JEE Advanced 2025 — Paper 2 — PHY-I solution

JEE Advanced2025Paper 2PHY-I

Two co-axial conducting cylinders of same length $ℓ$ with radii $2 R$ 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 $κ = 5$ . Consider an imaginary plane of the same length $ℓ$ 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 ( $ϵ_{0}$ is the permittivity of free space):

A
$\frac{Q}{30 ϵ _{0}}$
B
$\frac{Q}{15 ϵ _{0}}$
C
$\frac{Q}{60 ϵ _{0}}$
D
$\frac{Q}{120 ϵ _{0}}$

Solution

Treat the cylinder as effectively long so the field is purely radial. Since the outer grounded cylinder confines all field lines inside the annulus, the imaginary plane (at perpendicular distance $R$ from the axis) intercepts field only over the two arcs where it lies inside the annulus (radii from $2 R$ to $2 R$ ); the segment passing through the inner cylinder contributes nothing.

For a line of charge density $λ = Q / ℓ$ inside dielectric $κ$ , the radial field is $E = \frac{λ}{2 π ϵ _{0} κ r}$ . Taking an element of length $d y$ on the plane at radial distance $r = R sec θ$ , with $d y = R sec^{2} θ d θ$ and the component perpendicular to the plane being $E cos θ$ :

$d ϕ = E ℓ cos θ d y = \frac{λ ℓ}{2 π ϵ _{0} κ} d θ$

The chord at distance $R$ meets the inner cylinder ( $r = 2 R$ ) at $cos θ = 1/ 2$ , so $θ = π /4$ , and meets the outer cylinder ( $r = 2 R$ ) at $cos θ = 1/2$ , so $θ = π /3$ . Integrating one arc:

$ϕ_{arc} = \frac{Q}{2 π ϵ _{0} κ} (\frac{π}{3} - \frac{π}{4}) = \frac{Q}{2 π ϵ _{0} \cdot 5} \cdot \frac{π}{12} = \frac{Q}{120 ϵ _{0}}$

By symmetry, the plane crosses two such arcs (upper and lower), so the total flux is

$ϕ = 2 \times \frac{Q}{120 ϵ _{0}} = \frac{Q}{60 ϵ _{0}}$

$∴$ Option (C) is correct.

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