Electromagnetism

The electromagnetic field splits into an

• \(\color{purple}\phi\) electric potential (the time part) (here denoted by color, from cyan for negative, to red for positive)
• \(\color{purple}\mathbf{A}\) magnetic vector potential (the space part) (here shown as tiny arrows)

often written as the electric and magnetic fields: \[\color{purple}\mathbf{E}=-\nabla\phi-\partial_t\mathbf{A}, \qquad \mathbf{B}=\nabla\times\mathbf{A}\]Charges create the electric potential, while their motion generates the magnetic potential. Positive charges increase the potential; negatives decrease it. They move towards lower/higher potentials respectively, but their motion is "twisted" by the magnetic potential.

Notice how:

• Similar charges repel, opposites attract (but diminishing with their separating distance).
• In the simulation, opposite charges sometimes spiral in to link up as dipoles and 'disappear'.
• The electromagnetic field is able to propagate changes as electromagnetic waves at the maximum speed \(c\).

The general motion of charges and their associated electromagnetic field is complicated, because

• charges create the field \(\color{purple}\mathrm{div}\,F=J\) where \(\color{purple}F=dA\); i.e., \(\color{purple}\square\phi=-\rho,\ \square\mathbf{A}=-\mathbf{j}\); and
• the field creates a force on each charge \(\color{purple}\textrm{Force}=FJ\), i.e., \(\color{purple}\mathbf{f}=\rho\mathbf{E}+\mathbf{j}\times\mathbf{B}\).

Electric Fields

An unmoving point charge \(q\) has an electric potential around it of \(\color{purple}\phi = -\frac{q}{r}.\)

A dipole is the limiting case of two point charges of opposite charge placed very close together. It is effectively chargeless but has a characteristic dipole electric field, which is much weaker than a comparable point charge. Dipoles are hardly moved by an electric field, but turn around.

More complex configurations of charged particles have more complex electric fields, but when they are bunched together they look like a point charge from afar, or, when neutral, a dipole or more rarely, a quadrupole, etc.

For example, three positive charges and two negative charges close together appear as one positive charge from afar; three positives and three negatives appear (usually) as a dipole.

An electric potential gradient produces a force on a charge.
(Click on each charged particle. See Mechanics for more on forces.)

An external electric field exerts a slight force on an electric dipole, \(\color{purple}\mathbf{F} = \mathbf{p}\cdot\nabla\mathbf{E}\), plus a torque \(\color{purple}\mathbf{\tau} = \mathbf{p}\times\mathbf{E}\).

Magnetic Fields

A purely magnetic field can be created by having equal numbers of particles of opposite charge move relative to each other (a current). Overall the "fluid" is neutral, so there is no electric field.

There are no magnetic charges, so the simplest magnetic field is a magnetic dipole \(\mathbf{m}\), generated by a current loop; in particular that of a spinning charged particle.

An external magnetic field exerts a force on a moving charge, perpendicular to both the current and the magnetic field.

Just like electric fields on electric dipoles, a magnetic field on a magnetic dipole exerts a force \(\color{purple}\mathbf{F} = \mathbf{m}\cdot\nabla\mathbf{B}\), plus a torque \(\color{purple}\mathbf{\tau} = \mathbf{m}\times\mathbf{B}\), which causes the dipole to twist and precess.

Electromagnetic Waves

Changes in the electromagnetic field propagate as waves. An electromagnetic wave carries its own energy, momentum, and angular momentum at the maximum speed.

The simplest examples of such an emission of electromagnetism are:

Individual waves are polarized — they have a dominant direction.

An electromagnetic wave ('photon') seen head-on, travelling at speed \(c\):

The Electromagnetic Spectrum

When the electromagnetic radiation is of

• low frequency and low energy, then it appears as a wave
• high frequency and low energy, then it appears as a ray (geometric optics)
• high frequency and high energy, then it appears as a photon (particle)

A "black-body" in thermal equilibrium emits electromagnetic radiation at a peak frequency, which increases with temperature; the total power output increases with temperature as (\(T^4\)).

Conductors and Insulators

A conductor is a region in which charges can move freely inside but not beyong the boundary. There can be no electric field inside a neutral conductor; only at its boundary can there be a perpendicular electric field proportional to the charge surface density. An insulator is a region which does not allow movement of charges.

The amount of charge \(q\) that a conductor can hold per unit electric potential \(\phi\) is called the capacitance of the conductor. It depends on its shape only, e.g., the capacitance of a sphere is \(r\), of parallel plates is \(\frac{A}{4πd}\). The potential energy stored in a charged conductor turns out to be \(\frac{1}{2}C\phi^2 = \frac{1}{2}\frac{1}{C}q^2\).

A static electric field causes the charges in the conductor to move, which in turn create a magnetic field. The electromagnetic field is absorbed as \(\rho|\mathbf{j}|^2\), where \(\rho\) is the resistivity of the conductor and \(\mathbf{j}\) is the electric current vector.

An alternating electric field with frequency \(f\) is able to pass through a conductor only if the frequency is much larger than the ratio of the conductivity to the permittivity.

Parallel conductors allow for standing waves that reflect from both sides (e.g., a 10cm can has microwave standing waves).