Example

Question:

(a) Calculate the potential at a point P due to a charge of \( 4 \times 10^{-7}\,\text{C} \) located 9 cm away.
(b) Hence obtain the work done in bringing a charge of \( 2 \times 10^{-9}\,\text{C} \) from infinity to the point P. Does the answer depend on the path along which the charge is brought?

Solution:

(a) \[ V = \frac{1}{4\pi\varepsilon_0}\frac{Q}{r} = 9 \times 10^9\,\text{Nm}^2\,\text{C}^{-2} \times \frac{4 \times 10^{-7}\,\text{C}}{0.09\,\text{m}} = 4 \times 10^4\,\text{V} \] (b) \[ W = qV = 2 \times 10^{-9}\,\text{C} \times 4 \times 10^4\,\text{V} = 8 \times 10^{-5}\,\text{J} \] No, work done will be path independent. Any arbitrary infinitesimal path can be resolved into two perpendicular displacements: one along \( \mathbf{r} \) and another perpendicular to \( \mathbf{r} \). The work done corresponding to the latter will be zero.