Question

A small hollow conducting sphere radius r₁ is given a charge Q. It is surrounded by a concentric conducting spherical shell of inner radius r₂ and outer radius r₃, having charge −3q. 

If a point charge 2q were kept at the centre, find:

1. The electric flux through a concentric spherical Gaussian surfaces of radius x for (1) x < r₁, and (2) r₁ < x <  r₂.
2. Electric field at a point distant x from the centre for (1) x > r₃, and (2) r₁ < x < r₂.
3. Surface charge density on inner surfaces of (1) sphere, and (2) shell.

Answer 1

Given: Point charge at center = +2q

Inner sphere (r₁) = Q

Outer sphere (r₂, r₃) = −3q

(I) Electric Flux (Gauss Law): The electric flux through a Gaussian surface is determined by the total enclosed charge q.

Φ = `q_"enclosed"/ε_0`

(1) For x < r₁: The only charge enclosed is the point charge at the center (+2q).

Φ = `(2q)/ε_0`

(2) For r₁ < x <  r₂: The enclosed charge is the point charge (2q) plus the total charge on the inner sphere (Q).

Φ = `(Q + 2q)/ε_0`

(II) Electric Field (E): The electric field at a distance x from the centre is given by

E =`1/(4 pi ε_0) (q_"enclosed")/x^2`

(1) For x > r₃: The Gaussian surface encloses all charges: the point charge 2q, the inner sphere charge Q, and the outer shell charge −3q.

`q_"enclosed" = 2q + Q + (−3q)`

= Q − q

E = `1/(4 pi ε_0) (Q − q)/x^2`

= `(Q − q)/(4 pi ε_0 x^2)`

(2) For r₁ < x < r₂: The enclosed charge is the point charge 2q and the charge Q on the inner sphere.

`q_"enclosed" = 2q + Q`

= Q + 2q

E = `1/(4 pi ε_0) (Q + 2q)/x^2`

= `(Q + 2q)/(4 pi ε_0 x^2)`

(III) Surface Charge Density (σ)

σ = `"Induced Charge"/"Surface Area"`

(1) Inner surface of sphere (r₁): To make the field zero within the conducting wall of the sphere, the charge on its inner surface must exactly neutralize the point charge 2q at the centre.

Induced charge is −2q.

`σ_(r_(1)) = (-2q)/(4pir_1^2)`

= `-(q)/(2pir_1^2)`

(2) Inner surface of shell (r₂): To make the field zero within the conducting shell, the charge on its inner surface must neutralize the total charge inside it (the point charge 2q and the total charge Q on the inner sphere).

Induced charge is −(Q + 2q).

`σ_(r_(2)) = (-Q + 2q)/(4pir_2^2)`