1) Introduction

A Capacitor is a device designed to store electrical Charge. When a capacitor is charged, the energy is stored in the form of electrical potential energy between its plates. This energy results from the work done to move charge onto the plates against the electrostatic forces.

2) Derivation

a) In terms of Charge

The energy stored in a capacitor can be derived in terms of the charge $Q$ placed on the plates.

i) Initial Potential Difference

Initially, when the capacitor is uncharged, the potential difference across the plates is zero.

ii) Final Potential Difference

Once the capacitor is connected to a source of Electric Potential Difference $V$, it becomes charged. The charge deposited on the plates is $+Q$ and $-Q$, and the potential difference between the plates is $V$.

iii) Average Potential Difference

The average potential difference during the charging process is the average of the initial and final potential differences:

$$\text{Average Potential Difference} = \frac{0 + V}{2} = \frac{V}{2}$$

Energy Stored in the Capacitor

The energy $U$ stored in the capacitor is the product of the average potential difference and the charge on the plates:

$$U = \text{Average potential difference} \times \text{Charge}$$

Substituting the average potential difference:

$$U = \left( \frac{V}{2} \right) Q$$

This simplifies to:

$$U = \frac{1}{2} QV$$

Now, using the relationship $V = \frac{Q}{C}$ (where $C$ is the capacitance of the capacitor), we can express the energy in terms of charge and capacitance:

$$U = \frac{1}{2} Q \times \frac{Q}{C} = \frac{Q^2}{2C}$$

Alternatively, by substituting $Q = C \times V$ into the equation:

$$U = \frac{(C V)^2}{2C} = \frac{C^2 V^2}{2C} = \frac{1}{2} C V^2$$

Thus, the energy stored in a capacitor can be expressed as:

$$U = \frac{1}{2} C V^2$$

The potential energy is stored in the capacitor because work is required to deposit charge onto the plates. As more charge is deposited, the potential difference between the plates increases, and more work is needed to continue adding charge.

b) In terms of Electric Field

It is also possible to express the energy stored in a capacitor in terms of the electric field between the plates. The capacitance $C$ of a parallel plate capacitor is given by:

$$C = \frac{A \varepsilon_o \varepsilon_r}{d}$$

where $A$ is the area of the plates, $\varepsilon_o$ is the permittivity of free space, $\varepsilon_r$ is the relative permittivity of the material between the plates, and $d$ is the separation between the plates.

The electric field $E$ between the plates is related to the potential difference $V$ and the plate separation $d$ by:

$$V = E d$$

Substituting this into the energy formula:

$$U = \frac{1}{2} \left( \frac{A \varepsilon_o \varepsilon_r}{d} \right) (E d)^2$$

Simplifying the expression:

$$U = \frac{1}{2} \left( \frac{A \varepsilon_o \varepsilon_r}{d} \right) E^2 d^2$$ $$U = \frac{1}{2} A \varepsilon_o \varepsilon_r E^2 d$$

Thus, the energy stored in the electric field between the plates is:

$$U = \frac{1}{2} E^2 \varepsilon_o \varepsilon_r A d$$

c) Energy Density

Energy density is the energy stored per unit volume. From the previous expression for energy, we have:

$$U = \frac{1}{2} E^2 \varepsilon_o \varepsilon_r A d$$

The term $A d$ represents the volume between the plates. Therefore, the energy density $u$ is:

$$u = \frac{U}{V} = \frac{1}{2} E^2 \varepsilon_o \varepsilon_r$$

This shows that the energy density of a capacitor is directly proportional to the square of the electric field $E$, and also depends on the permittivity of free space $\varepsilon_o$ and the relative permittivity $\varepsilon_r$.

Summary

This section provides an overview of the concepts related to the energy stored in a capacitor, including derivations in terms of charge, electric field, and energy density.

Key Points:

1. Core Concept: A capacitor stores energy in the form of electrical potential energy, either in terms of the charge on the plates or the electric field between them.
2. Important Definitions:
   • Capacitance $C$: The ability of a capacitor to store charge, given by $C = \frac{A \varepsilon_o \varepsilon_r}{d}$.
   • Energy Density $u$: The energy stored per unit volume, given by $u = \frac{1}{2} E^2 \varepsilon_o \varepsilon_r$.
3. Key Relationships/Processes:
   • Energy in a capacitor can be expressed as $U = \frac{1}{2} QV$, $U = \frac{1}{2} C V^2$, or in terms of the electric field $U = \frac{1}{2} E^2 \varepsilon_o \varepsilon_r A d$.

Formula	Description
$U = \frac{1}{2} QV$	Energy in terms of Charge and Electric Potential Difference
$U = \frac{1}{2} C V^2$	Energy in terms of capacitance of Capacitor and potential difference
$U = \frac{1}{2} E^2 \varepsilon_o \varepsilon_r A d$	Energy in terms of Electric Field

The concept of energy stored in a capacitor is essential for understanding how capacitors function in electronic circuits, from energy storage in power supply systems to their use in filters and oscillators.

References