PHY132H1/Unit 3
Unit 3 · Introduction to Physics II

Electric Potential

The energy view of the field, and a large practical simplification: potential is a scalar. No components, no angles, just addition. Whenever a problem can be done with potential rather than field, it should be.

By the end you can
  • Distinguish potential energy from potential.
  • Compute potential from point charges by simple addition.
  • Relate field and potential, and use equipotential surfaces.
  • Compute capacitance and stored energy.
Why it matters & where it comes from
Why it exists

Adding vectors is laborious and error-prone. Because the electric force is conservative, it has a potential energy, and dividing by charge gives a scalar that carries the same information as the field with none of the bookkeeping.

In the real world

Voltage is potential difference. Every battery, outlet and electronic component is specified by it. Defibrillators and camera flashes are capacitors dumping stored energy quickly.

Story & open questions

Volta built the first battery in 1800 by stacking zinc and copper discs with brine-soaked cloth, showing that electricity could be produced chemically rather than only by friction. The volt is named for him.

Potential energy and potential

Definition
Potential energy of two point charges: $U = k\dfrac{q_1q_2}{r}$ — note $r$, not $r^2$.
Potential is potential energy per unit charge: $V = \dfrac{U}{q}$, measured in volts.
For a point charge, $V = k\dfrac{Q}{r}$, and this keeps its sign: negative charges give negative potential.
Rule The parallel with Unit 2
Force relates to field as potential energy relates to potential: divide by the charge placed there.
$F = qE$ and $U = qV$.
Field and force are vectors; potential and potential energy are scalars.
Potentials add with sign, and that is the point

Unlike fields, potentials from several charges add as plain numbers. Two equal charges either side of a point give zero field by cancellation, but a doubled potential. Do not import cancellation intuition from vectors.

Example Potential at a point

Charges $+2\,\mu$C and $-3\,\mu$C, each $0.1$ m from a point $P$.

$V = k(2 \times 10^{-6})/0.1 + k(-3 \times 10^{-6})/0.1$
$= (8.99 \times 10^9)(-1 \times 10^{-6})/0.1 = -8.99 \times 10^4$ V.
One line, no components. The corresponding field calculation would need vector addition and an angle.

Relating field and potential

Rule Potential difference and work
Moving charge $q$ through a potential difference $\Delta V$ changes its potential energy by $\Delta U = q\Delta V$.
For a uniform field over a distance $d$ along the field, $\Delta V = -Ed$.
The minus sign says potential decreases along the field direction. Positive charges accelerate from high to low potential.
Definition

An equipotential surface joins points at the same potential. Moving a charge along one takes no work at all, since $\Delta V = 0$.

Field lines are always perpendicular to equipotentials.
A conductor's surface is an equipotential, and so is its whole interior — zero field means no potential change.
Example Accelerating an electron

An electron accelerates from rest through $500$ V.

Energy gained: $qV = (1.6 \times 10^{-19})(500) = 8 \times 10^{-17}$ J.
All of it becomes kinetic: $v = \sqrt{2 \times 8 \times 10^{-17} / 9.11 \times 10^{-31}} = 1.3 \times 10^7$ m/s.
About four per cent of light speed. This is how an old television tube worked, and the calculation is why the unit electronvolt exists.

Capacitance

Definition

A capacitor stores charge on two conductors separated by an insulator. Its capacitance is

$C = \dfrac{Q}{V}$, measured in farads.

For parallel plates of area $A$ separated by $d$: $C = \dfrac{\varepsilon_0 A}{d}$ — depending only on geometry, not on the charge stored.

Rule Stored energy
$U = \tfrac12 CV^2 = \tfrac12 QV = \dfrac{Q^2}{2C}$

The factor of a half appears because the potential rises as charge accumulates: the first charge moves across for free, the last across the full voltage, and the average is half.

Example A camera flash

A $100\,\mu$F capacitor charged to $300$ V.

$U = \tfrac12 (10^{-4})(300)^2 = 4.5$ J.
Released in a millisecond, that is $4500$ W — far more power than the battery could supply directly.
Capacitors do not store much energy; they release it quickly. That distinction is why they are used for flashes and defibrillators rather than as batteries.
Example Combining capacitors
Parallel: $C = C_1 + C_2$. The plate areas effectively add.
Series: $\dfrac{1}{C} = \dfrac{1}{C_1} + \dfrac{1}{C_2}$. The separations effectively add, so the total is less than either.

