A positively charged thin metal ring of radius $R$ is fixed in the $xy - $ plane with its centre at the $O$. A negatively charged particle $P$ is released from rest at the point $(0,\,0,\,{z_0})$, where ${z_0} > 0$. Then the motion of $P$ is

What will be the magnitude of electric field at point $O$ as shown in figure ? Each side of the figure is $I$ and perpendicular to each other.

Charges $q$, $2q$, $3q$ and $4q$ are placed at the corners $A$,$ B$,$ C$ and $D$ of a square as shown in the following figure. The direction of electric field at the centre of the square is along

A uniformly charged rod of length $4\,m$ and linear charge density $\lambda  = 30\,\mu C/m$ is placed as shown in figure. Calculate the $x-$ component of electric field at point $P$.

A positively charged pendulum is oscillating in a uniform electric field pointing upwards. Its time period as compared to that when it oscillates without electric field

The charge distribution along the semi-circular arc is non-uniform . Charge per unit length $\lambda $ is given as $\lambda  = {\lambda _0}\sin \theta $ , with $\theta $ measured as shown in figure. $\lambda_0$ is a positive constant. The radius of arc is $R$ . The electric field at the center $P$ of semi-circular arc is $E_1$ . The value of $\frac{{{\lambda _0}}}{{{ \in _0}{E_1}R}}$ is