If Coulomb's law involved `1/r^3` instead of `1/r^2`, would Gauss's law still be true?

Options

A. there were magnetic monopoles
B. the inverse square law were not exactly true
C. the velocity of light were not a universal constant
D. None of the above

Answer

To determine whether Gauss's law would still hold true if Coulomb's law involved \( \frac{1}{r^3} \) instead of \( \frac{1}{r^2} \), we can analyze the implications step by step.

### Step 1: Understand Coulomb's Law
Coulomb's law states that the electric field \( E \) due to a point charge \( Q \) at a distance \( r \) is given by:
\[ E = \frac{KQ}{r^2} \]
where \( K \) is a proportionality constant. This shows that the electric field decreases with the square of the distance.

**Hint:** Recall that the dependence on \( r^2 \) is crucial for the behavior of electric fields in three-dimensional space.

### Step 2: Modify Coulomb's Law to \( \frac{1}{r^3} \)
If we modify Coulomb's law to involve \( \frac{1}{r^3} \), the electric field at a distance \( r \) from a point charge \( Q \) would become:
\[ E = \frac{KQ}{r^3} \]

**Hint:** Consider how the electric field changes with distance when the exponent is modified.

### Step 3: Calculate Electric Flux
The electric flux \( \Phi \) through a closed surface is given by:
\[ \Phi = E \cdot A \]
For a sphere of radius \( r \), the surface area \( A \) is:
\[ A = 4\pi r^2 \]
Thus, the total electric flux through the sphere would be:
\[ \Phi = E \cdot A = \left(\frac{KQ}{r^3}\right) \cdot (4\pi r^2) = \frac{4\pi KQ}{r} \]

**Hint:** Remember that the electric field is uniform over the surface of the sphere, and the area is constant for a given radius.

### Step 4: Compare with Gauss's Law
Gauss's law states that the total electric flux through a closed surface is equal to the charge enclosed divided by the permittivity of free space \( \epsilon_0 \):
\[ \Phi = \frac{Q_{\text{enc}}}{\epsilon_0} \]
In our case, if we apply Gauss's law, we would have:
\[ \frac{4\pi KQ}{r} = \frac{Q}{\epsilon_0} \]

**Hint:** Consider how the dependence on \( r \) affects the validity of Gauss's law.

### Step 5: Conclusion
From the comparison, we see that the electric flux depends on \( \frac{1}{r} \) when using \( \frac{1}{r^3} \) in Coulomb's law, which contradicts the form of Gauss's law that does not involve \( r \). Therefore, Gauss's law would not hold true if Coulomb's law were based on \( \frac{1}{r^3} \).

**Final Answer:** No, Gauss's law would not be valid if Coulomb's law involved \( \frac{1}{r^3} \).