A small spherical ball having charge \( q \) and mass \( m \), is tied to a thin massless non-conducting string of length \( l \). The other end of the string is fixed to an infinitely extended thin non-conducting sheet with uniform surface charge density \( \sigma \). Under equilibrium, the string makes an angle of 45° with the sheet as shown in the figure. Then \( \sigma \) is given by \[ g \text{ is the acceleration due to gravity and } \epsilon_0 \text{ is the permittivity of free space.} \]
Given a spherically symmetric charge density $\rho(r)=\begin{cases}kr^2, & r<R \\ 0, & r>R\end{cases}$ (k being a constant), the electric field for $r<R$ is (take the total charge as $Q$)
Let the electric field in some region $R$ be given by $\vec{E} = e^{-y^2}\hat{i} + e^{-x^2}\hat{j}$. From this we conclude that
A sphere of radius \( R \) has a uniform charge density \( \rho \). A sphere of smaller radius \( \frac{R}{2} \) is cut out from the original sphere, as shown in the figure below. The center of the cut-out sphere lies at \( z = \frac{R}{2} \). After the smaller sphere has been cut out, the magnitude of the electric field at \( z = - \frac{R}{2} \) is \( \frac{\rho R}{2 \epsilon_0} \). The value of the integer \( n \) is:
In a coaxial cable, the radius of the inner conductor is 2 mm and that of the outer one is 5 mm. The inner conductor is at a potential of 10 V, while the outer conductor is grounded. The value of the potential at a distance of 3.5 mm from the axis is:
A sphere of radius \( R \) has a uniform charge density \( \rho \). A sphere of smaller radius \( \frac{R}{2} \) is cut out from the original sphere, as shown in the figure. The center of the cut-out sphere lies at \( z = \frac{R}{2} \). After the smaller sphere has been cut out, the magnitude of the electric field at \( z = -\frac{R}{2} \) is \( \frac{\rho R}{n \epsilon_0} \). The value of the integer \( n \) is:
