recall and use W = 21QV = 21CV2

Capacitors and Capacitance

Learning Objective

Recall and use the energy stored in a capacitor: \$W = \frac{1}{2} QV = \frac{1}{2} CV^{2}\$

Key Concepts

• Capacitance definition
• Relationship between charge, voltage and capacitance
• Energy stored in a capacitor

Definitions

The capacitance \$C\$ of a device is the ability to store charge per unit potential difference:

\$C = \frac{Q}{V}\$ where \$Q\$ is the charge on one plate and \$V\$ is the potential difference between the plates.

Energy Stored in a Capacitor

Derivation:

1. Work required to move a small charge \$dq\$ onto a plate when the existing charge is \$q\$: \$dW = V\,dq = \frac{q}{C}\,dq\$.
2. Integrate from \$0\$ to \$Q\$: \$W = \int_{0}^{Q} \frac{q}{C}\,dq = \frac{1}{2}\frac{Q^{2}}{C} = \frac{1}{2} QV = \frac{1}{2} C V^{2}.\$

Units and Typical \cdot alues

Example Problem

Problem: A 47 µF capacitor is charged to 12 V. Calculate the energy stored.

1. Identify the given values: \$C = 47 \times 10^{-6}\,\text{F}\$, \$V = 12\,\text{V}\$.
2. Use the formula \$W = \frac{1}{2} C V^{2}\$.
3. Calculate: \$W = \frac{1}{2} (47 \times 10^{-6}) (12)^{2} \approx 3.4 \times 10^{-3}\,\text{J}.\$
4. Interpretation: The capacitor stores about 3.4 mJ of energy.

Common Mistakes

• Confusing \$W = \frac{1}{2} QV\$ with \$W = QV\$ – the factor \$\frac{1}{2}\$ arises from the integration.
• Using the voltage of the source instead of the voltage across the capacitor when it is partially charged.
• Mixing units – always convert µF to F and m \cdot to \cdot before substitution.

Further Applications

The energy formula is useful for:

• Estimating discharge energy in flash lamps.
• Designing timing circuits (RC circuits) where energy considerations affect component choice.
• Understanding energy density in capacitors versus batteries.

Quick Revision Checklist

1. Write down the definition \$C = Q/V\$.
2. Remember the three equivalent forms of the energy equation.
3. Check units: \$[C] = \text{F}\$, \$[W] = \text{J}\$.
4. Practice converting between \$Q\$, \$V\$, and \$C\$ using the energy formula.