Question

A short dipole is placed at origin of coordinate system in $\text{x-y}$ plane as shown in figure. Find the electric field at point $\text{P}(0,y)$. $(y>>$ length of dipole$)$

• A. $\frac{Kp}{2y^3}[-2\hat{i}-\hat{j}]$
• B. $\frac{2Kp}{y^3}[-2\hat{i}-\hat{j}]$
• C. $\frac{\sqrt{2}Kp}{y^3}[-\hat{i}-2\hat{j}]$
• D. $\frac{Kp}{\sqrt{2}{y}^3}[-\hat{i}-2\hat{j}]$

Answer 1

The correct option is D $\frac{Kp}{\sqrt{2}{y}^3}[-\hat{i}-2\hat{j}]$

We can resolve that dipole moment into two components: along $\text{x}$ and $\text{y}$ directions respectively.


Dipole moment along $\text{x-}$axis,

$p_x=p\cos {45}^o=\frac{p}{\sqrt{2}}$

Dipole moment along $\text{y-}$ axis,

$p_y=p\sin {45}^o=\frac{p}{\sqrt{2}}$

Component of electric field parallel to $\text{x-}$axis at point $\text{P}$,

${\overrightarrow{E}}_{P_x}=\frac{-K{p}_x\hat{i}}{y^3}=\frac{-Kp\hat{i}}{\sqrt{2}{y}^3}$

Component of electric field parallel to $\text{y-}$axis at point $\text{P}$,

${\overrightarrow{E}}_{P_y}=\frac{-2K{P}_y\hat{j}}{y^3}=\frac{-2Kp\hat{j}}{\sqrt{2}{y}^3}$

Net electric field at point $\text{P}$,

${\overrightarrow{E}}_{net}={\overrightarrow{E}}_{P_x}+{\overrightarrow{E}}_{P_y}$

Substituting the values of ${\overrightarrow{E}}_{P_x}$ and ${\overrightarrow{E}}_{P_y}$ in the above equation, we get

${\overrightarrow{E}}_{net}=\frac{Kp}{\sqrt{2}{y}^3}[-\hat{i}-2\hat{j}]$

Hence, option (d) is the correct answer.