Two electric dipoles of moments $p$ and $27 p$ are placed in opposite directions on a line at a distance of $24 \,cm$. The electric field will be zero at a point between the dipoles whose distance from the dipole of moment $p$ is

The electric field $\vec{E}$ at a point at a distance $r$ from the axis of a dipole is given by:

If the electric field intensity at a distance $x$ on the axis of a dipole is equal to the electric field intensity at a distance $y$ on its equatorial line,then the ratio $x:y$ is:

$A$ small electric dipole $\vec{p}_0$,having a moment of inertia $I$ about its center,is kept at a distance $r$ from the center of a spherical shell of radius $R$. The surface charge density $\sigma$ is uniformly distributed on the spherical shell. The dipole is initially oriented at a small angle $\theta$ as shown in the figure. While staying at a distance $r$,the dipole is free to rotate about its center. If released from rest,then which of the following statement$(s)$ is (are) correct? [$\varepsilon_0$ is the permittivity of free space.]
$(A)$ The dipole will undergo small oscillations at any finite value of $r$.
$(B)$ The dipole will undergo small oscillations at any finite value of $r > R$.
$(C)$ The dipole will undergo small oscillations with an angular frequency of $\sqrt{\frac{\sigma p_0}{4 \varepsilon_0 I}}$ at $r = 2R$.
$(D)$ The dipole will undergo small oscillations with an angular frequency of $\sqrt{\frac{\sigma p_0}{100 \varepsilon_0 I}}$ at $r = 10R$.

$A$ given charge is situated at a certain distance from an electric dipole in the axial position and experiences a force $F$. If the distance of the charge is doubled,the force acting on the charge will be:

An electric dipole having each charge of magnitude $2 \mu C$ is placed in an electric field of intensity $8 \times 10^{4} \ N/C$. If the maximum torque acting on the dipole is $4 \times 10^{-3} \ N \cdot m$,the length of the dipole is: (in $mm$)