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 electric field at a point distant x from the centre for (1) $x > r_3$, and (2) $r_1 < x < r_2$.

Hint:

For any point strictly located outside a perfectly spherically symmetric charge distribution, the entire complex system behaves mathematically exactly as if all its combined charge were tightly concentrated at the central origin.
Always remember the surface area of a sphere is strictly $4\pi x^2$, completely avoiding common confusion with the circle's area $\pi x^2$.

Answer 1

Concept:
• Gauss's law serves as a powerful mathematical tool to decisively determine the electric field generated by highly symmetric continuous charge distributions.

• For universally symmetric spherical systems, the electric field vector perfectly aligns radially with the area vector everywhere precisely on a concentric spherical Gaussian surface.

• This elegant geometric alignment simplifies the complex flux integral strictly into the algebraic product of the constant electric field magnitude and the entire surface area: $E \cdot 4\pi x^2 = \frac{Q_{enc}}{\epsilon_0}$.

Step 1: Determine Electric Field for region (1) $x > r_3$
For this completely outermost region, the mathematical Gaussian surface is drawn broadly to safely encompass the entire physical apparatus.
The overarching Gaussian surface universally encloses all localized point and shell charges contained anywhere in the entire physical system.
We must meticulously sum every single charge to firmly find the absolute total enclosed charge:
The central point charge is $+2q$.
The net charge purposefully given to the inner hollow sphere is $+Q$.
The net charge purposefully given to the massive outer conducting shell is $-3q$.
The absolute total enclosed charge is mathematically computed by combining these algebraically:
\[ Q_{enc} = 2q + Q + (-3q) = Q - q \]
Using the standard robust spherical symmetry relation, we easily derive the external radial electric field:
\[ E \cdot 4\pi x^2 = \frac{Q - q}{\epsilon_0} \]
\[ E_{(x > r_3)} = \frac{Q - q}{4\pi\epsilon_0 x^2} \]

Step 2: Determine Electric Field for region (2) $r_1 < x < r_2$
For this specific intermediate region cleanly located between the nested conductors, the Gaussian surface is rigidly restricted in radius.
As previously proven definitively in the prior flux calculation, the comprehensively enclosed total charge here consists strictly of the central charge and the inner sphere's net charge.
\[ Q_{enc} = 2q + Q \]
Applying identically the exact same geometric symmetry arguments directly yields the electric field strictly residing within this empty gap:
\[ E \cdot 4\pi x^2 = \frac{2q + Q}{\epsilon_0} \]
\[ E_{(r_1 < x < r_2)} = \frac{2q + Q}{4\pi\epsilon_0 x^2} \]
This accurately establishes the precise magnitude of the electric fields in the requested distinct spatial domains.