Two electric dipoles of dipole moments $1.2 \times 10^{-30} \, C \cdot m$ and $2.4 \times 10^{-30} \, C \cdot m$ are placed in two different uniform electric fields of strengths $5 \times 10^{4} \, N \cdot C^{-1}$ and $15 \times 10^{4} \, N \cdot C^{-1}$ respectively. The ratio of maximum torque experienced by the electric dipoles will be $\frac{1}{x}$. The value of $x$ is $.....$

$A$ neutral ammonia $(NH_3)$ molecule in its vapour state has an electric dipole moment of magnitude $5 \times 10^{-30} \ C \cdot m$. How far apart are the molecule's centres of positive and negative charge?

$A$ positive charge $Q$ is distributed uniformly over a circular ring of radius $R$. $A$ point particle having a mass $m$ and a negative charge $-q$ is placed on its axis at a distance $x$ from the centre. Assuming $x \ll R$,find the time period of oscillation of the particle,if it is released from there [neglect gravity].

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. . . . . .

Three point charges $+q$,$-2q$ and $+q$ are placed at points $(x = 0, y = a, z = 0)$,$(x = 0, y = 0, z = 0)$ and $(x = a, y = 0, z = 0)$ respectively. The magnitude and direction of the electric dipole moment vector of this charge assembly are

$A$ molecule with a dipole moment $p$ is placed in an electric field of strength $E$. Initially,the dipole is aligned parallel to the field. If the dipole is to be rotated to be anti-parallel to the field,the work required to be done by an external agent is: