Animated Solution for Physics - Electromagnetic Induction: The time taken for the magnetic energy to reach $25\%$ of its maximum value, when a solenoid of resistance $R$, inductance $L$ is connected to a battery, is

Unraveling the L-R Circuit

Imagine an inductor as a heavy, stubborn water wheel in a river. When you first open the floodgates (close the switch in an electrical circuit), the wheel doesn't instantly spin at top speed. It resists the sudden rush of water. In an $L-R$ circuit, the inductor fights the sudden change in current, causing the current to grow gradually over time rather than jumping instantly to its maximum value.

As this "electrical water wheel" starts spinning, it stores kinetic energy. For an inductor, this is magnetic potential energy. The question challenges us to find the exact moment when this stored energy reaches $25\%$ of its absolute maximum capacity.

The most common trap students fall into is assuming that $25\%$ energy means $25\%$ time, or perhaps $25\%$ current. This is a fatal flaw! The energy $U$ stored in an inductor is not linearly related to the current. It is governed by a quadratic relationship:

At steady state, after a long time has passed, the current reaches its maximum value $I_0$, and the maximum stored energy is:

We are looking for the moment when $U=0.25U_0$. Let's substitute our formulas into this condition:

Notice how beautifully the $\frac{1}{2}L$ terms cancel out on both sides. We are left with a pure relationship between the instantaneous current and the maximum current:

Taking the square root of both sides reveals the hidden truth of the problem: to reach $25\%$ of the maximum energy, the current must reach exactly $50\%$ of its maximum value ($I=\frac{I_0}{2}$). The physics problem has now been translated into a pure math problem!

Now that we know our target current is $\frac{I_0}{2}$, we need to find out when this happens. The growth of current in an $L-R$ circuit connected to a DC battery is governed by the classic exponential equation:

Here, the term $\frac{L}{R}$ is known as the time constant ($\tau$) of the circuit. It dictates how "sluggish" the inductor is. Let's substitute our target current into this master equation:

The $I_0$ terms on both sides cancel out immediately, leaving us with a clean algebraic equation:

Rearranging the terms to isolate the exponential function gives:

To bring the time variable $t$ down from the exponent, we must take the natural logarithm ($\ln$) of both sides. Remember your logarithm rules: $\ln \left(\frac{1}{2}\right)=-\ln 2$.

The negative signs cancel out perfectly. Multiplying both sides by $\frac{L}{R}$ yields our final, elegant answer:

This is the exact moment the magnetic energy hits the $25\%$ mark! Always remember: when dealing with energy in inductors (or capacitors), the relationship to current (or voltage) is squared. Find the linear variable first, and the rest of the problem will unfold naturally.