Two electric dipoles of moment $P$ and $64 P$ are placed in opposite directions on a line at a distance of $25 \ cm$. The electric field will be zero at a point between the dipoles whose distance from the dipole of moment $P$ is.....$cm$.

An electric dipole is situated in an electric field as shown in the figure. The dipole and electric field are both in the plane of the paper. The dipole is rotated about an axis perpendicular to the paper at point $A$ in an anti-clockwise direction. If the angle of rotation is measured with respect to the direction of the electric field,then the torque for different values of the angle of rotation $\theta$ is correctly represented by which graph among $a, b, c, d$ given in the figure?

Two charges $+q$ and $-q$ are placed at a distance $b$ apart as shown in the figure. The electric field at a point $P$ on the perpendicular bisector as shown is

At any point on the perpendicular bisector of the line joining two equal and opposite charges,which of the following is true?

Three charges $+2q$, $+3q$, and $-4q$ are situated at $(0, -3a)$, $(2a, 0)$, and $(-2a, 0)$ respectively in the $xy$ plane. The resultant dipole moment about the origin is . . . . . .

One end of a spring of negligible unstretched length and spring constant $k$ is fixed at the origin $(0,0)$. $A$ point particle of mass $m$ carrying a positive charge $q$ is attached at its other end. The entire system is kept on a smooth horizontal surface. When a point dipole $\overrightarrow{p}$ pointing towards the charge $q$ is fixed at the origin,the spring gets stretched to a length $\ell$ and attains a new equilibrium position. If the point mass is now displaced slightly by $\Delta \ell \ll \ell$ from its equilibrium position and released,it is found to oscillate at frequency $\frac{1}{\delta} \sqrt{\frac{k}{m}}$. The value of $\delta$ is. . . . . .