Electrostatic Potential and Capacitance

Lesson

Part 1 — describing the electric field with a single number at each point, the potential, and the energy stored in arrangements of charges.

In this lesson you will

Define electric potential and potential difference and find the potential of a point charge, a dipole and a system of charges
Relate equipotential surfaces to field lines and use E = −dV/dr
Calculate the potential energy of a system of charges and of a dipole in a uniform field

Concept

Electric potential and potential difference

The electrostatic force is conservative: the work done in moving a charge between two points does not depend on the path. So we can define $V_B - V_A = \dfrac{W_{A \to B}}{q},$ the work done by an external agent per unit positive charge in moving it slowly from A to B. The potential at a point is the work per unit charge to bring a test charge from infinity ( $V_\infty = 0$ ). Unit: volt, $1\ \text{V} = 1\ \text{J C}^{-1}$ .

Potential is a scalar: potentials of several charges simply add.
A positive charge moves on its own from high to low potential; a negative charge from low to high.

Concept

Potential of a point charge

Bringing a unit positive charge from infinity to distance $r$ from $q$ : $V(r) = \dfrac{1}{4\pi\varepsilon_0}\,\dfrac{q}{r} = \dfrac{kq}{r}$ Positive near a positive charge, negative near a negative one; it falls as $1/r$ , more slowly than the field ( $1/r^2$ ).

$V$ has a sign; $E$ is a magnitude with a direction.
For a system of charges: $V = k\sum_i \dfrac{q_i}{r_i}$ (algebraic sum).

V ∝ 1/r
E ∝ 1/r²

Point charge: both are 1 at r = 1; the field falls off faster.

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