Gauss's law of electrostatics would be invalid if

Options

A. there were magnetic monopoles
B. the speed of light was not a universal constant
C. the inverse square law was not exactly true
D. the electrical charge was not quantized

Explanation

To determine under what conditions Gauss's law of electrostatics would be invalid, we can analyze the options provided in the question.

### Step-by-Step Solution:

1. **Understanding Gauss's Law**: Gauss's law states that the electric flux through a closed surface is proportional to the charge enclosed within that surface. Mathematically, it is expressed as:
\[ \Phi_E = \oint \mathbf{E} \cdot d\mathbf{A} = \frac{q_{\text{enclosed}}}{\epsilon_0} \]
where \(\Phi_E\) is the electric flux, \(\mathbf{E}\) is the electric field, \(d\mathbf{A}\) is the differential area vector, \(q_{\text{enclosed}}\) is the charge enclosed, and \(\epsilon_0\) is the permittivity of free space.

2. **Identifying the Conditions**: The question provides four conditions under which Gauss's law might be invalid:
- A) Existence of magnetic monopoles
- B) The speed of light is not a universal constant
- C) The inverse square law is not exactly true
- D) Electrical charge is not quantized

3. **Analyzing Each Condition**:
- **A) Magnetic Monopoles**: The existence of magnetic monopoles would affect magnetic fields but would not invalidate Gauss's law for electric fields, which pertains to electric charges.
- **B) Speed of Light**: The speed of light being variable would impact electromagnetic theory but does not directly invalidate Gauss's law.
- **C) Inverse Square Law**: The inverse square law states that the force between two point charges is inversely proportional to the square of the distance between them. If this law is not exactly true, then the relationship between electric field and distance would also change, leading to a breakdown of Gauss's law, which relies on this principle.
- **D) Charge Quantization**: The quantization of charge is a fundamental aspect of electric charge and does not affect the validity of Gauss's law.

4. **Conclusion**: The only condition that would directly invalidate Gauss's law is if the inverse square law is not exactly true. Therefore, the correct answer is:
\[ \text{C) The inverse square law is not exactly true.} \]