Animated Solution for Physics - Electromagnetic Induction: A solenoid has an inductance of $10\text{ H}$ and a resistance of $2\Omega$. It is connected to a $10\text{ V}$ battery. How long will it take for the magnetic energy to reach $1/4$ of its maximum value ?

The beauty of an inductor lies in its stubbornness. It is the electrical equivalent of mass—it resists any change in the flow of current. When you connect a battery to an L-R circuit, the current doesn't just jump to its maximum value instantly. Instead, it fights its way up, building a magnetic field as it goes. This magnetic field is where the energy is stored.

The magnetic energy $U$ stored in an inductor is given by the elegant formula:

Notice the squared term? This is the key to unlocking the problem. The energy is directly proportional to the square of the current ($U\propto i^2$). The question asks us to find the time when the energy reaches exactly $\frac{1}{4}$ of its maximum possible value.

If the energy is $\frac{1}{4}$ of the maximum, what must the current be? Since $(\frac{1}{2}{)}^2=\frac{1}{4}$, the current must be exactly half of its maximum steady-state value ($i=\frac{i_0}{2}$).

Now that we know we are looking for the moment when the current reaches half its maximum, we need to understand how current grows over time. In an L-R circuit, the current follows an exponential growth curve governed by the equation:

Here, $i_0$ is the maximum current (which would be $\frac{V}{R}$), and ${\tau }_L$ is the inductive time constant. The time constant dictates how fast the circuit responds. It is simply the ratio of inductance to resistance:

We now have all the pieces of the puzzle. We substitute our target current $i=\frac{i_0}{2}$ and our time constant ${\tau }_L=5$ into the growth equation:

The beauty of this step is that the maximum current $i_0$ cancels out perfectly from both sides. We don't even need to know the battery voltage to solve for the time! This leaves us with a pure exponential equation:

Rearranging the terms, we isolate the exponential part:

To bring the time variable $t$ down from the exponent, we take the natural logarithm ($\ln$) on both sides. Remember that $\ln (e^x)=x$ and $\ln (\frac{1}{2})=-\ln (2)$:

Finally, we multiply by 5 to isolate $t$. Using the standard approximation $\ln (2)\approx 0.693$:

And there we have it! It takes exactly $3.465$ seconds for the inductor to build up enough magnetic field to hold one-fourth of its maximum energy capacity.