Question

Are the 8 Maxwell's equations enough to derive the formula for the electromagnetic field created by a stationary point charge, which is the same as the law of Coulomb

F = keq1q2r2 ?
If I am not mistaken, due to the fact that Maxwell's equations are differential equations, their general solution must contain arbitrary constants. Aren't some boundary conditions and initial conditions needed to have a unique solution. How is it possible to say without these conditions, that a stationary point charge does not generate magnetic field, and the electric scalar potential is equal to

Φ(r)=er.

If the conditions are needed, what kind of conditions are they for the situation described above (the field of stationary point charge)?

electromagnetismforceselectrostatics

Answer 1

Maxwell's equations are a set of four differential equations that describe the behavior of electric and magnetic fields. To determine the electromagnetic field created by a stationary point charge, we can use the electrostatic approximation. Specific boundary conditions are needed to obtain a unique solution.

Step-by-step explanation:

Maxwell's equations are a set of four differential equations that describe the behavior of electric and magnetic fields. These equations, along with the Lorentz force law, encompass all the laws of electricity and magnetism.

Although the general solution of Maxwell's equations contains arbitrary constants, specific boundary conditions and initial conditions are needed to obtain a unique solution.

To determine the electromagnetic field created by a stationary point charge, we can use the electrostatic approximation.

In this approximation, we assume that the charge is not moving, so there is no changing electric field and no magnetic field is generated.

Therefore, the electric field can be described by Coulomb's law, F = kq1q2/r^2, and the electric scalar potential is given by Φ(r) = q1/r.

The specific boundary conditions for the situation of a stationary point charge would depend on the geometry and surrounding medium.

For example, if the point charge is located near a conducting surface, the boundary condition would involve the behavior of the electric field at the surface.