Question

Two identical electric dipoles are arranged on $x$-axis as shown in figure. Electric field at the origin will be ( Assume $\frac{1}{4\pi {ϵ}_0}=K$ )

• A. zero
• B. $\frac{Kp\sqrt{2}}{r^3}\hat{j}$
• C. $\frac{-Kp\sqrt{2}}{r^3}\hat{j}$
• D. $\frac{2Kp}{r^3}\hat{i}-\frac{kp}{r^3}\hat{j}$

Answer 1

The correct option is C $\frac{-Kp\sqrt{2}}{r^3}\hat{j}$

The dipoles can be resolved along $x$ and $y$ axes as shown in figure.

The formulae for electric field at axial and equatorial points of a dipole are:
$E\left(\text{axial}\right)=\frac{2Kp}{r^3}$ along the dipole
$E\left(\text{equatorial}\right)=\frac{Kp}{r^3}$ opposite to the direction of dipole.
Electric field at centre due to horizontal components of dipoles will cancel out.
Total electric field at centre due to vertical components will be $E=\frac{Kp/\sqrt{2}}{r^3}\times 2$ in vertically downward direction.
$\therefore E=\frac{-Kp\sqrt{2}}{r^3}\hat{j}$