Electric potential of point charges

The electric potential due to a point charge is given by

⃗ 𝐄 (⃗ 𝐫) = 1 4 𝜋 𝜖 0 𝑞 𝔯

and for a collection of point charges

⃗ 𝐄 (⃗ 𝐫) = 1 4 𝜋 𝜖 0 𝑛 \sum 𝑖 = 1 𝑞 𝑖 𝔯 𝑖

Derivation

Since by convention, we define the zero of an Electrostatic potential field to be the point infinitely far away in which there is no electrostatic force, we can derive a scalar field for the potential at each point due to a point charge.

𝑉 = l i m ⃗ 𝐫 0 \to ⃗ \infty 𝑞 4 𝜋 𝜖 0 \int 𝑟 𝑟 0 ˆ 𝐫 \cdot 𝑑 ⃗ 𝐫 𝑟 2 = l i m 𝑟 0 \to \infty 𝑞 4 𝜋 𝜖 0 \int 𝑟 𝑟 0 𝑑 𝑟 𝑟 2 = l i m 𝑟 0 \to \infty 𝑞 4 𝜋 𝜖 0 [1 𝑟 - 1 𝑟 0] = 1 4 𝜋 𝜖 0 𝑞 𝑟

This is sometimes called the Coulomb potential

#state/tidy | #SemBr | #lang/en