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# Derive an Expression for Energy Stored in a Capacitor

## Derive an Expression for Energy Stored in a Capacitor

The energy stored in a capacitor is given by the equation.

E = (1/2) * C * V^2

Where: E is the energy stored in the capacitor, C is the capacitance of the capacitor, and V is the voltage across the capacitor.

This formula represents the energy stored in an ideal capacitor without considering factors like dielectric losses or resistance.

Derivation

The energy stored in a capacitor can be derived using calculus and the concept of work.

Consider a capacitor with a charge q on its plates and a voltage difference V across it. The capacitance C of the capacitor is defined as the ratio of the charge on the plates to the voltage across them:

C = q / V

To calculate the work done in moving a small increment of charge dq from one plate to the other against the electric field, we can use the formula for work:

dW = V * dq

Here, dW represents the work done and dq is a small increment of charge. Since the voltage V remains constant during the process, we can rearrange the equation to isolate dq:

dq = C * dV

Now, we integrate both sides of the equation to find the total charge q:

∫dq = ∫C * dV q = ∫C * dV

Next, we substitute the expression for C from the initial definition:

q = ∫(q / V) * dV

We can simplify this equation by multiplying both sides by V:

q * V = ∫q * dV

Now, we integrate both sides of the equation:

∫(q * V) = ∫q * dV

The left-hand side of the equation represents the energy stored in the capacitor, denoted as E. The right-hand side is the integral of q with respect to V, which can be written as:

E = ∫q * dV

To solve this integral, we express q as a function of V. Since q = C * V, we substitute this in:

E = ∫(C * V) * dV

Now we can integrate the right-hand side:

E = C * ∫V * dV E = C * (V^2 / 2)

Finally, substituting the expression for C = q / V, we obtain the formula for the energy stored in a capacitor:

E = (1/2) * C * V^2

That’s how the formula for energy stored in a capacitor is derived.