Question

Explain axial field or electric field at a point in end-on position, i.e. the axial position of an electric dipole.

Answer 1

Step 1: To Find

Axial field or electric field at a point in end-on position.

Step 2: Calculate the axial field.

Let,

The charge of the electric dipole be $q$ and length be $2d$.

The electric field is at an axial point $P$ which is at a distance $r$ from the center of the dipole.

$$\begin{gathered} {\mathrm{E}}_{\mathrm{P}}={\mathrm{E}}_{+}-{\mathrm{E}}_{-} \\ =\frac{\mathrm{Kq}}{{\left(\mathrm{r}-\mathrm{d}\right)}^2}-\frac{\mathrm{Kq}}{{\left(\mathrm{r}+\mathrm{d}\right)}^2} \\ =\mathrm{Kq}\left(\frac{{\left(\mathrm{r}+\mathrm{d}\right)}^2-{\left(\mathrm{r}-\mathrm{d}\right)}^2}{{\left(\mathrm{r}-\mathrm{d}\right)}^2{\left(\mathrm{r}+\mathrm{d}\right)}^2}\right) \\ =\mathrm{Kq}\left(\frac{4\mathrm{rd}}{{\left(\mathrm{r}-\mathrm{d}\right)}^2{\left(\mathrm{r}+\mathrm{d}\right)}^2}\right) \\ =\frac{2\mathrm{Kpr}}{{\left(\mathrm{r}-\mathrm{d}\right)}^2{\left(\mathrm{r}+\mathrm{d}\right)}^2} \end{gathered}$$

Where, $\mathrm{p}=2\mathrm{qd}$ (Dipole moment).