Potential due to a System of Charges

The electrostatic potential V at a point in an electric field is defined as the work done by an external force in bringing a unit positive charge (without acceleration) from infinity to that point.

V = \[\frac{W_{\infty\to P}}{q_{0}}\] (Work done per unit positive charge)

SI Unit: Volt (V) = Joule/Coulomb (J/C);
Dimensional Formula: [M¹L²T⁻³A⁻¹]

When multiple point charges are present, the total potential at any point equals the algebraic sum of potentials due to each individual charge. This follows directly from the superposition principle. Since potential is a scalar, we simply add numbers — no vector components or angles required.

Derivation (Step-by-Step)

Setup: Consider nn point charges q₁,q₂,…,q_n​ at distances r_1P,r_2P,…,r_nP from a point P.

Step 1: Potential at P due to charge q₁​ alone:

Step 2: Similarly, for each charge q_i​:

Step 3: By the superposition principle, the total potential at P:

For a continuous charge distribution with volume charge density ρ(r):

• Divide the distribution into infinitesimal volume elements, each of size Δv carrying charge ρ⋅Δv
• Find potential due to each element
• Integrate over the entire distribution
  V(r) = \[\frac{1}{4\pi\varepsilon_0}\int\frac{\rho(\mathbf{r}^{\prime})}{|\mathbf{r}-\mathbf{r}^{\prime}|}dv^{\prime}\]

4.1 Three Cases

4.2 Key Equations

Outside (r ≥ R):

Inside (r < R):

V = \[{\frac{1}{4\pi\varepsilon_0}\cdot\frac{q}{R}}\] = constant   ...[2.19(b)]