Question

Consider the electric dipole moment $(p=2qa)$ shown in the figure. The electric field at a point $P(x,y,0)$ in the polar coordinate system is :

• A. $\frac{k_{e}p}{r^3}\left[\right(2\cos \theta )\hat{r}-(\sin \theta \left)\hat{\theta }\right]$
• B. $\frac{k_{e}p}{r^3}\left[\right(2\cos \theta )\hat{r}+(\sin \theta \left)\hat{\theta }\right]$
• C. $\frac{k_{e}p}{r^3}[-(2\cos \theta )\hat{r}+(\sin \theta \left)\hat{\theta }\right]$
• D. $\frac{k_{e}p}{r^3}[-(2\cos \theta )\hat{r}-(\sin \theta \left)\hat{\theta }\right]$

Answer 1

The correct option is C $\frac{k_{e}p}{r^3}[-(2\cos \theta )\hat{r}+(\sin \theta \left)\hat{\theta }\right]$

Electric dipole moment $P=2qa$

The electric field at a point $P(x,y,0)$

Now in figure,

$\overline{E}={\overline{E}}_1+{\overline{E}}_2$

$E_1^2=E_1^2+E_2^2$

or, $E=\sqrt{E_1^2+E_2^2}$

then, $E_1=\frac{2P\cos \theta }{{kr}^3}$

$E_2=\frac{P\sin \theta }{{kr}^3}$

The resultant of $P$

$E=\sqrt{{\left(\frac{2P\cos \theta }{{kr}^3}\right)}^2+{\left(\frac{P\sin \theta }{{kr}^3}\right)}^2}$

or, $E=\frac{P}{{kr}^3}\frac{P}{{kr}^3}\sqrt{3{\cos }^2\theta +1}$

at, $P=2qa$

or, $E=\frac{k_{c}P}{r^3}[-\left(2\cos \theta \right)\hat{r}+\left(\sin \theta \right)\hat{\theta }]$