Question

Derive an expression for electrostatic potential due to a point charge.

Answer 1

1. Consider a positive charge ‘q’ kept fixed at the origin. Let P be a point at distance r from the charge ‘q’.
   
   Electrostatic potential at a point P
2. The electric potential at the point P is
   V = `int_∞^"r" (- vec"E")* "d"vec"r" = - int_∞^"r" vec"E" * vec"dr"`
3. Electric field due to positive point charge is
   `vec"E" = 1/(4piε_0) "q"/"r"^2 hat"r"`
   V = `- 1/(4piε_0) int_∞^"r" "q"/"r"^2 hat"r" * "d" vec"r"`
4. The infinitesimal displacement vector,
   `"d"vec"r" = "dr" hat"r"` and using `hat "r" hat "r"` = 1, we have
   V = `- 1/(4piε_0) int_∞^"r" "q"/"r"^2 hat"r" * "dr" hat"r" = - 1/(4piε_0) int_∞^"r" "q"/"r"^2 "dr"`
5. After the integration,
   V = `- 1/(4piε_0) "q"{- 1/"r"}_∞^"r" = 1/(4piε_0) "q"/"r"`
6. Hence the electric potential due to a point charge q at a distance r is
   V = `1/(4piε_0) "q"/"r"`