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Potential energy and voltage

Potential energy tracks the work stored in charge positions, and voltage is that energy per charge — the height of the electrical hill.

Potential energy

Bringing two charges together from far away stores or releases energy. The potential energy of a pair is U = kq1q2/r, negative for opposite charges (they are bound) and positive for like charges, with zero chosen at infinite separation.

U = k\frac{q_1 q_2}{r}

Potential

Potential V is potential energy per unit charge: V = U/q. For a point charge, V = kq/r, positive near positive charge. The electric field always points from high potential toward low potential, so positive charges roll downhill on the voltage landscape while electrons climb it.

V = \frac{U}{q} = k\frac{q}{r}

Worked example

Hold q1 = 3.0 microcoulombs fixed and place q2 = -2.0 microcoulombs 0.15 m away. The pair's energy is U = (8.99e9)(3.0e-6)(-2.0e-6)/0.15 = -0.36 J, negative because the pair is bound. The potential from q1 at that point is V = (8.99e9)(3.0e-6)/0.15, about 1.8e5 V — the same number you get dividing U by q2.

U = \frac{(8.99\times 10^9)(3.0\times 10^{-6})(-2.0\times 10^{-6})}{0.15} \approx -0.36\ \text{J}

Flashcard

U = k\frac{q_1 q_2}{r}

Potential energy of two point charges

This is the work needed to assemble the pair from infinite separation, with zero chosen at infinity. A negative value means the charges are bound together and energy must be supplied to pull them apart.

Check your understanding

What is the electric potential 0.50 m from a point charge of +5.0 nanocoulombs?

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