Expression for electric potential due to a point charge

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

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Answer:  

The ‘electric potential’ due to a point charge is kq/r.

Solution:

Let us consider q’ be the point charge placed at a distance r from the charge q.

By coulomb law, the force exerted on the point charge due to the positive charge q is

F=k \frac{q q^{\prime}}{r^{2}} \rightarrow(1)

The electric potential is defined as the work done to bring a point charge to a particular position.

V=\frac{W}{q^{\prime}} \rightarrow(2)

The work done can be determined by

W=-\int_{\infty}^{r} F . d r=-\int_{\infty}^{r} k \frac{q q^{\prime}}{r^{2}} d r

W=-k q q^{\prime}\left[-\frac{1}{r}\right]_{\infty}^{r}=\frac{k q q^{\prime}}{r}

Thus the ‘electric potential’ will be

V=\frac{\frac{k q q^{\prime}}{r}}{q^{\prime}}=\frac{k q}{r}

The electric potential for a point charge is kq/r.


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