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Step-by-Step Explanation
Step 1: Recognize the Physical Situation
We have a sphere of radius $R$ uniformly charged with total charge $Q$. We want the dependence of the electric field $E$ on the distance $r$ from the center, both inside ($r < R$) and outside ($r > R$) the sphere.
Step 2: Apply Gauss's Law Inside the Sphere ($r < R$)
• Consider a Gaussian surface (a sphere) of radius $r < R$ centered at the same point as the charged sphere.
• Because the charge distribution is uniform, the amount of charge enclosed $q_{\text{enc}}$ is proportional to the volume enclosed:
$$
q_{\text{enc}} = Q \times \frac{\frac{4}{3}\pi r^3}{\frac{4}{3}\pi R^3} = Q \left(\frac{r^3}{R^3}\right).
$$
• Gauss’s law states $ \oint \mathbf{E}\cdot d\mathbf{A} = \frac{q_{\text{enc}}}{\epsilon_0} $. Because of symmetry, $E$ is the same everywhere on this Gaussian surface and points radially outward, so
$$
E \times 4\pi r^2 = \frac{Q\,\frac{r^3}{R^3}}{\epsilon_0}.
$$
• Solving for $E$ inside the sphere:
$$
E_{\text{in}} = \frac{Q}{4\pi \epsilon_0 R^3} \, r.
$$
• Thus, $E_{\text{in}} \propto r$, which means the electric field grows linearly from zero at the center ($r=0$) to a maximum at the surface ($r=R$).
Step 3: Apply Gauss's Law Outside the Sphere ($r \ge R$)
• Now consider a Gaussian surface of radius $r > R$ that encloses the entire charged sphere.
• The total charge enclosed is $Q$, independent of $r$ (as long as $r > R$).
• Using Gauss’s law again:
$$
E \times 4\pi r^2 = \frac{Q}{\epsilon_0}.
$$
• Solving for the electric field outside gives:
$$
E_{\text{out}} = \frac{Q}{4\pi \epsilon_0\, r^2}.
$$
• Hence $E_{\text{out}} \propto \frac{1}{r^2}$.
Step 4: Shape of the Graph
• For $r < R$, the field increases linearly with $r$ (straight line starting from zero).
• For $r > R$, the field decreases as $1/r^2$ (hyperbolic decrease).
Conclusion
The correct plot of $E$ versus $r$ starts at $E=0$ at the center ($r=0$), increases linearly inside the sphere, and then decreases as $1/r^2$ outside. Therefore, the correct answer is:
Correct Answer :
