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

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

$\begin{array}{l} E_{inside} \propto r \\ E_{outsides} \propto \frac{1}{r^{2}} \end{array}$

To determine the correct graph of electric field E versus distance r from the center for a uniformly charged sphere, we need to analyze the behavior of E both inside and outside the sphere.

Step 1: Electric Field Outside the Sphere (r ≥ R)

For points outside the sphere (r ≥ R), the sphere behaves as if all its charge Q is concentrated at its center. Thus, the electric field is given by Coulomb's law:

E = \frac{1}{4 π ε_{0}} \frac{Q}{r^{2}}

This shows that E ∝ 1/r² for r ≥ R. So, as r increases, E decreases rapidly.

Step 2: Electric Field Inside the Sphere (r < R)

For points inside the sphere (r < R), only the charge enclosed within a Gaussian surface of radius r contributes to the electric field. The charge enclosed is:

Q_{enc} = Q \frac{r^{3}}{R^{3}}

Using Gauss's law, the electric field inside is:

E = \frac{1}{4 π ε_{0}} \frac{Q_{enc}}{r^{2}} = \frac{1}{4 π ε_{0}} \frac{Q}{R^{3}} r

This shows that E ∝ r for r < R. So, inside the sphere, E increases linearly with r.

Step 3: Behavior at r = R

At the surface (r = R), both expressions give the same value:

E = \frac{1}{4 π ε_{0}} \frac{Q}{R^{2}}

So, the graph is continuous at r = R.

Step 4: Graph Characteristics

The correct graph should show:

A straight line passing through the origin for r < R (E ∝ r).
A curve that decreases as 1/r² for r > R.
Maximum value at r = R, with no discontinuity.

Among the options, the graph that matches this is the one where E increases linearly from 0 at r=0 to a maximum at r=R, and then decreases hyperbolically for r>R.

Detailed Solution

Step 1: Electric Field Outside the Sphere (r ≥ R)

Step 2: Electric Field Inside the Sphere (r < R)

Step 3: Behavior at r = R

Step 4: Graph Characteristics

Related Topics and Formulae

Detailed Solution

Step 1: Electric Field Outside the Sphere (r ≥ R)

Step 2: Electric Field Inside the Sphere (r < R)

Step 3: Behavior at r = R

Step 4: Graph Characteristics

Related Topics and Formulae

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