Derive an expression for the electric field intensity at a point on the equatorial line of an electric dipole of dipole moment $\overrightarrow{\text{P}}$and length 2a. What is the direction of this field?
CBSE OUTSIDE DELHI - SET 1 SOUTH 2016
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$\text{E}_{+q} = \text{kq}/ (\text{r}^{2} + \text{a}^{2})\text{ and }\text{E}_{-q} =\text{Kq}/ (\text{r}^{2} + \text{a}^{2})$
The two Electric fields have equal magnitudes and their directions are as shown in diagram. Components along dipole axis get added up while normal components cancel each other.
$\therefore\text{E} = - [\text{E}_{-q} + \text{E}_{+q}]\cos\theta\hat{\text{ r}}\text{ so }\text{ E } = - \frac{\text{K}2\text{qa}}{[\text{r}^{2} + \text{a}^{2}]\frac{3}{2}}\hat{\text{r}}$
$ = \frac{\text{kp}}{[\text{r}^{2} + \text{a}^{2}]\frac{3}{2}}(\text{p} = 2 \text{qa}\hat{\text{r}}) = \frac{-1}{4\pi\in_{o}}\frac{\text{p}}{[\text{r}^{2} + \text{a}^{2}]\frac{3}{2}}$
$\therefore$ Direction of electric field is opposite to that of dipole moment.
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