The Misunderstanding of Equivalence Between Gauss's Law and Coulomb's Law

It is often incorrectly claimed in academic discussions and literature that Gauss's Law and Coulomb's Law are equivalent. This misunderstanding arises due to the historical development of classical electromagnetism and the interplay between different laws and concepts over time.

Historical Context and Development

To properly understand the relationship between Gauss's Law and Coulomb's Law, it is crucial to trace the historical origins and evolution of these concepts. Coulomb's law, formulated by Charles-Augustin de Coulomb in 1785, describes the electrostatic force between two point charges. It is an action-at-a-distance force law, stating that the force between charges ( q ) and ( Q ) is given by the equation ( F frac{1}{4pi varepsilon_0} frac{qQ}{r^2} ). Here, ( r ) is the distance between the charges, and ( varepsilon_0 ) is the electric constant.

Starting in the 1850s, James Clerk Maxwell introduced a more comprehensive framework for electromagnetism. Maxwell extended Coulomb's law by redefining it in terms of fields. He introduced the concept of the electric field ( mathbf{E} ), defined as ( mathbf{E} frac{1}{4pi varepsilon_0} frac{Q}{r^2} hat{mathbf{r}} ). This field was defined on all of space and represented the force per unit charge at any point. Maxwell's equations showed that the force ( mathbf{F} ) on a charge ( q ) is given by ( mathbf{F} q mathbf{E} ), where ( mathbf{E} ) is evaluated at the location of ( q ) where the field is in contact with ( q ).

From Coulomb's Law to Gauss's Law

To connect Gauss's Law to Coulomb's Law, it is necessary to consider a more advanced field concept. Gauss's Law, formulated in 1835, gives the total flux through a closed surface containing a total charge ( q ) as ( oint_{partial V} mathbf{E} cdot dmathbf{a} frac{q}{varepsilon_0} ). This is also known as the divergence theorem in vector calculus.

For the equivalence between Gauss's Law and Coulomb's Law, it is essential to recognize that Gauss's original version could not be directly linked to Maxwell's electric field ( mathbf{E} ). Instead, it was linked to the concept of total electric flux. When assuming a spherically symmetric point charge source and a Gaussian surface as a sphere, the symmetry allows us to simplify the field to ( mathbf{E} frac{1}{4pi varepsilon_0} frac{q}{r^2} hat{mathbf{r}} ). Integrating this over the surface of a sphere gives:

$$oint_{partial V} mathbf{E} cdot dmathbf{a} oint_{partial V} left( frac{q}{4pi varepsilon_0 r^2} hat{mathbf{r}} right) cdot left( r^2 sintheta dtheta dphi hat{mathbf{r}} right) frac{q}{varepsilon_0}$$

This equation confirms that the total electric flux through the surface is ( frac{q}{varepsilon_0} ), which is indeed consistent with Gauss's Law.

Why They Are Not Equivalent

Despite the similarities, Gauss's Law and Coulomb's Law are fundamentally different in nature and origin:

Nature of the Laws: Coulomb's Law describes a specific force between point charges, while Gauss's Law describes the total electric flux through a closed surface. Coulomb's Law is more specific in its application, while Gauss's Law is a more general equation that can be applied to any closed surface, regardless of the distribution of charges. Historical Development: Coulomb's Law was formulated based on experimental observations in the late 18th century, while Gauss's Law emerged from Maxwell's theoretical framework in the mid-19th century. The development of field theory by Maxwell provided a more comprehensive understanding of the relationship between macroscopic and microscopic phenomena. Utility and Scope: Coulomb's Law is the most direct relationship between charges and forces, which is useful in many practical applications. Gauss's Law, on the other hand, is more versatile and is used to simplify complex problems involving large or irregularly distributed charges by converting them into boundary value problems.

Conclusion

While there is no denying the superficial similarities and the fact that Gauss's Law can be derived from Coulomb's Law in certain cases, it is important to recognize that these two laws are not equivalent. They have different origins and serve different purposes in the field of electromagnetism. It is crucial for students and researchers to understand the historical context and development of these laws to avoid the common misconception that they are equivalent.