19.2 Energy Stored in a Capacitor

Mechanism of Charging a Capacitor

Microscopic Analysis: Why is Work Done?

The battery acts as an electron pump that actively separates charge against an opposing Electric Field.

1. The Initial State ($t=0$)

• The plates are neutral ($Q=0,V=0$).
• The first electron moved by the battery faces negligible repulsion from the destination plate.
• Virtually zero work is required to move the very first increment of charge.

2. The Intermediate State ($0<t<\text{charged}$)

• As electrons accumulate on the negative plate, they create a net negative charge.
• This charge creates an Electric Field between the plates that exerts a repulsive electrostatic force on any new incoming electrons.
• Simultaneously, the positive plate becomes more positive, attracting the electrons back (trying to prevent them from leaving).
• Crucial Physics: To move the next electron, the battery must exert a force $F_{emf}$ to overcome this electrostatic repulsion/attraction.
• Work Done: Since $W=Fd$, work is transferred from the chemical energy of the battery into the Electric Potential Energy of the charge configuration.

3. The Rising Difficulty (Why $\frac{1}{2}QV$?)

• As the charge $Q$ on the plates increases, the Potential Difference $V$ increases ($V\propto Q$).
• Recall definition: $V=\frac{\text{Work}}{\text{Charge}}$.
• Therefore, the work required to deposit each subsequent unit of charge increases linearly.
• The last electron requires the most work; the first required none.
• This is why the total energy is the average of the work done: $\frac{1}{2}QV$.

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Graphical Determination ($V-Q$ Graph)

If we plot the Potential Difference ($\mathrm{V}$) across a capacitor against the Charge stored ($\mathrm{Q}$):

The electric potential energy stored in the capacitor is the area under the potential-charge graph.

The V-Q Relationship

Since $V=\frac{1}{C}Q$ and $C$ is constant, the graph is a straight line through the origin.

• Gradient: $1/C$ (inverse of capacitance).
• Area Under Graph: Work Done / Energy Stored.

Why is it $\frac{1}{2}QV$ and not $QV$?

• Common Mistake: Using $W=VQ$.
• The Physics: $W=VQ$ applies when moving charge through a constant potential difference (like in a resistor).
• The Capacitor: When charging starts, $V=0$. As charge builds up, $V$ increases linearly to its final value. The “average” potential difference during the process is only $\frac{1}{2}{V}_{final}$.
• Therefore, Energy = Area of Triangle = $\frac{1}{2}\times \text{base}\times \text{height}$.

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

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3. Energy Formulae

You must be able to recall the primary formula and derive the alternatives using $C=Q/V$.

Primary Formula (Area under graph)

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

• Use when: You know Charge and Voltage (rare in lab questions, common in theory).

Derived Formula 1 (The “Standard”)

Substitute $Q=CV$: $W=\frac{1}{2}(CV)V$ $W=\frac{1}{2}C{V}^2$

• Use when: You know Capacitance and Voltage. This is the most common formula because $C$ is fixed and $V$ is easily measured with a voltmeter.

Derived Formula 2 (The “Isolated”)

Substitute $V=Q/C$: $W=\frac{1}{2}Q\left(\frac{Q}{C}\right)$ $W=\frac{1}{2}\frac{Q^2}{C}$

• Use when: $Q$ is constant (e.g., an isolated capacitor disconnected from the supply).

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Dynamic Scenarios

Scenario A: Connected to Battery ( $V$ is constant)

Action: The plates are pulled further apart (distance $d$ increases).

• $C$ decreases (since $C\propto 1/d$).
• $V$ is constant (fixed by battery).
• Formula Choice: Use $W=\frac{1}{2}C{V}^2$.
• Result: Since $C$ drops, Energy Stored decreases.
• Where did the energy go? It flowed back into the battery.

Scenario B: Isolated / Disconnected ( $Q$ is constant)

Action: The plates are pulled further apart (distance $d$ increases).

• $C$ decreases (since $C\propto 1/d$).
• $Q$ is constant (charge is trapped).
• Formula Choice: Use $W=\frac{1}{2}\frac{Q^2}{C}$.
• Result: Since $C$ drops (and is in the denominator), Energy Stored increases.
• Where did the energy come from? Work was done by the external mechanical force pulling the attractive plates apart.

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Energy in Combinations

Total Energy: Whether in Series or Parallel, the total energy stored is simply the sum of the individual energies. $E_{total}=E_1+E_2+...$