Two co-axial conducting cylinders of same length with radii 2 R and...

Two co-axial conducting cylinders of same length

\ell

with radii

\sqrt{2} R

and

2 R

are kept, as shown in Fig. 1. The charge on the inner cylinder is

Q

and the outer cylinder is grounded. The annular region between the cylinders is filled with a material of dielectric constant

\kappa=5

. Consider an imaginary plane of the same length

\ell

at a distance R from the common axis of the cylinders. This plane is parallel to the axis of the cylinders. The cross-sectional view of this arrangement is shown in Fig. 2. Ignoring edge effects, the flux of the electric field through the plane is (

\epsilon_0

is the permittivity of free space): Two co-axial conducting cylinders of same length with radii 2 R and 2 R are kept, as shown in Fig. 1. The charge on the

Options

A $\frac{\mathrm{Q}}{30 \epsilon_0}$
B $\frac{\mathrm{Q}}{15 \epsilon_0}$
C $\frac{\mathrm{Q}}{60 \epsilon_0}$
D $\frac{\mathrm{Q}}{120 \epsilon_0}$

Correct answer

C. $\frac{\mathrm{Q}}{60 \epsilon_0}$

Step-by-step solution

Here we are assuming that "

\ell

" is very large just for the sake of symmetry. Outside cylinder will have zero electric field inside, so the flux generated on the plate will be due to inner cylinder only in sections AB and CD , as section be will be at that place where electric field is zero.flux through element will be

\begin{aligned} \mathrm{d} \phi & =\overrightarrow{\mathrm{E}} \cdot \overrightarrow{\mathrm{dS}} \\ \mathrm{d} \phi & =\frac{2 \mathrm{k} \lambda}{} \mathrm{dy} \cdot \ell \cdot \cos \theta ..... \mathrm{(1)}\end{aligned}

from figure we can say that

\begin{aligned} & \cos \theta=\frac{\mathrm{R}}{\mathrm{r}} \Rightarrow \mathrm{r}=\mathrm{R} \sec \theta \\ & \tan \theta=\frac{\mathrm{y}}{\mathrm{R}} \Rightarrow \mathrm{y}=\mathrm{R} \tan \theta \\ & \Rightarrow \mathrm{dy}=\mathrm{R} \sec ^2 \theta \mathrm{ d} \theta \\ & \mathrm{ d} \phi=\frac{2 \mathrm{k} \lambda}{\mathrm{R} \sec \theta} \mathrm{R} \cdot \sec ^2 \theta \cdot \ell \cdot \cos \theta \mathrm{ d} \theta \\ & \mathrm{ d} \phi=2 \mathrm{k} \lambda \ell \mathrm{dq} \\ & \int_0^{\phi_{\mathrm{AB}}} \mathrm{d} \phi=2 \mathrm{k} \lambda \ell \int_{\pi / 4}^{\pi / 3} \mathrm{ d} \theta \\ & \phi_{\mathrm{AB}}=2 \mathrm{k} \lambda \ell\left[\frac{\pi}{3}-\frac{\pi}{4}\right]\end{aligned}

\begin{aligned} & \phi_{\mathrm{AB}}=2 \mathrm{kQ}\left[\frac{\pi}{12}\right] \\ & \phi_{\mathrm{AB}}=2 \times \frac{1}{4 \pi \epsilon_0 \epsilon_{\mathrm{r}}} \mathrm{Q}\left[\frac{\pi}{12}\right] \\ & \phi_{\mathrm{AB}}=\frac{\mathrm{Q}}{120 \epsilon_0} \\ & \phi_{\text{plate }}=\phi_{\mathrm{AB}}+\phi_{\mathrm{BC}}+\phi_{\mathrm{CD}} \quad\left(\phi_{\mathrm{CD}}=\phi_{\mathrm{AB}}\right) \\ & \phi_{\text{plate }}=\frac{\mathrm{Q}}{60 \epsilon_0}\end{aligned}

Practice Electrostatics on Quantrex Academy →

More from Electrostatics

Two charges Q_1 = q and Q_2 = mq are placed at the points P_1(a, b) and P_2(ma, mb)… JEE Advanced 2026 Consider an electric dipole comprising two charges +q and -q each with mass m , separated… JEE Advanced 2026 Two point charges q_1=3\,μ C and q_2=-4\,μ C are placed at points (2 i +3 j +3 k ) and (… JEE Main 2026 The electric potential as a function of x, y is given by V = 5(x^2 - y^2) V. The electric… JEE Main 2026 A thin half ring of radius 35 cm is uniformly charged with a total charge of Q coulomb.… JEE Main 2026 A three coulomb charge moves from the point (0, -2, -5) to the point (5, 1, 2) in an… JEE Main 2026 A particle of charge q and mass m is projected from origin with an initial velocity v = (… JEE Main 2026 A rigid dipole undergoes a simple harmonic motion about its centre in the presence of an… JEE Main 2026 Full Electrostatics list All JEE Advanced PYQs