Question

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

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

Answer 1

Consider a positive charge ‘q’ kept fixed at the origin. Let P be a point at a distance r from the charge ‘q’.

Electrostatic potential at a point P

The electric potential at the point P is:

V = `int_infty^r (-vec E) * d vec r`

= `-int_infty^r vec E * d vec r`

The electric field due to a positive point charge is:

`vec E = 1/(4 pi epsilon_0) q/r^2 hat r`

V = `-1/(4 pi epsilon_0) int_infty^r q/r^2 hat r * d vec r`

The infinitesimal displacement vector,

`d vec r = dr hat r` and using `hat r hat r` = 1, we have,

V = `-1/(4 pi epsilon_0) int_infty^r q/r^2 hat r * dr hat r`

= `-1/(4 pi epsilon_0) int_infty^r q/r^2 dr`

After the integration,

V = `-1/(4 pi epsilon_0) q {-1/r}_infty^r`

= `1/(4 pi epsilon_0) q/r`

Hence, the electric potential due to a point charge q at a distance r is:

V = `1/(4 pi epsilon_0) q/r`