Animated Solution for Physics - Electrostatics: A battery is used to charge a parallel plate capacitor till the potential difference between the plates becomes equal to the electromotive force of the battery. The ratio of the energy stored in the capacitor and the work done by the battery will be

The process of charging a capacitor is one of the most fascinating and fundamental concepts in electrostatics and circuit theory. It is a beautiful interplay of energy, work, and thermodynamics. Let's embark on a journey to understand exactly what happens when a battery breathes life into an uncharged capacitor.

Visualizing the Circuit

Imagine you have a simple electrical circuit. On one side, you have a battery, a powerhouse of electromotive force (EMF) denoted by $V$. On the other side, you have a parallel plate capacitor, initially completely uncharged and devoid of any stored energy.

The moment you close the switch, the circuit comes alive. The battery acts like an electrical pump. It begins to pull electrons from one plate of the capacitor and pushes them onto the other plate. This movement of charge creates a growing potential difference across the capacitor's plates.

Pumping Charge

Let's look at this process from the perspective of the battery. The battery's sole purpose in life is to maintain a constant potential difference $V$ across its terminals and to push charge through the circuit against the building electrical pressure of the capacitor.

If the battery successfully pumps a total charge $q$ from its negative terminal to its positive terminal, how much work has it done? The definition of electrical work is straightforward: it is the product of the charge moved and the potential difference across which it is moved.

Therefore, the total work done by the battery, let's call it $W$, is given by the elegant equation:

This represents the total energy supplied by the battery to the entire circuit.

Storing Energy

Now, let's shift our focus to the capacitor. As the battery pumps charge, the capacitor plates accumulate $+q$ and $-q$. This separation of charge creates an electric field between the plates, and within this electric field, energy is stored.

However, the capacitor doesn't charge instantly. The first tiny bit of charge $dq$ is easy to move because the capacitor is empty. But as more charge accumulates, the capacitor pushes back with its own potential difference $v=\frac{q}{C}$. The battery has to do increasingly more work to push each subsequent bit of charge.

When the capacitor is fully charged, its potential difference matches the battery's EMF, $V$. The total electrostatic potential energy $U$ stored in the capacitor is the integral of the work done to build up this charge. This gives us the famous formula:

Alternatively, using $q=CV$, this can also be written as $U=\frac{1}{2}C{V}^2$.

A Beautiful Cancellation

The question asks us for a very specific relationship: the ratio of the energy stored in the capacitor to the work done by the battery.

We have our two master equations: 1. Energy stored: $U=\frac{1}{2}qV$ 2. Work done: $W=qV$

Let's set up the ratio:

Substituting our expressions into the ratio:

Notice how beautifully the $qV$ terms cancel out. The specific values of the charge, the capacitance, or the voltage don't matter at all. The physics distills down to a pure, elegant constant:

The Heat Loss Mystery

This result is mathematically simple, but physically profound. The battery did $W$ amount of work, but the capacitor only stored $\frac{1}{2}W$.

Where did the other $50\%$ of the energy go? Did it just vanish?

In physics, energy is never lost; it only changes form. The missing half of the energy was dissipated as heat! When the charge $q$ flowed through the connecting wires, it encountered electrical resistance. Even if the wires are incredibly good conductors, the rapid surge of current generates Joule heating ($I^{2}R$ loss).

Remarkably, the total heat dissipated is always exactly half of the work done by the battery, regardless of the actual resistance of the wires. This $50\%$ efficiency is a universal signature of charging a capacitor from a constant voltage source. It is a brilliant reminder that in the real world, moving energy around always comes with a thermodynamic tax!