2.14. ENERGY STORED IN A CAPACITOR

A capacitor, as we have seen above, is a system of two conductors with charge Q and –Q. To determine the energy stored in this configuration, consider initially two uncharged conductors 1 and 2. Imagine next a process of transferring charge from conductor 2 to conductor 1 bit by bit, so that at the end, conductor 1 gets charge Q. By charge conservation, conductor 2 has charge –Q at the end. In transferring positive charge from conductor 2 to conductor 1, work will be done externally, since at any stage conductor 1 is at a higher potential than conductor 2.

To understand how a capacitor stores energy, imagine starting with two uncharged plates. As you begin to add positive charge to one plate (conductor 1) from the other uncharged plate (conductor 2), you must do work against the electric field created by the charge on conductor 1. The more positive charge you move onto conductor 1, the higher the potential of that conductor becomes. This process of moving charge requires energy, which is then stored in the capacitor. When all the charge is transferred, conductor 1 has charge Q, and conductor 2 has charge -Q.

Think of this process like pumping air into a balloon. As you blow air into the balloon (charging the conductor), you have to work against the balloon’s elasticity (the electric field) to inflate it. The energy you use while pumping the air is stored in the stretched rubber of the balloon. Similarly, the energy used to transfer charge into the capacitor is stored as electric potential energy.

To calculate the total work done, we first calculate the work done in a small step involving transfer of an infinitesimal (i.e., vanishingly small) amount of charge. Consider the intermediate situation when the conductors 1 and 2 have charges Q′ and –Q′ respectively. At this stage, the potential difference V′ between conductors 1 to 2 is Q′/C, where C is the capacitance of the system. Next imagine that a small charge dQ′ is transferred from conductor 2 to 1. Work done in this step (dW), resulting in charge Q′ on conductor 1 increasing to Q′ + dQ′, is given by dW = V′ dQ′ = (Q′/C) dQ′.

In this phase, while moving a tiny charge dQ from conductor 2 to conductor 1, the amount of work done is directly related to the potential difference V' between the two conductors. Since V' = Q' / C, substituting gives you dW = (Q' / C) * dQ. This formula allows you to accumulate the total work done by integrating this expression as charge is gradually transferred. Consider all possible values of Q' from 0 to Q, and you will arrive at the total work done (energy) stored in the capacitor.

Imagine filling a bucket with water from another larger bucket. The small amount of work you do to lift a little water against gravity is similar here to moving a small charge. The more water (charge) you transfer, the more effort (work) you need to exert against the force (potential) trying to keep it in place.

Integrating eq. (2.68) W = Q (cid:242) Q′ δQ′ = 1 Q′ 2 Q = Q2/(2C). We can write the final result, in different ways Q2/ (2C) = (1/2)CV2 = (1/2)QV.

After performing the integral, we find the total work done in charging the capacitor given by W = Q²/(2C). This formula reveals the relationship between charge (Q), capacitance (C), and the energy stored in the capacitor. Additionally, it's interesting to note that this energy can also be expressed in terms of the capacitance and the voltage across the capacitor, showing that U = (1/2)CV², which tells you how the energy is affected by the parameters of the capacitor.

This analogy can be likened to a spring. When a spring is compressed or stretched, it stores energy. Similarly, when a capacitor is charged, it 'stretches' its electric field, storing energy that you can release when needed, such as lighting a bulb.

The energy stored in the capacitor as seen earlier can also be viewed as being stored in the electric field between the plates. For a parallel plate capacitor [of area A (of each plate) and separation d between the plates]. Energy stored in the capacitor = 1/2 (ε₀)(E²)·A·d.

Breaking down the energy storage, we can look at it as a function of the electric field (E) created between two plates of the capacitor. This expression captures how energy density relates to the strength of the electric field and the configuration of the capacitor. The energy density formula shows how much energy is stored per unit volume in the electric field between the plates, which is a critical aspect to understand how capacitors function in circuits.

Imagine the electric field between the capacitor's plates is like the pressure of water building up in a hose. The tighter you hold the hose, the more pressure builds up. Similarly, the stronger the electric field between the plates, the more energy is stored. When you release the end of the hose, that energy rushes out just like the potential energy in a charged capacitor when it's connected to a circuit.

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