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Step-by-Step Solution
Step 1: Identify the given situation
We have three concentric spherical shells of radii $a$, $b$, and $c$, with $a < b < c$. A total charge $Q$ is distributed over these three shells such that their surface charge densities are equal.
Step 2: Express the individual charges on each shell
Let $\sigma$ be the common surface charge density on each shell. Then the charge on:
• Shell of radius $a$ is
$$
Q_a = \sigma \cdot 4\pi a^2
$$
• Shell of radius $b$ is
$$
Q_b = \sigma \cdot 4\pi b^2
$$
• Shell of radius $c$ is
$$
Q_c = \sigma \cdot 4\pi c^2
$$
Since the total charge is $Q$, we have
$$
Q_a + Q_b + Q_c = Q
\quad \Longrightarrow \quad
\sigma \cdot 4\pi (a^2 + b^2 + c^2) = Q.
$$
Thus,
$$
\sigma = \frac{Q}{4\pi \left(a^2 + b^2 + c^2\right)}.
$$
Step 3: Write down the potential at a point inside the innermost shell
We need the potential at a distance $r$ from the center such that $r < a$. Inside a uniformly charged thin spherical shell, the entire charge of that shell behaves like a point charge located at the center (by symmetry and the shell theorem). Thus, the potential at $r < a$ is the sum of potentials due to all three shells:
$$
V_\text{total} = \frac{1}{4\pi \varepsilon_0}
\left( \frac{Q_a}{a} + \frac{Q_b}{b} + \frac{Q_c}{c} \right).
$$
Step 4: Substitute the individual charges in terms of $Q$ and simplify
From Step 2,
$$
Q_a = \frac{Q \, a^2}{a^2 + b^2 + c^2}, \quad
Q_b = \frac{Q \, b^2}{a^2 + b^2 + c^2}, \quad
Q_c = \frac{Q \, c^2}{a^2 + b^2 + c^2}.
$$
Substitute these into the expression for $V_{\text{total}}$:
\[
V_\text{total}
= \frac{1}{4\pi \varepsilon_0} \left(
\frac{Q \, a^2}{(a^2 + b^2 + c^2) \, a}
+ \frac{Q \, b^2}{(a^2 + b^2 + c^2) \, b}
+ \frac{Q \, c^2}{(a^2 + b^2 + c^2) \, c}
\right).
\]
Simplify each term inside the parentheses:
\[
\frac{Q \, a^2}{a (a^2 + b^2 + c^2)} = \frac{Q \, a}{a^2 + b^2 + c^2},
\quad
\frac{Q \, b^2}{b (a^2 + b^2 + c^2)} = \frac{Q \, b}{a^2 + b^2 + c^2},
\quad
\frac{Q \, c^2}{c (a^2 + b^2 + c^2)} = \frac{Q \, c}{a^2 + b^2 + c^2}.
\]
Hence,
\[
V_{\text{total}} = \frac{1}{4\pi \varepsilon_0} \,
\frac{Q (a + b + c)}{a^2 + b^2 + c^2}.
\]
Step 5: Final expression for the potential
The total potential at a point inside the innermost shell ($r < a$) is
$$
V_{\text{total}} = \frac{Q (a + b + c)}{4\pi \varepsilon_0 (a^2 + b^2 + c^2)}.
$$
This matches the given correct answer.
