Charge, field, and potential

Why this matters

Every circuit, motor, sensor, battery and chip runs on electric charge. Electrical engineers design with voltages every day; chemists use the same ideas for electrochemistry and batteries; materials scientists study how charges move through solids. The single most useful idea in this lesson is that voltage is energy per unit charge. Once that clicks, batteries, circuits and power stop being mysterious.

Electric charge

Charge comes in two kinds, positive and negative. Like charges repel; opposite charges attract. Protons carry +e and electrons carry −e, where

e = 1.602 × 10⁻¹⁹ C     (coulombs)

A coulomb is a lot of charge: about 6.24 × 10¹⁸ electrons. Charge is conserved: it can move from one object to another, but the total never changes. Rub a balloon on your hair and electrons move from your hair to the balloon; the balloon becomes negative and your hair positive by the same amount.

Coulomb's law

The force between two point charges q₁ and q₂ a distance r apart is

F = k · |q₁ · q₂| / r²       k = 8.99 × 10⁹ N·m²/C²

Two charges of 1.0 µC (10⁻⁶ C) held 0.10 m apart push or pull with

F = 8.99 × 10⁹ × (10⁻⁶ × 10⁻⁶) / 0.10² = 0.90 N

It is an inverse-square law, just like gravity: double the distance and the force falls to a quarter.

The electric field

Rather than working out every pair of charges, physicists describe the effect of charges with an electric field E: the force per unit charge that a small test charge would feel at a point.

E = F / q          units: N/C, which are the same as V/m

The field points in the direction a positive charge would be pushed: away from positive charges and toward negative ones. Around a single point charge, E = k·|q| / r². At 0.10 m from a 1.0 µC charge, E = 8.99 × 10⁵ N/C.

Between two large parallel metal plates with opposite charges, the field is nearly uniform: the same size and direction everywhere between them. That arrangement is a capacitor, and it appears in almost every circuit board.

Potential energy and voltage

Lifting a mass against gravity stores potential energy. Pushing a positive charge toward another positive charge, against the electric force, stores electric potential energy U in the same way.

Electric potential V is potential energy per unit charge:

V = U / q          units: volts, where 1 V = 1 J/C

What matters in practice is the potential difference ΔV between two points, usually just called the voltage. Moving a charge q through a potential difference ΔV changes its energy by

ΔU = q · ΔV

So a 9 V battery gives every coulomb that passes through it 9 J of energy; a 1.5 V AA cell gives each coulomb 1.5 J. The battery does not supply charge (the charges are already in the wires); it supplies energy to the charges.

The gravity analogy

Water at the top of a dam has energy per kilogram because of its height; charge at the + terminal of a battery has energy per coulomb because of its potential.

Field and voltage in a uniform field

For two plates a distance d apart with voltage ΔV between them:

E = ΔV / d

12 V across a 2.0 mm gap gives E = 12 / 0.002 = 6,000 V/m. This is why the field is quoted in V/m and why thin insulation layers in chips must withstand huge fields even at small voltages.

The electron-volt

Physicists often measure tiny energies in electron-volts (eV): the energy an electron gains moving through 1 V.

1 eV = 1.602 × 10⁻¹⁹ J

Chemical bonds are a few eV; visible-light photons carry about 1.7–3.1 eV; the electron beams in X-ray tubes carry tens of keV.

Common mistakes

• Thinking voltage "flows". Voltage is a difference between two points; current flows (next lesson).
• Confusing E and V. E is a field (force per charge, a vector). V is potential (energy per charge, a scalar). In a uniform field they are linked by E = ΔV / d.
• Dropping powers of ten. µC is 10⁻⁶ C, nC is 10⁻⁹ C, mm is 10⁻³ m.
• Assuming low voltage means low danger everywhere, or high voltage always kills. Harm depends on current through the body, path and duration. Treat every mains or high-voltage source as dangerous; work on them requires training, supervision and, in many jobs, a licence.

Quick recap

• Charge is quantised in units of e and is conserved.
• Coulomb's law: F = k·q₁·q₂ / r².
• Electric field E = F / q, in N/C or V/m.
• Voltage = potential energy per unit charge: 1 V = 1 J/C, and ΔU = q·ΔV.
• In a uniform field, E = ΔV / d.

Check yourself

This lesson has no coding activity. Use the quiz in the Do tab to check that you can link voltage, energy and charge.