Gauss.s law is true only if force due to a charge varies as:

Options

A. `r^(-1)`
B. `r^(-2)`
C. `r^(-3)`
D. `r^(-4)`

Answer

To solve the question regarding Gauss's law and the variation of force due to a charge, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Gauss's Law**:
Gauss's law states that the total electric flux (Φ) through a closed surface is equal to the charge (q) enclosed by that surface divided by the permittivity of free space (ε₀). Mathematically, it is expressed as:
\[ \Phi = \frac{q}{\epsilon_0} \]

2. **Relating Electric Flux to Electric Field**:
The electric flux can also be expressed in terms of the electric field (E) and the area (A) of the surface:
\[ \Phi = E \cdot A \]

3. **Combining the Two Expressions**:
By equating the two expressions for flux, we have:
\[ E \cdot A = \frac{q}{\epsilon_0} \]

4. **Finding the Electric Field (E)**:
For a point charge, the electric field (E) at a distance (r) from the charge is given by:
\[ E = \frac{1}{4\pi \epsilon_0} \frac{q}{r^2} \]
Here, we see that the electric field varies inversely with the square of the distance (r).

5. **Relating Electric Field to Force**:
The force (F) experienced by a charge (q) in an electric field (E) is given by:
\[ F = qE \]
Substituting the expression for E, we get:
\[ F = q \left(\frac{1}{4\pi \epsilon_0} \frac{q}{r^2}\right) = \frac{q^2}{4\pi \epsilon_0 r^2} \]

6. **Conclusion on Force Variation**:
From the expression \( F \propto \frac{1}{r^2} \), we conclude that the force varies as \( r^{-2} \). Therefore, the correct answer is that Gauss's law is true only if the force due to a charge varies as \( r^{-2} \).

### Final Answer:
The correct option is \( r^{-2} \).

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