Flux `(ph)` as a function of distance `(r)` from centre of uniformly charged solid sphere of charge `Q` and radius `R`, is best represented by

Options

A.
B.
C.
D. (D)

Explanation

To find the relationship between the electric flux (Φ) and the distance (r) from the center of a uniformly charged solid sphere of charge Q and radius R, we can use Gauss's Law. Let's go through the solution step by step:

### Step 1: Understanding Gauss's Law
Gauss's Law states that the electric flux through a closed surface is equal to the charge enclosed (Q_enc) divided by the permittivity of free space (ε₀):
\[ \Phi = \frac{Q_{\text{enc}}}{\epsilon_0} \]

### Step 2: Case 1 - Inside the Sphere (r < R)
When we are inside the sphere (r < R), we need to find the charge enclosed (Q_enc) within a Gaussian surface of radius r.

1. The charge density (ρ) of the sphere can be calculated as:
\[ \rho = \frac{Q}{\frac{4}{3}\pi R^3} \]

2. The volume of the Gaussian surface of radius r is:
\[ V = \frac{4}{3}\pi r^3 \]

3. The charge enclosed (Q_enc) within this Gaussian surface is:
\[ Q_{\text{enc}} = \rho \cdot V = \left(\frac{Q}{\frac{4}{3}\pi R^3}\right) \cdot \left(\frac{4}{3}\pi r^3\right) = Q \cdot \frac{r^3}{R^3} \]

4. Now, substituting Q_enc into Gauss's Law:
\[ \Phi = \frac{Q \cdot \frac{r^3}{R^3}}{\epsilon_0} = \frac{Q}{\epsilon_0} \cdot \frac{r^3}{R^3} \]
This shows that the flux (Φ) is proportional to \(r^3\) when \(r < R\).

### Step 3: Case 2 - Outside the Sphere (r ≥ R)
When we are outside the sphere (r ≥ R), the entire charge Q is enclosed within the Gaussian surface.

1. The charge enclosed is simply Q.
2. Applying Gauss's Law:
\[ \Phi = \frac{Q}{\epsilon_0} \]
This indicates that the flux (Φ) is constant and does not depend on r when \(r \geq R\).

### Step 4: Summary of Results
- For \(r < R\): \(\Phi \propto r^3\)
- For \(r \geq R\): \(\Phi = \frac{Q}{\epsilon_0}\) (constant)

### Conclusion
The relationship between the electric flux and the distance from the center of the uniformly charged solid sphere is best represented by a graph that shows:
- A cubic relationship for \(r < R\) (increases as \(r^3\))
- A constant value for \(r \geq R\)

Thus, the correct option is **C**.

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