(a). Derive the expression for electric field at a point on the equatorial line of an electric dipole. 

(b) Depict the orientation of the dipole in 

(i) stable,           

(ii) unstable equilibrium in a uniform electric field.

Electric dipole moment: It is the product of the magnitude of either charge and distance and distance between them

$\overrightarrow{\mathrm{q}}=\mathrm{q}\times 2\mathrm{l}$

$\mathrm{it} \mathrm{is} \mathrm{a} \mathrm{vector} \mathrm{quantity} \mathrm{whose} \mathrm{direction} \mathrm{is} \mathrm{from} \mathrm{negative} \mathrm{to} \mathrm{positive} \mathrm{charge}.$

$\mathrm{Electric} \mathrm{field} \mathrm{intensity} \mathrm{at} \mathrm{P} \mathrm{due} \mathrm{to} +\mathrm{q} \mathrm{charge} \mathrm{is}$

$$\begin{gathered} \overrightarrow{{\mathrm{E}}_{+}}=\frac{1}{4\mathrm{\pi }{\in }_0}\frac{\mathrm{q}}{{\mathrm{BP}}^2} \mathrm{along} \mathrm{PD} \\ =\frac{1}{4\mathrm{\pi }{\in }_0}\frac{\mathrm{q}}{\left({\mathrm{r}}^2+{\mathrm{l}}^2\right)} \mathrm{along} \mathrm{PD} ...\left(\mathrm{i}\right) \end{gathered}$$

$\mathrm{Electric} \mathrm{field} \mathrm{intensity} \mathrm{at} \mathrm{P} \mathrm{due} \mathrm{to} -\mathrm{q} \mathrm{charge} \mathrm{is}$

$$\begin{gathered} \overrightarrow{{\mathrm{E}}_{-}}=\frac{1}{4\mathrm{\pi }{\in }_0}\frac{\mathrm{q}}{{\mathrm{AP}}^2} \mathrm{along} \mathrm{PC} \\ =\frac{1}{4\mathrm{\pi }{\in }_0}\frac{\mathrm{q}}{\left({\mathrm{r}}^2+{\mathrm{l}}^2\right)} \mathrm{along} \mathrm{PC} ...\left(\mathrm{ii}\right) \end{gathered}$$

$\mathrm{From} \left(\mathrm{i}\right) \mathrm{and} \left(\mathrm{ii}\right), \left|\overrightarrow{{\mathrm{E}}_{+}}\right|=\left|\overrightarrow{{\mathrm{E}}_{-}}\right|=\frac{1}{4\mathrm{\pi }{\in }_0}\frac{\mathrm{q}}{\left({\mathrm{r}}^2+{\mathrm{l}}^2\right)} ...\left(\mathrm{iii}\right)$$\mathrm{Net} \mathrm{electric} \mathrm{field} \mathrm{intensity} \mathrm{due} \mathrm{to} \mathrm{the} \mathrm{electric} \mathrm{dipole} \mathrm{at} \mathrm{point} \mathrm{P}$

$$\begin{gathered} \therefore \mathrm{E}=\sqrt{{\mathrm{E}}_{+}^2 + {\mathrm{E}}_{-}^2 - 2{\mathrm{E}}_{+}{\mathrm{E}}_{-}\cos 2\mathrm{\theta }} \\ \Rightarrow \mathrm{E}=\sqrt{{\mathrm{E}}_{+}^2 + {\mathrm{E}}_{-}^2 + 2{\mathrm{E}}_{+}^2\cos 2\mathrm{\theta }} (\because {\mathrm{E}}_{+}={\mathrm{E}}_{-}) \\ \Rightarrow \mathrm{E}=\sqrt{2{\mathrm{E}}_{+}^2 + {\mathrm{E}}_{+}^2\cos 2\mathrm{\theta }} \\ \Rightarrow \mathrm{E}=\sqrt{2{\mathrm{E}}_{+}^2(1+\cos 2\mathrm{\theta })} \\ \Rightarrow \mathrm{E}=\sqrt{2{\mathrm{E}}_{+}^{2}2{\cos }^2\mathrm{\theta }} \left(\because 1+\cos 2\mathrm{\theta }=2 {\cos }^2\mathrm{\theta }\right) \\ \therefore \mathrm{E}=2{\mathrm{E}}_{+} \cos \mathrm{\theta }=2\times \frac{1}{4\mathrm{\pi }{\in }_0}\frac{\mathrm{q}}{\left({\mathrm{r}}^2+{\mathrm{l}}^2\right)} \cos \mathrm{\theta } \left[\mathrm{Using} \mathrm{equation} \left(\mathrm{iii}\right)\right] \end{gathered}$$$$\begin{gathered} \mathrm{Now} \mathrm{from} ∆\mathrm{OAP}, \mathrm{cos\theta }=\frac{1}{\sqrt{{\mathrm{r}}^2+{\mathrm{l}}^2}} \\ \mathrm{E}=2\times \frac{1}{4\mathrm{\pi }{\in }_0}\frac{\mathrm{q}}{\left({\mathrm{r}}^2+{\mathrm{l}}^2\right)}\times \frac{\mathrm{l}}{{\left({\mathrm{r}}^2+{\mathrm{l}}^2\right)}^{1/2}} \\ \Rightarrow \mathrm{E}=\frac{\mathrm{q}\times 2\mathrm{l}}{4\mathrm{\pi }{\in }_0{\left({\mathrm{r}}^2+{\mathrm{l}}^2\right)}^{3/2}} \\ \mathrm{Since} \mathrm{q}\times 2\mathrm{l}=\mathrm{p} ...(\mathrm{p} \mathrm{is} \mathrm{dipole} \mathrm{moment}) \end{gathered}$$

$$\begin{gathered} \mathrm{E}=\frac{\mathrm{p}}{4\mathrm{\pi }{\in }_0{\left({\mathrm{r}}^2+{\mathrm{l}}^2\right)}^{3/2}} \mathrm{along} (-)\mathrm{x}-\mathrm{axis} \\ \mathrm{If} \mathrm{l} << \mathrm{r} \mathrm{i}.\mathrm{e}. \mathrm{dipole} \mathrm{is} \mathrm{short}, \mathrm{then} {\mathrm{l}}^2 \mathrm{can} \mathrm{be} \mathrm{negleted} \mathrm{as} \mathrm{compared} \mathrm{to} {\mathrm{r}}^2 \\ \mathrm{Hence} \boxed{\mathrm{E}=\frac{\mathrm{p}}{4\mathrm{\pi }{\in }_0{\mathrm{r}}^3 }} \mathrm{along} (-)\mathrm{x}-\mathrm{axis} \end{gathered}$$

$\left(\mathrm{ii}\right) \left(\mathrm{a}\right) \mathrm{For} \mathrm{stable} \mathrm{equilbrium}, \mathrm{the} \mathrm{angle} \mathrm{between} \mathrm{p} \mathrm{and} \mathrm{E} \mathrm{is} 0^{\circ },$

$\left(\mathrm{b}\right) \mathrm{For} \mathrm{unstable} \mathrm{equilibrium}, \mathrm{the} \mathrm{angle} \mathrm{between} \mathrm{p} \mathrm{and} \mathrm{E} \mathrm{is} {180}^{\circ },$

$\mathrm{For} \mathrm{unstable} \mathrm{equilibrium}, \mathrm{the} \mathrm{angle} \mathrm{between} \mathrm{p} \mathrm{and} \mathrm{E} \mathrm{is} {180}^{\circ },$