Electric Potential
Core concepts
Electric potential is potential energy per unit charge, (V = U/q), and is a scalar, which makes it far easier to superpose than the field: (V = \sum kq_i/r_i). The field and potential are linked by (\vec E = -\nabla V), so (\Delta V = -\int \vec E\cdot d\vec s). Potential difference is path-independent because the electrostatic field is conservative. Equipotential surfaces are always perpendicular to field lines, and no work is done moving a charge along one.
For a point charge (V = kq/r) with (V \to 0) at infinity. Potential energy of a pair is (U = kq_1q_2/r).
Traps
- Sign errors: potential from a negative charge is negative everywhere, but the magnitude falls with distance.
- Adding potentials as scalars is correct; adding fields as scalars is wrong. Questions often test whether you know the difference.
- Choosing the wrong reference point. Inside a charged shell, (V = kQ/R) (constant, not zero) even though (E = 0).
- Confusing potential with potential energy: a proton and electron at the same point have the same (V) but opposite (U).
- Forgetting that (\vec E) points toward decreasing (V).
Worked example
Two charges, (+q) at (x = -a) and (-q) at (x = +a). Find (V) on the perpendicular bisector at distance (y) from the midpoint. Each contributes (kq/\sqrt{a^2+y^2}), but with opposite signs, so (V = 0) everywhere on that line. Note that (\vec E) is not zero there - it points along (+x). Rule of thumb: zero potential does not mean zero field, and zero field does not mean zero potential.