The ultimate guide to learn the Maxwell’s equations

James Clerk Maxwell (1831-1879) unified the theories about the electricity and the magnetism. He developed four representative equations that establish the cause-effect relations between the electric and magnetic fields and the sources that create them (charges, electric currents or temporary variations of one or another).

In this post, the vector form of the Maxwell’s equations is obtained by using the following two theorems.

Divergence Theorem

Being S and closed surface and Ā a vector field, with ñ a normal vector exterior to S, then:

image002 image003

The flux of a vector field towards the exterior of a closed surface is equal to the divergence integral of that vector field over the volume surrounded by the surface.image004



Stokes’ Theorem

Being S a surface with boundary C and Ā a vector field defined in a region of the space which contains S, with a vector ñ, normal to S, then:

image005 image006

The circulation of a vector field along a closed line is equal to the rotational flux of that vector field, calculated over a surface which boundary is that line.




Now that we have the basic tools that we need to work with the Maxwell’s equations, we can express them:


Gauss’ Law

This theorem links the electric flux through a closed surface with the charge that the surface contains, q. Therefore, its integral form is:

image008where Є0 is the dielectric constant or permittivity in the vacuum.

By applying the divergence theorem, we can obtain the corresponding differential form:

image009where ρ is the volume charge density. image010In addition, by observing the Gauss’ law in its differential form, we can observe that the field’s sources are positive charges while the sinks are the negative charges. The associated electric field to a charge “emerges” from it (if it’s positive) or “ends” on it (if it’s negative). image011

Gauss’ Law for the magnetic fields

The probable nonexistence of magnetic monopoles as magnetic field’s sources or sinks make us consider that these lines are closed, with no starting or ending point. Therefore, if we consider that the lines cross a closed surface, the number of incoming lines is equal to the number of outgoing lines, so the net flux of any closed surface is always equal to zero:

image012 image013

If we apply the divergence theorem to the previous expression, we will obtain:


This means that, in the magnetic field, there are no sources or sinks. This fact highlight a fundamental difference between the magnetic and electric fields.


Faraday-Lenz’s Law

The electric field circulation is the definition of induced electromotive force, ɛ. Thus, the Faraday’s law, in its integral form, indicates that the induced electromotive force around any closed curve, C, is equivalent to the rate of change of the magnetic flux,ΦB , that crosses any surface bounded by the curve:


Faraday's experiment
Faraday’s experiment
Increasing Flux
Increasing Flux
Flux decreasing
Flux decreasing

In other words, if we apply the Stokes’ theorem to the previous expression, we will obtain:

image021This means that the electric field’s rotational is equal to the time variation of the magnetic field.

Therefore, we can conclude that the time variation of the magnetic field implies the existence of an electric field.


Ampère’s Law

The magnetic field circulation of any closed curve C is equal to the current that goes through C multiplied by the permeability in the vacuum, µ0:image022Equivalently, we can say:

The magnetic field circulation around C is equal to µ0 multiplied by the flux of J, through any surface S defined by the curve Cimage024

Where J is the current density.image025

However, these expressions are only valid for stationary currents. As explained earlier, the Faraday’s law links the electric field circulation with the change in the magnetic flux. Therefore, we would expect that a changing electric field would create a magnetic field, but the Ampère’s law, in its static form, only links the magnetic field circulation with the current. Maxwell proved that an additional term is needed to make the Ampère’s law consistent. This term is the displacement current, ID:image026This expression is obtained by applying the “charge’s conservation principle” and the “continuity equation”.

Given any closed surface and calculating the flux through it,J , we obtain:

image027By applying the divergence theorem, we obtain the continuity equation:


According to the Gauss’ law:image029


Substituting and re-ordering the terms, we end up with:

image031Due to the fact that the divergence of a rotor is always null, the only way to make the Ampère’s law and the continuity equation both feasible is by adding the electric field time variation in the second term of the equation that, as a result, states the complete form of the Ampère-Maxwell’s law:

image032The magnetic field’s rotational is equal to the current density multiplied by µ0 plus the electric field’s time variation multiplied by µ0 and Є0.

By using the Stokes’ theorem we obtain:

image033Thus, an electric field that changes with the time produces a magnetic field. In addition, this law is consequent with the charge conservation principle.


Now you know all the Maxwell’s equations in the free-space but, we can’t use these in a material medium… don’t worry, we will explain the Maxwell’s equations for material medium in the next post!

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