t=\frac{L}{R}ln\left(2\right)$

The correct opt: B