Electric Potential Energy and Electric Potential

Energy without arrows

Forces and fields are vectors, and adding vectors takes work. Energy is a scalar: you just add numbers (with signs). In this lesson you will meet electric potential energy and electric potential, the quantity behind every "volt" printed on a battery. Using energy often turns a hard force problem into a few lines of algebra.

Remember from Grade 11 (Electricity and Magnetism): potential difference (voltage) is energy per unit charge, ΔV = E/Q, measured in volts. Now you will see where that idea comes from.

Electric potential energy of two charges

By the same reasoning that gave Eg = −GMm/r, with the zero at infinite separation:

Ee = k q₁ q₂ / r

Here you do keep the signs of the charges, because the sign of Ee tells you something:

• Like charges: Ee > 0. They repel, so you had to push them together. Release them and they fly apart, turning Ee into kinetic energy.
• Unlike charges: Ee < 0. They attract, so they are bound, like a planet and the Sun. You must add energy to pull them apart.

Compare: Eg = −GMm/r is always negative because gravity always attracts.

For three or more charges, add Ee for every pair. Three charges have three pairs; four charges have six.

Electric potential

Just as the field is force per unit charge, electric potential is potential energy per unit charge:

V = Ee/q (unit: the volt, 1 V = 1 J/C)

For a point charge Q, dividing Ee = kQq/r by q:

V = kQ/r

• V is a scalar. Keep the sign of Q: positive charges make positive potential around them, negative charges make negative potential.
• For several charges, the potential at a point is the algebraic sum V = kQ₁/r₁ + kQ₂/r₂ + ... No vectors!
• Like g and E, V describes the place, not the charge you put there.

Only differences in potential matter physically. When a charge q moves through a potential difference ΔV, the change in its potential energy is

ΔEe = qΔV

and the work needed (with no change in kinetic energy) is W = qΔV. If only electric forces act, ΔEk = −qΔV: positive charges speed up moving toward lower potential, and negative charges speed up moving toward higher potential.

The electron volt

In atomic and particle physics, joules are awkwardly large. The electron volt is the energy one elementary charge gains moving through 1 V:

1 eV = (1.60 × 10⁻¹⁹ C)(1 V) = 1.60 × 10⁻¹⁹ J

An electron accelerated through 100 V gains 100 eV. Chemical bonds involve a few eV; alpha particles from radioactive decay have a few MeV (10⁶ eV); the Large Hadron Collider gives protons several TeV (10¹² eV).

Equipotential surfaces

An equipotential is a surface where V is the same everywhere. Around a point charge, V = kQ/r depends only on r, so the equipotentials are spheres centred on the charge (circles in a 2D drawing).

• Equipotentials are always perpendicular to field lines.
• Moving a charge along an equipotential takes no work (ΔV = 0).
• The field points from high to low potential.
• The surface of a conductor in equilibrium is an equipotential (otherwise charges would move).

They are like contour lines on a topographic map: the field is the "downhill" direction, and the field is strongest where equipotentials are closest together.

Rutherford's experiment: energy at the atomic scale

In 1909, Hans Geiger and Ernest Marsden, working with Ernest Rutherford, fired alpha particles (charge +2e) at a very thin gold foil. Most went straight through, but about 1 in 8000 bounced back at large angles. Rutherford, who had done earlier Nobel-winning work on radioactivity at McGill University in Montréal, said it was as if a shell fired at tissue paper had come back and hit you. In 1911 he explained the result: the atom's positive charge and almost all of its mass are concentrated in a tiny nucleus.

Energy tells us how close an alpha gets. In a head-on approach, the alpha slows until all its kinetic energy has become electric potential energy: Ek = k(2e)(Ze)/r_min. This distance is an upper limit on the size of the nucleus.