Maxwell's equations

References:

• https://photonlines.substack.com/p/an-intuitive-guide-to-maxwells-equations

Electric and magnetic fields: the electromagnetic (or Lorentz) force on a particle with charge $q$ moving with velocity $\mathbf{v}$ is given by

where $\mathbf{E}$ is the electric field and $\mathbf{B}$ the magnetic field. In particular:

1. In the absence of a magnetic field, charged particles simply flow according to the electric field.
2. In the absence of an electric field (if this were possible), a charged particle at rest (zero velocity) would stay at rest. But a charged particle moving through a magnetic field would tend to curl perpendicular to the field lines and to its own motion.

Maxwell's equations describe how these electric and magnetic fields are generated by charges and currents, and how they interact with each other. They are as follows.

Gauss's law for electric fields: the divergence of the field $E$ is proportional to the charge density $\rho$ at each point:

where the constant $e_0$ is the "electric permittivity of free space". In other words, point charges act as sources (positive charge) or sinks (negative charge) in the electric field.

When we talk about the strength of an electric field, this refers to the electric flux density, ie the flux per unit area of an enclosing surface. Using the divergence theorem we can express Gauss's law in integral form,

stating that the total flux through any closed surface (sliced into infinitesimal regions $a$ with unit normal vectors $\hat{n}$) is proportional to the total enclosed charge $q_\text{enc}$. This implies the inverse square law in three dimensions, since the surface area of a sphere (or other surface) increases with the square of the radius, so the flux density at any given point on the surface must decrease correspondingly in order for the integral to remain constant.

Gauss's law for magnetic fields: a magnetic field $B$ has zero divergence everywhere:

Equivalently:

• the magnetic flux across any enclosed surface is always zero
• there is no magnetic equivalent to a point charge, i.e., no "magnetic monopole's

Similarly to the electric case, we can write this in integral form

saying that the total magnetic flux across any closed surface is zero.

Faraday's law says that a time-varying magnetic field induces a spatially-varying electric field:

Note that this is a circulating electric field, different from what would be produced by a point charge. The circulating electric field opposes the magnetic flux, in the sense that it 'curls against' the magnetic flux (the curl of the field is the negation).

Finally, the Ampere-Maxwell law states that a circulating magnetic field is induced by either an electric current (of density $J$) or a change in the electric field:

An electric current is a movement of electric charge (the $\rho$ in Gauss's law above): for example, charged particles like electrons or protons. So current flowing through a wire induces a magnetic field that circulates around the wire (in the direction corresponding to the right-hand rule, i.e., counterclockwise as you look 'down' the wire in the direction of current flow). The current density $J$ is the number of coulombs ($\approx 6.2 \times 10^{18}$ elementary charges) passing through a unit cross-sectional area (perpendicular to the current) per second.

Note that we can have an electric current even with no change in the electric field, and vice versa. The electric field provides a force acting on charged particles. A constant field means a constant set of forces, and these forces will move any charged particles that happen to be around. In general this will change the field (which arises, in part, from the charged particles), but there are configurations where the field is stable: for example, particles flowing through an infinitely long wire, where the 'movement' of particles within the wire is a no-op from the field's point of view. Conversely, the electric field can change due to magnetic effects even in the absence of any charged particles to create a current.