These combine oppositely to resistors in Unit 4, which is a frequent source of confusion — worth noting now.

Common pitfalls

• Using $r^2$ in the potential formula. Potential goes as $1/r$, field as $1/r^2$.
• Dropping the sign of a negative charge when adding potentials.
• Confusing potential with potential energy. One is per unit charge, the other is not.
• Expecting potentials to cancel the way fields do.
• Combining capacitors with the resistor rules.

Practice

Work each fully before opening the answer. Original exercises written for this guide in the style of the course's problem sets.

Practice 1 · Potential

Find the potential $0.3$ m from a $-4\,\mu$C charge.

Answer
$V = (8.99 \times 10^9)(-4 \times 10^{-6})/0.3 = -1.2 \times 10^5$ V. Negative, and it stays negative.
Practice 2 · Field zero, potential not

Two equal positive charges. At the midpoint, what are $E$ and $V$?

Answer
$E = 0$ by cancellation; $V$ is twice one charge's contribution and is not zero. Vectors cancel, scalars do not.
Practice 3 · Work

How much work moves $+2\,\mu$C through $50$ V?

Answer
$W = qV = 10^{-4}$ J.
Practice 4 · Equipotential

How much work moves a charge along an equipotential surface?

Answer
None. $\Delta V = 0$, so $\Delta U = 0$ regardless of the path length.
Practice 5 · Capacitor energy

Find the energy in a $50\,\mu$F capacitor at $200$ V.

Answer
$U = \tfrac12(5 \times 10^{-5})(4 \times 10^4) = 1$ J.
Practice 6 · Changing separation

A charged capacitor is disconnected and its plates pulled apart. What happens to $V$?

Solution
$Q$ is fixed. Doubling $d$ halves $C$, so $V = Q/C$ doubles. The stored energy $Q^2/(2C)$ also doubles — supplied by the work you did pulling against the plates' attraction.
Practice 7 · Series capacitors

Two $10\,\mu$F capacitors in series. Find the total.

Answer
$1/C = 1/10 + 1/10$, so $C = 5\,\mu$F — less than either, the opposite of resistors in series.
Practice 8 · Why the half

Why is the stored energy $\tfrac12 QV$ rather than $QV$?

Answer
Because the voltage grows from zero to $V$ as charge accumulates. Each increment crosses the voltage at that moment, and the average over the whole process is $V/2$.

Expanded topic checklist

Course coverage. This unit is part of the complete PHY132H1 topic map. It should cover the following items before you move on:

  • Potential energy and potential. The distinction, and the $1/r$ dependence.
  • Scalar addition. Why potential is easier than field, and where the intuitions differ.
  • Field and potential. $\Delta V = -Ed$ and equipotential surfaces.
  • Capacitance. Definition, geometry dependence, stored energy, and combinations.
Textbook/source alignment

The calendar's "electricity" topic includes potential and capacitance. Check the syllabus on Quercus for assigned problems.

Core idea map

Field is force per charge; potential is energy per charge. The second is scalar and therefore easier. Unit 4 puts potential differences to work driving currents.

Study routine. Ask whether the question can be answered with potential before reaching for the field. Scalar addition beats vector addition every time it is available.

Deep practice extensions

Use these after the main practice set. They push on definitions and edge cases rather than routine calculation.

Extension 1 · Which power of r

How do field and potential fall off with distance?

Guide
Field as $1/r^2$, potential as $1/r$. Mixing them is the commonest algebraic slip.
Extension 2 · Scalar advantage

Why is potential easier to work with?

Guide
It is a scalar, so contributions add as signed numbers with no components or angles.
Extension 3 · Equipotentials

What is the angle between field lines and equipotentials?

Guide
Ninety degrees. Any component of field along an equipotential would mean a potential change along it.
Extension 4 · Capacitance dependence

Does capacitance depend on the charge stored?

Guide
No. It depends only on geometry and the insulating material. $Q$ and $V$ rise together, keeping the ratio fixed.
← Unit 02The electric field Unit 04 →Current & circuits