Electric potential due to a uniformly charged thin spherical shell

Let the shell have radius $R$ and total charge $Q$.

The electric potential at a distance $r$ from the centre is:

• Outside the shell $(r \ge R)$: $V(r)=\frac{1}{4\pi\varepsilon_0}\frac{Q}{r}$

• On the surface $(r=R)$: $V(R)=\frac{1}{4\pi\varepsilon_0}\frac{Q}{R}$

• Inside the shell $(r<R)$: $V(r)=\frac{1}{4\pi\varepsilon_0}\frac{Q}{R}$

So, inside the shell the potential is constant, and outside it decreases as $\dfrac{1}{r}$.

Nature of the graph

Therefore, the graph of $V$ versus radial distance $r$ is:

• a horizontal line from $r=0$ to $r=R$,
• then from $r=R$ onward, a decreasing hyperbola proportional to $1/r$.

Also, the graph is continuous at $r=R$.

Matching with the given options

The correct option must show:

• constant potential inside the shell,
• smoothly joining to a decreasing $1/r$ curve outside.

This corresponds to Option A.

Conclusion

$\boxed{A}$