Question

A small hollow conducting sphere of radius $r_1$ is given a charge Q. It is surrounded by a concentric conducting spherical shell of inner radius $r_2$ and outer radius $r_3$, having charge $-3q$. If a point charge 2q were kept at the centre, find surface charge density on the inner surface of (1) sphere, and (2) shell.

Hint:

Conductors invariably manipulate their vast supply of free charges to effectively eliminate internal electric fields; analyzing nested shells is purely a rigorous logical exercise in successfully balancing inner enclosed charges step-by-step from the inside out.
Surface charge density rigorously requires mathematically dividing the specific bounded charge by the exact corresponding geometrical spherical surface area, not the volume.

Answer 1

Concept:
• A fundamental cornerstone property of solid conductors in stable electrostatic equilibrium is that the electric field deeply robustly inside the actual solid metallic material is strictly zero.

• This mandatory zero internal field phenomenon inherently forces mobile induced charges to perfectly and dynamically arrange themselves entirely on the available inner and outer surfaces of the conductors.

• By strategically placing a mathematical Gaussian surface entirely within the conductive thickness, we mathematically mandate that the net enclosed charge must total exactly zero.

Step 1: Analyze the Inner Surface of the Primary Sphere (1)
Consider the innermost hollow conducting sphere which possesses an exact radius $r_1$.
To absolutely guarantee that the internal electric field deeply inside the solid shell of this conducting sphere is completely zero, a Gaussian surface drawn precisely inside its metallic body must logically enclose a net zero total charge ($Q_{enc} = 0$).
The spatial region enclosed essentially contains the central point charge ($+2q$) and whatever new charge ($q_{inner1}$) aggressively induces on the exact inner surface at radius $r_1$.
Therefore, the sum must perfectly equate to zero:
\[ 2q + q_{inner1} = 0 \implies q_{inner1} = -2q \]
This physically requires that an induced charge of precisely $-2q$ must form perfectly on its inner surface to actively counteract the central charge.
The resulting surface charge density $\sigma$ is calculated by dividing this induced charge by the precise spherical inner surface area:
\[ \sigma_{\text{sphere, inner}} = \frac{q_{inner1}}{A_1} = \frac{-2q}{4\pi r_1^2} \]

Step 2: Analyze the Inner Surface of the Secondary Shell (2)
Now consider the massive outer conducting spherical shell which strictly possesses an inner radius $r_2$.
Similarly, a Gaussian surface flawlessly drawn completely inside the outer conducting shell's solid metallic thickness must absolutely also enclose exactly zero net charge.
The total enclosed charge inside this vast hollow cavity space is clearly the algebraic sum of the central point charge and the total net charge safely residing on the entire primary inner sphere (both its inner and outer surfaces combined).
The combined inner charge total is exactly $(+2q) + (+Q)$.
Let the desperately needed induced charge on the shell's inner surface be $q_{inner2}$.
To maintain zero net enclosed charge, the sum must precisely equal zero:
\[ (+2q + Q) + q_{inner2} = 0 \implies q_{inner2} = -(2q + Q) \]
Therefore, a massive compensating induced charge of $-(2q + Q)$ must systematically form entirely on the inner surface of the outer shell.
The corresponding surface charge density decisively becomes the induced charge divided by that specific surface area:
\[ \sigma_{\text{shell, inner}} = \frac{q_{inner2}}{A_2} = \frac{-(2q + Q)}{4\pi r_2^2} \]