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The electric field produced by a thin spherical shell of radius $R$ with a uniform charge $Q$ can be analyzed in two regions: inside the shell and outside the shell.
1. Electric Field Inside the Shell
According to Gauss's law, the electric field inside a uniformly charged spherical shell is zero. This is because the electric field contributions from different parts of the shell cancel each other out at any point inside the shell. Therefore, for any point where the distance $r$ from the center of the shell is less than $R$ (i.e., $0 \leq r < R$), we have:
$$ E(r) = 0 \quad \text{for } r < R $$
2. Electric Field Outside the Shell
For points outside the shell (i.e., where $r > R$), the shell behaves like a point charge located at its center. The electric field at a distance $r$ from the center of the shell is given by the formula:
$$ E(r) = k \frac{Q}{r^2} $$
where $k$ is the electrostatic constant (Coulomb's constant), and $Q$ is the total charge on the shell. This means that as you move further away from the shell, the electric field decreases with the square of the distance.
3. Graphical Representation
In the graph representing the electric field $E(r)$ as a function of the distance $r$, we observe the following characteristics:
For $0 \leq r < R$, the electric field is zero.
For $r = R$, the electric field starts to increase as $E(r) = k \frac{Q}{R^2}$.
For $r > R$, the electric field decreases with $1/r^2$ as you move further away from the shell.
Thus, the correct graph that represents this behavior is graph $(a)$, which shows the electric field being zero inside the shell and following the $1/r^2$ relationship outside the shell.
