Classical Electromagnetism

Classical electrodynamics (or '''classical electromagnetism''') is a theory of Electromagnetism that was developed over the course of the 19th Century , most prominently by James Clerk Maxwell . It provides an excellent description of electromagnetic phenomena whenever the relevant Length scales and field strengths are large enough that Quantum Mechanical effects are negligible (see Quantum Electrodynamics ).

Mathematically it follows from applying Lorentz Transformation to Coulomb Force of point Electric Charge (in order to find the force moving charges interact with).

LORENTZ FORCE

The electromagnetic field exerts the following force (often called the Lorentz force) on Charged particles:

:

\mathbf{F} = q\mathbf{E} + q\mathbf{v} 	imes \mathbf{B}

where all boldfaced quantities are at q's location, v is q's velocity, B is the strength of the Magnetic Field at q's position.

This description of the force between charged particles, unlike Coulomb's Force Law , does not break down under Relativity and in fact, the magnetic force is seen as part of the relativistic interaction of fast moving charges that Coulomb's law neglects.

THE ELECTRIC FIELD E

The Electric Field E is defined such that, on a stationary charge:

:

\mathbf{F} = q_0 \mathbf{E}

where q₀ is what is known as a test charge. The size of the charge doesn't really matter, as long as it is small enough as to not influence the electric field by its mere presence. What is plain from this definition, though, is that the unit of E is N/C, or Newtons per Coulomb . This unit is equal to V/m ( Volts per Meter ), see below.

The above definition seems a little bit circular but, in electrostatics, where charges are not moving, Coulomb's law works fine. So what we end up with is:

:

where ρ is the Charge Density as a function of position, r_unit is the unit vector pointing from dV to the point in space E is being calculated at, and r is the distance from the point E is being calculated at to the point charge.

Both of the above equations are cumbersome, especially if one wants to calculate E as a function of position. There is, however, a scalar function called the Electrical Potential that can help. Electric potential, also called voltage (the units for which are the volt), which is defined thus:

:

\phi_\mathbf{E} = - \int_s \mathbf{E} \cdot d\mathbf{s}

where φ_E is the electric potential, and s is the path over which the integral is being taken.

Unfortunately, this definition has a caveat. From Maxwell's equations, it is clear that

abla 	imes \mathbf{E}

is not always zero, and hence the scalar potential alone is insufficient to define the electric field exactly. As a result, one must resort to adding a correction factor, which is generally done by subtracting the time derivative of the A vector potential. Whenever the charges are quasistatic, however, this condition will be essentially met, so there will be few problems. (As a side note, by using the appropriate gauge transformations, one can define V to be zero and define E entirely as the negative time derivative of A, however, this is rarely done because a) it's a hassle and more importantly, b) it no longer satisfies the requirements of the Lorenz gauge and hence is no longer relativistically invariant).

From the definition of charge, it is trivial to show that the electric potential of a point charge as a function of position is:

:

\phi rac{1}{4 \pi \epsilon_0} rac{q}{\left \mathbf{r} - \mathbf{r}_q(t_{ret}) ight-rac{\mathbf{v}(t_{ret})}{c} \cdot \mathbf{r}_q(t_{ret})}

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