AIIMS 2004 Physics Electric Field MCQ Question

To understand the electric field due to a uniformly charged sphere of radius $R$, we need to consider two regions: inside the sphere (where $r < R$) and outside the sphere (where $r \geq R$).

Electric Field Inside the Sphere $(r < R)$

For a uniformly charged sphere, the electric field inside the sphere can be derived using Gauss's Law. According to Gauss's Law, the electric field $E$ at a distance $r$ from the center of the sphere is given by:

$$\Phi_E = \oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\varepsilon_0}$$

Where:

• $\Phi_E$ is the electric flux,
• $Q_{\text{enc}}$ is the charge enclosed by the Gaussian surface,
• $\varepsilon_0$ is the permittivity of free space.

If the sphere has a total charge $Q$, the charge enclosed $Q_{\text{enc}}$ by a Gaussian surface of radius $r$ (inside the sphere) is proportional to the volume enclosed:

$$Q_{\text{enc}} = Q \cdot \frac{V_{\text{enc}}}{V_{\text{total}}} = Q \cdot \frac{\frac{4}{3}\pi r^3}{\frac{4}{3}\pi R^3} = Q \cdot \frac{r^3}{R^3}$$

Thus, Gauss's law becomes:

$$E \cdot 4\pi r^2 = \frac{Q \cdot r^3}{R^3 \varepsilon_0}$$

Solving for $E$:

$$E = \frac{Qr}{4\pi \varepsilon_0 R^3}$$

This shows that the electric field inside the uniformly charged sphere increases linearly with the distance $r$ from the center.

Electric Field Outside the Sphere $(r \geq R)$

For points outside the sphere, the electric field behaves as if all the charge were concentrated at the center of the sphere. Thus, the electric field is given by:

$$E = \frac{Q}{4\pi \varepsilon_0 r^2}$$

This shows that the electric field outside the sphere decreases with the square of the distance from the center.

Graphical Representation

Now, let's consider the graphical representation of the electric field as a function of the distance from the center:

1. For $r < R$: The electric field increases linearly from $0$ at $r = 0$ to its maximum value at $r = R$.
2. For $r \geq R$: The electric field decreases with $\frac{1}{r^2}$ behavior, starting from its value at $r = R$.

The correct answer, therefore, corresponds to a graph that shows:

• A linear increase of $E$ for $r < R$,
• A decrease of $E$ following a $\frac{1}{r^2}$ relation for $r \geq R$.

The option labeled B correctly represents this relationship.

Why Other Options Are Incorrect

• Option A: This might suggest a constant electric field inside the sphere, which is incorrect.
• Option C: This could imply an abrupt change or a non-linear relationship that does not reflect the linear increase followed by a $\frac{1}{r^2}$ decrease.
• Option D: Similar to option C, it could misrepresent the actual behavior of the electric field in the two regions.

In conclusion, the correct answer is B because it accurately reflects the behavior of the electric field due to a uniformly charged sphere as derived from electrostatics principles.