JEE Main 2025 · Physics · Magnetic Effects of Current · hard

A coil of inductance L and resistance R is connected at t=0 across an ideal battery. The time at which the energy stored in its magnetic field…

Single correct option

A coil of inductance

L

and resistance

R

is connected at

t=0

across an ideal battery. The time at which the energy stored in its magnetic field becomes half of its final (maximum) value is

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Answer: (B) $\dfrac{L}{R}\ln\left(2+\sqrt{2}\right)$

Current grows as

I = I_0\left(1-e^{-t/\tau}\right)

with

\tau = L/R

. Energy

\tfrac12 LI^2

equals half of its final value

\tfrac12LI_0^2

when

I = I_0/\sqrt2

. Thus

1 - e^{-t/\tau} = \dfrac{1}{\sqrt2}

, so

e^{-t/\tau} = 1-\dfrac{1}{\sqrt2}

and

e^{t/\tau} = \dfrac{\sqrt2}{\sqrt2-1} = \sqrt2(\sqrt2+1) = 2+\sqrt2

. Therefore

t = \dfrac{L}{R}\ln\left(2+\sqrt2\right)\approx1.23\,\dfrac{L}{R}

. Choosing

\ln 2

wrongly assumes the current, not the energy, is halved.

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