Energy Stored in a Capacitor

A capacitor stores electrical energy in the electric field created between its plates.

When a capacitor is connected to a battery, work is done in transferring charge from one plate to the other.

This work done is stored as electrical potential energy.

Why Does a Capacitor Store Energy?

Initially, an uncharged capacitor has no charge on its plates.

As charge is transferred from one plate to the other, a potential difference develops between the plates.

Further transfer of charge requires work to be done against the increasing electric field.

This work is stored in the capacitor as electrical energy.

Derivation of Energy Stored in a Capacitor

Consider a capacitor of capacitance C.

Suppose a small charge dq is transferred to the capacitor.

If q is the charge already present on the capacitor, then the potential difference at that instant is:

The small amount of work done is:

Substituting the value of V:

The total work done in charging the capacitor from 0 to Q is:

Taking C as constant:

Integrating:

This work done is stored as electrical energy.

Therefore,

Alternative Forms of Energy Stored

Using:

The energy expression can be written in different forms.

First Form

Second Form

Third Form

All three expressions represent the same stored energy.

Graph Between Charge and Potential

For a capacitor:

The graph between charge and potential difference is a straight line passing through the origin.

The area under the graph represents the energy stored in the capacitor.

The area of the triangle is:

This confirms the energy formula.

Energy Stored in the Electric Field

The energy of a capacitor is actually stored in the electric field between its plates.

The field exists even when the battery is removed.

Thus, the electric field acts as a reservoir of energy.

Energy Density of a Capacitor

Energy density is the energy stored per unit volume.

It is represented by u.

For a parallel plate capacitor:

Where:

• u = energy density
• ε₀ = permittivity of free space
• E = electric field intensity

This expression shows that energy is distributed throughout the electric field.

Factors Affecting Stored Energy

From the equation:

The stored energy increases when:

• Capacitance increases
• Applied voltage increases

Since energy depends on V², a small increase in voltage produces a large increase in stored energy.

Solved Example 1

A capacitor of capacitance 10 μF is connected across a 100 V supply.

Find the energy stored.

Given:

Using:

Substituting values:

Therefore,

Solved Example 2

A capacitor stores a charge of 20 μC at a potential difference of 50 V.

Find the stored energy.

Given:

Using:

Substituting values:

Hence,

Applications of Energy Stored in Capacitors

• Camera flash units
• Defibrillators
• Power backup systems
• Electronic filters
• Pulse circuits
• Energy storage devices

Important Points to Remember

• A capacitor stores energy in its electric field.
• The work done in charging a capacitor becomes stored energy.
• The energy stored is always positive.
• The standard formula is U = ½CV².
• Energy density is given by u = ½ε₀E².
• Stored energy increases with capacitance and voltage.
• Energy is distributed throughout the electric field between the plates.

Frequently Asked Questions

Where is energy stored in a capacitor?

Energy is stored in the electric field between the capacitor plates.

What is the formula for energy stored in a capacitor?

Can capacitor energy be negative?

No. Energy stored in a capacitor is always positive.

What is energy density?

Energy density is the energy stored per unit volume.

Why does stored energy increase rapidly with voltage?

Because energy is proportional to the square of voltage.