Proper application of which of the following theorems to Coulomb's law results in Gauss's law? 

Explanation:

Proper Application of the Divergence Theorem to Coulomb's Law Results in Gauss's Law

Introduction:

The relationship between Coulomb's law and Gauss's law is a fundamental concept in electromagnetism. Coulomb's law describes the force between two point charges, whereas Gauss's law relates the electric flux through a closed surface to the charge enclosed by that surface. The correct mathematical tool to transition from Coulomb's law to Gauss's law is the Divergence Theorem, also known as Gauss's Theorem.

Coulomb's Law:

Coulomb's law states that the electric force \( \mathbf{F} \) between two point charges \( q_1 \) and \( q_2 \) separated by a distance \( r \) is given by:

\(\mathbf{F} = k_e \frac{q_1 q_2}{r^2} \hat{\mathbf{r}}\)

where \( k_e \) is Coulomb's constant and \( \hat{\mathbf{r}} \) is the unit vector in the direction of the force.

Electric Field:

The electric field \( \mathbf{E} \) due to a point charge \( q \) at a distance \( r \) is given by:

\(\mathbf{E} = k_e \frac{q}{r^2} \hat{\mathbf{r}}\)

For multiple charges, the electric field is the vector sum of the fields due to each charge.

Gauss's Law:

Gauss's law states that the total electric flux \( \Phi_E \) through a closed surface \( S \) is equal to the charge enclosed \( Q_{\text{enc}} \) divided by the permittivity of free space \( \epsilon_0 \):

\(\Phi_E = \oint_S \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{\text{enc}}}{\epsilon_0}\)

Divergence Theorem:

The Divergence Theorem relates the flux of a vector field through a closed surface to the volume integral of the divergence of the field. Mathematically, it is expressed as:

\(\oint_S \mathbf{E} \cdot d\mathbf{A} = \int_V (\nabla \cdot \mathbf{E}) dV\)

where \( \mathbf{E} \) is the electric field, \( S \) is the closed surface, and \( V \) is the volume enclosed by \( S \).

Derivation of Gauss's Law from Coulomb's Law using the Divergence Theorem:

To derive Gauss's law from Coulomb's law, we start with the expression for the electric field due to a point charge:

\(\mathbf{E} = k_e \frac{q}{r^2} \hat{\mathbf{r}}\)

The divergence of the electric field for a point charge can be found using the Dirac delta function \( \delta(\mathbf{r}) \):

\(\nabla \cdot \mathbf{E} = \nabla \cdot \left( k_e \frac{q}{r^2} \hat{\mathbf{r}} \right) = \frac{q}{\epsilon_0} \delta(\mathbf{r})\)

Applying the Divergence Theorem:

\(\oint_S \mathbf{E} \cdot d\mathbf{A} = \int_V (\nabla \cdot \mathbf{E}) dV = \int_V \frac{q}{\epsilon_0} \delta(\mathbf{r}) dV = \frac{q}{\epsilon_0}\)

This shows that the electric flux through a closed surface \( S \) is proportional to the charge enclosed by the surface, which is Gauss's law.

Conclusion:

The Divergence Theorem is the key mathematical tool that allows the transition from Coulomb's law to Gauss's law. By applying the Divergence Theorem to the electric field derived from Coulomb's law, we obtain the integral form of Gauss's law, which relates the electric flux through a closed surface to the charge enclosed within that surface.

Analysis of Other Options:

Option 1: Stokes's Theorem

Stokes's Theorem relates the surface integral of the curl of a vector field to the line integral of the vector field around the boundary of the surface. It is given by:

\(\oint_{\partial S} \mathbf{F} \cdot d\mathbf{r} = \int_S (\nabla \times \mathbf{F}) \cdot d\mathbf{A}\)

Stokes's Theorem is not directly related to the derivation of Gauss's law from Coulomb's law as it deals with the curl of a vector field and line integrals, rather than the divergence and flux through a closed surface.

Option 3: Helmholtz's Theorem

Helmholtz's Theorem states that a vector field is uniquely determined by its divergence and curl, given appropriate boundary conditions. While this theorem is important in vector calculus, it does not directly relate to the derivation of Gauss's law from Coulomb's law.

Option 4: Uniqueness Theorem

The Uniqueness Theorem in electrostatics states that the solution to Poisson's or Laplace's equation for the electric potential is unique if the boundary conditions are specified. This theorem ensures that the electric field and potential are uniquely determined by the charge distribution and boundary conditions, but it is not directly involved in deriving Gauss's law from Coulomb's law.