Why Doubling the Voltage on a Capacitor Stores Four Times the Energy

Electric field tells you the force on a charge; electric potential tells you the energy per unit charge at a point. Potential is a scalar, so charges simply add their contributions without any vector diagrams.

This lesson covers the potential of charges and dipoles, electrostatic potential energy, capacitors and their combinations, and how dielectrics change capacitance and stored energy.

**The potential of a point charge is $V=\frac{1}{4\pi {\epsilon }_0}\frac{q}{r}$, a system's potential is the algebraic sum of these, a dipole gives $V=\frac{1}{4\pi {\epsilon }_0}\frac{p\cos \theta }{r^2}$, and field and potential are linked by $E=-\frac{dV}{dr}$.

Key results:**

- Potential is work done per unit charge in bringing a test charge from infinity, in volts
- Dipole: maximum on the axis, where $\theta =0^{\circ }$, and zero on the equatorial line, where $\theta =90^{\circ }$
- Field points from high to low potential, and equipotential surfaces are perpendicular to field lines

Worked example 1 — a system. Charges of $+4.0\times 10^{-9}$ C and $-2.0\times 10^{-9}$ C are 0.20 m apart. At the midpoint, 0.10 m from each:

$$V=\frac{9.0\times 10^9\times (4.0-2.0)\times 10^{-9}}{0.10}=180\text{ V}$$

Worked example 2 — a dipole. For $p=4.0\times 10^{-10}$ C m at 0.20 m on the axis:

$$V=\frac{9.0\times 10^9\times 4.0\times 10^{-10}}{0.20^2}=90\text{ V}$$

Worked example 3 — field from potential. A 120 V difference across plates 4.0 mm apart gives $E=\frac{120}{4.0\times 10^{-3}}=3.0\times 10^4$ V m${}^{-1}$.

An everyday example. A bird perched on a single live wire is safe because both its feet are at almost the same potential, so no current flows through it.

The substance. Zero potential does not mean zero field — at the dipole's midpoint V is zero, yet the field there is strong.

**The potential energy of two charges is $U=\frac{1}{4\pi {\epsilon }_0}\frac{q_1{q}_2}{r}$, summed over every pair for a larger system, and a dipole in a uniform field has $U=-pE\cos \theta$.

Systems of charges. U is the work done in assembling the charges from infinity. A negative U means the charges attract and the system is bound.

Dipole in a uniform field.** The dipole feels a torque $\tau =pE\sin \theta$ but no net force. Its energy is least at $\theta =0^{\circ }$, the stable position, and greatest at $180^{\circ }$.

Worked example 1. Charges of $+2.0\times 10^{-6}$ C and $-4.0\times 10^{-6}$ C are 0.30 m apart:

$$U=\frac{9.0\times 10^9\times 2.0\times 10^{-6}\times (-4.0\times 10^{-6})}{0.30}=-0.24\text{ J}$$

So 0.24 J of work is needed to pull them apart to infinity.

Worked example 2 — a dipole. With $p=3.0\times 10^{-9}$ C m in $E=2.0\times 10^4$ N C${}^{-1}$, $pE=6.0\times 10^{-5}$ J. The work to turn it from $0^{\circ }$ to $180^{\circ }$ is

$$W=U_{180}-U_0=pE-(-pE)=2pE=1.2\times 10^{-4}\text{ J}$$

An everyday example. Water molecules in a microwave oven are dipoles that keep turning to line up with a rapidly reversing field, and that motion heats the food.

The substance. Only differences in U are physical — choosing $U=0$ at $90^{\circ }$ is a convention, which is why the work between two angles never depends on it.

**A parallel-plate capacitor has $C=\frac{{\epsilon }_{0}A}{d}$; in series the reciprocals add, $\frac{1}{C_s}=\frac{1}{C_1}+\frac{1}{C_2}$, and in parallel the capacitances add, $C_p=C_1+C_2$.

Rules for combinations:

- Series: every capacitor carries the same charge, and the voltages add
- Parallel: every capacitor has the same voltage, and the charges add

Worked example 1.** Plates of area 0.020 m${}^2$, 1.0 mm apart:

$$C=\frac{8.85\times 10^{-12}\times 0.020}{1.0\times 10^{-3}}=1.77\times 10^{-10}\text{ F}=177\text{ pF}$$

Worked example 2. Capacitors of 3.0 $\mu$F and 6.0 $\mu$F are joined to a 12 V battery.

$$C_s=\frac{3.0\times 6.0}{3.0+6.0}=2.0\mu \text{F}Q=2.0\times 12=24\mu \text{C}$$

The voltages are $24/3.0=8.0$ V and $24/6.0=4.0$ V, adding to 12 V. In parallel, $C_p=9.0$ $\mu$F.

An everyday example. A ceiling fan's starting capacitor is a small capacitor; when it weakens, the fan hums but struggles to start.

The substance. In series, the smallest capacitor takes the largest voltage — here the 3.0 $\mu$F one takes 8.0 V, so it breaks down first.

**Filling a capacitor with a dielectric of constant K multiplies its capacitance by K, and the stored energy $U=\frac{1}{2}C{V}^2=\frac{Q^2}{2C}$ then rises or falls depending on whether the battery stays connected.

Why capacitance rises. The dielectric's molecules polarise and set up an opposing field, reducing the net field and the voltage for the same charge.

Worked example.** A 10 $\mu$F capacitor is charged to 20 V, so $Q=200$ $\mu$C and

$$U=\frac{1}{2}\times 10\times 10^{-6}\times 20^2=2.0\times 10^{-3}\text{ J}$$

A dielectric with $K=5$ is now inserted.

- Battery disconnected, so Q stays fixed: $C=50$ $\mu$F, $V=4.0$ V, and $U=\frac{(2.0\times 10^{-4}{)}^2}{2\times 50\times 10^{-6}}=4.0\times 10^{-4}$ J — energy falls by K
- Battery connected, so V stays fixed: $U=\frac{1}{2}\times 50\times 10^{-6}\times 20^2=1.0\times 10^{-2}$ J — energy rises by K

An everyday example. Capacitors in mobile phone chargers use thin ceramic dielectrics, which pack large capacitance into tiny components.

The substance. Energy grows with the square of voltage, so doubling V at fixed C stores four times the energy.