Animated Solution for Physics - Electrostatics: In a uniformly charged sphere of total charge $Q$ and radius $R$, the electric field $E$ is plotted as function of distance from the centre. The graph which would correspond to the above will be

Imagine a solid sphere of radius $R$ with a total charge $Q$ uniformly distributed throughout its volume. Our goal is to understand how the electric field $E$ behaves as we move from the center of the sphere all the way out to infinity, and then plot this behavior on a graph.

Let's start our journey right at the center of the sphere and move outwards. Using Gauss's Law, we can determine the electric field at any point inside the sphere (where $r\le R$). The formula is given by:

Look closely at this equation. The terms $Q$, $R$, and ${ϵ}_0$ are all constants. The only variable is $r$, the distance from the center. This means that the electric field $E_{in}$ is directly proportional to $r$ ($E_{in}\propto r$).

In graphical terms, a direct proportionality represents a straight line passing through the origin. So, as we move from the center to the surface, the electric field increases linearly.

Now, let's step outside the sphere. For any point outside (where $r\ge R$), the entire uniformly charged sphere behaves exactly as if all its charge were concentrated at a single point at its center. The formula for the electric field is the familiar Coulomb's Law expression:

Here, the electric field $E_{out}$ is inversely proportional to the square of the distance ($E_{out}\propto \frac{1}{r^2}$). This means that as we move further away from the sphere, the electric field decreases rapidly, forming a curve that asymptotically approaches zero.

To get the final graph, we simply combine these two behaviors. From $r=0$ to $r=R$, we draw a straight line with a positive slope. From $r=R$ onwards, we draw a curve that decreases as $1/r^2$.

This combined shape perfectly matches the graph shown in option (c). It's a beautiful demonstration of how physical laws translate into elegant geometric curves.