Animated Solution for Physics - Electromagnetic Induction: The self-induced emf of a coil is 25 V. When the current in it is changed at uniform rate from 10 A to 25 A in 1s, the change in the energy of the inductance is

Imagine you are trying to push a heavy boulder. At first, it strongly resists your push due to its physical inertia. An inductor in an electrical circuit behaves exactly like that boulder, but instead of physical mass, it possesses 'electrical inertia'. When you try to change the current flowing through it, it fights back by inducing a voltage—an electromotive force (emf)—that directly opposes your action. This beautiful phenomenon is known as self-induction.

In our problem, we are told that the current is ramping up from $10\text{ A}$ to $25\text{ A}$ in exactly $1\text{ s}$. The coil fights this rapid change by generating a self-induced emf of $25\text{ V}$. But notice what is missing? The problem doesn't tell us the actual 'mass' of our electrical boulder—the inductance $L$.

To find this hidden parameter, we invoke Faraday's Law of Induction, specifically tailored for self-inductance:

Let's plug in the values we know. The induced emf $\epsilon$ is $25\text{ V}$. The change in current, $dI$, is the final current minus the initial current, which is $25\text{ A}-10\text{ A}=15\text{ A}$. The time interval, $dt$, is $1\text{ s}$. Substituting these into our equation gives:

Solving for $L$, we find that the inductance of our coil is $L=\frac{25}{15}=\frac{5}{3}\text{ H}$.

Now we enter the second phase of our journey. As we force the current to increase against the coil's opposition, we are actively doing work. Where does this work go? It doesn't just vanish; it gets stored in the invisible magnetic field blooming around the coil.

The magnetic potential energy $E$ stored in an inductor at any given instant is given by the elegant equation:

We are asked to find the change in energy, $\Delta E$. This is simply the final energy minus the initial energy:

Factoring out the common terms, we get a cleaner expression to work with:

Let's substitute our known values into the energy change formula. We have $L=\frac{5}{3}\text{ H}$, $I_2=25\text{ A}$, and $I_1=10\text{ A}$.

A word of caution: A classic trap many students fall into is calculating the change in current first and then squaring it. Remember, $(I_2)^2 - (I_1)^2 eq (I_2 - I_1)^2$. You must square the currents individually!

Calculating the squares, $25^2=625$ and $10^2=100$. Their difference is $625-100=525$. Now, we just need to multiply this by our constants:

Evaluating this fraction gives us exactly $437.5\text{ J}$. The magnetic field has absorbed a massive $437.5$ Joules of energy in just one second. This confirms that option (a) is the correct answer.