Charges and their Electric Fields

The electric field of a point charge and Coulombs law.If we would make a poll about the most fundamental question of electrostatics, the field of a point charge is very likely the winner. You may already know the answer but are you able to derive it directly from Maxwell's equations?

Problem Statement

Consider a point charge \(q\) in vacuum, e.g. an electron or proton. What is the resulting electric field \(\mathbf{E} \left( \mathbf{r}\right) \)?

Can you relate Coulomb's law to this force?

Background: From Forces to Fields

The electric field of a point charge is the most fundamental concept in electromagnetism. Historically, the force between two charged objects was found to scale with the product of the object's charges their inverse squared distance. This force is now known as Coulomb's law and was found in the second half of the 18th century.
The field concept is however a little different. Given a charge, one can calculate the force acting on a test object with unit charge. This force, in appropriate units, is then called the electric field. You can use the results from this problem to find the electric field of an arbitrary charge distribution using the principle of superposition see the electric field of two point charges.

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