For an electric dipole with charge $q = 2 \times 10^{-6} \ C$ and separation distance $d = 0.01 \ m$,if the external electric field is $E = 5 \times 10^{5} \ N/C$,calculate the maximum torque acting on the dipole.

If ${E_a}$ is the electric field strength of a short dipole at a point on its axial line and ${E_e}$ is the electric field strength on the equatorial line at the same distance,then:

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?

Match Column-$I$ with Column-$II$ related to an electric dipole of dipole moment $\vec{p}$ that is placed in a uniform electric field $\overrightarrow{E}$.

Column-$I$ (Angle between $\vec{p}$ and $\vec{E}$)	Column-$II$ (Potential energy of the dipole)
$a. 180^{\circ}$	$i. -pE$
$b. 120^{\circ}$	$ii. pE$
$c. 90^{\circ}$	$iii. \frac{1}{2} pE$
	$iv. 0$

An electric dipole is placed along the $x$-axis at the origin $O$. $A$ point $P$ is at a distance of $20 \, cm$ from this origin such that $OP$ makes an angle $\frac{\pi}{3}$ with the $x$-axis. If the electric field at $P$ makes an angle $\theta$ with the $x$-axis,the value of $\theta$ would be

Two point dipoles $p \hat{k}$ and $\frac{p}{2} \hat{k}$ are located at $(0, 0, 0)$ and $(1 \text{ m}, 0, 2 \text{ m})$ respectively. The resultant electric field due to the two dipoles at the point $(1 \text{ m}, 0, 0)$ is