Maxwell’s ##\nabla\cdot\vec{E}=\rho/\epsilon_0## can be rewritten in an integral form. Try applying that integral form to a spherical volume with a point charge in the center, and taking advantage of the symmetry of that configuration.MaestroBach said:Any ideas would be appreciated.
MaestroBach said:Summary:: Maxwell vs Coulomb
My textbook tells me that one of Maxwell's equations, namely divergence of E = 4pi * charge density (in cgs) or divergence of E = pi / epsilon nought (in SI) is exactly equivalent to Coulomb's Law.
How in the world is that so?
The relationship between Maxwell's equations and Coulomb's Law is that Coulomb's Law is one of the four equations that make up Maxwell's equations. Coulomb's Law describes the relationship between the electric field and the electric charge, while Maxwell's equations describe the fundamental laws of electromagnetism, including the relationship between electric and magnetic fields.
Maxwell's equations and Coulomb's Law are related mathematically through the equation for Gauss's Law, which is one of the four equations in Maxwell's equations. This equation is mathematically equivalent to Coulomb's Law, but it also takes into account the displacement current, which was not included in Coulomb's Law.
Yes, Coulomb's Law can be derived from Maxwell's equations. When the displacement current is assumed to be zero, the equation for Gauss's Law in Maxwell's equations reduces to Coulomb's Law. This means that Coulomb's Law is a special case of Maxwell's equations.
The relationship between Maxwell's equations and Coulomb's Law is crucial to our understanding of electromagnetism. Maxwell's equations provide a more comprehensive and accurate description of electromagnetism than Coulomb's Law alone. They allow us to understand the behavior of electric and magnetic fields and how they are related, leading to technological advancements such as electricity, telecommunications, and electronics.
While Maxwell's equations provide a more complete understanding of electromagnetism, they are not perfect and have limitations. For example, they do not take into account the effects of quantum mechanics at a microscopic level. Additionally, they are based on classical physics and do not fully explain phenomena such as relativity and gravitational forces. However, for most practical applications, the relationship between Maxwell's equations and Coulomb's Law is sufficient.