Merriam-Webster™ defines a field as a "region or space in which a given effect exists".

For example, a charged particle distorts its surrounding space in such a way that another charged particle is either repelled or attracted, depending on whether the charges have the same or the opposite polarity.

This electric force is directed along the straight line between the two charges.  Since the force has both a value and a direction, it's a vector.  The mathematical symbol for a force vector is:

The standard unit of force is the newton, abbreviated N (capital N because Newton was a person).

The "region or space" wherein the electric force exists is the electric field.

Electric Field Strength (E)

The electric field strength (symbol E) at any point in an electric field is simply defined as the force (F) that a unit of positive charge would feel there, per unit of the field's source charge (q):

Like force, E is a vector. The unit of charge (q) is the coulomb (C) so the unit of electric field strength is the newton per coulomb (N ⁄ C).

An equivalent unit of field strength is the volt per meter—the force of a one volt potential between two parallel conductive plates that are one meter apart.  One V ⁄ m is exactly one N ⁄ C.

By convention, electric fields point in the direction that a positive test charge would move—that is, away from positive and toward negative charge:

Electric Field Lines

The field lines in the above illustration are a crude but helpful way to picture a force field.  The electric field strength is greater where the lines are closer together.  The direction of the field at any particular point is tangent to the field line.

Coulomb's Law

Charles-Augustin de Coulomb determined that the magnitude of the electric force (F_E) between two charges (q₁ & q₂) that are separated by a distance r is:

This equation looks like Newton's law of gravitational force—just replace the two charges with two masses and put a different constant up front.

The electric constant is κ (kappa, also called Coulomb's constant).  Kappa equals 1/4πε₀ where ε₀ (epsilon zero) is the electric permittivity of free space (its ability to permit an electric field).

Epsilon zero equals 1/μ₀c² where μ₀ (mu zero) is the magnetic permeability of free space (its ability to hold a magnetic field) and c is the speed of light.

A magnetic field is generated whenever charge is in motion, either microscopically, like electrons spinning and orbiting in an atom, or macroscopically, like electric current in a wire.

Bar Magnets

The magnetic field lines of a bar magnet are shown here.

The magnetic force field arises from electrons in the bar, all spinning and orbiting in the same direction.

One end of the bar is called the north pole and the other, the south.  Poles come in pairs and, by convention, the field lines point from north to south.

As we can see from the magnetized compass needles in this drawing, opposite poles attract.  The north pole of each needle points toward the south pole of the bar magnet.

The planet Earth has its own magnetic field.  The field is generated by electric currents resulting from the convection of a molten mixture of iron and nickel in the Earth's outer core.  The convection is caused by heat escaping from the core.

Originally, a magnet's north pole was defined as the pole that is attracted by Earth's "North Magnetic Pole".  Now we know that Earth's "North Magnetic Pole" must be the south pole of Earth's magnetic field in order to attract the north pole of magnets.

Likewise, the Earth's "South Magnetic Pole" (where field lines point downward into the Earth) is actually up near the North Pole.  Geomagnetic reversals do randomly occur, however, about 100 thousand to 50 million years apart.  The last reversal occurred about 780,000 years ago.

Current-Carrying Wires

The magnetic field around a current-carrying wire is illustrated below.  The yellow disc is the cross section of a wire whose current flows into the page ("X" represents the tail of an arrow).  The magnetic field lines curl around the current.

If you point your right thumb in the direction of the current, your fingers will curl in the direction of the force field.  An opposite current produces an opposite curl.

NOTE :  By convention, electric current flows from positive to negative, opposite to the flow of electrons.

Unlike electric field lines, magnetic field lines have no beginning or end—they're all closed loops.  That's because the magnetic north and south poles are a united dipole, unlike the separated plus and minus electric charges called monopoles.

A moving charge not only creates a magnetic field but also reacts to one, including the one it created!

Electricity and magnetism are two tightly coupled aspects of a single electromagnetic force that pulls on moving charges.  The pull is perpendicular to both the charge velocity (v) and the strength of the surrounding magnetic field (B).

The illustration below shows the electromagnetic force (F) acting on a positively charged particle that's moving perpendicular to a magnetic field (B).

If you open your right hand and point its thumb in the direction of the charge's velocity and point its fingers in the direction of the magnetic field, then your palm will point in the direction of the electromagnetic force.

Magnetic Flux Density (B)

At any point in a magnetic field, magnetic flux density (B) is defined to be the electromagnetic force (F) that would be felt there by a particular charge (q) moving through the field at a particular velocity (v), per unit of charge and velocity.

Symbolically,

Solving for F gives us :

However, one additional variable affects the electromagnetic force :  the angle between the charge's motion (v) and the direction of the magnetic field (B).

To multiply two vectors (v and B) having separate directions, a vector cross product is required, written as follows :

This cross product is a force F whose direction is perpendicular to both v and B and whose magnitude takes into account the angle (θ, theta ) between v and B as follows :

What's Flux?

As an analogy, consider a hollow tube held in a waterfall.  If the opening of the tube is perpendicular to the waterfall, the maximum amount of water will flow through the opening.

But if the tube's opening is parallel to the waterfall, no water will flow across it.  At in-between angles, an in-between amount of water will cross the opening.

In this analogy, flux is the total amount of water entering the tube at a particular moment.

The amount depends on the strength of the waterfall, the angle it makes with the tube, and the area of the tube's opening.

In the same way, magnetic flux (capital Phi Φ) is the portion of a magnetic field B acting perpendicular to some surface area A.

This projection of B onto the direction of A is a vector dot product that returns a scalar (non-vector) value as follows :

Magnetic Units

The unit of magnetic flux (Φ) is the weber, abbreviated Wb (capital W because Weber was a person).

The unit of magnetic flux density (B) is the tesla, abbreviated T (capital T because Tesla was a person).  A subdivision of the tesla is the gauss, abbreviated G (capital G because Gauss was a person).  There are 10,000 gauss in a tesla.

Since Φ = B ⋅ A, flux density (B) describes a flux per area (Φ/A).  That is, one tesla is equivalent to one weber per square meter.

Although magnetic flux is mathematically convenient for calculating the effects of a magnetic field, flux has no physical reality beyond the mathematician's scratch pad.

In 1833, Wilhelm Weber and Carl Gauss built the first operative telegraph.  It ran between the Göttingen Observatory in Germany and the Institute of Physics, about 1 km away.

Magnetic Field Strength (H)

In materials like iron, nickel and cobalt, electronic orbits and their associated magnetic fields naturally align with nearby magnetic fields, increasing the total flux density.  These materials are said to be magnetically permeable.

Magnetic field strength (symbol H) is a raw field strength that does not include the effects of magnetic permeability.

The unit of magnetic field strength is the ampere per meter (A‌ ‌/‌ ‌m), which is the magnetic strength inside a coil of wire passing one ampere of current per meter of coil length (not wire length).

In free space, to convert a raw field strength H into a total flux density B, the permeability of free space (μ₀, mu zero:  the ability to hold a magnetic field) must also be factored in :

The factor μ₀ is a "dimensional" constant—that is, it has units.  Field strength (H) converts to flux density (B) as follows:

If a region's permeability isn't μ₀, another factor is needed to calculate B from H.  This dimensionless constant is called the relative permeability (μ_r) of the region, so that :

Relative permeability is determined by experiment.

Lorentz Force Law

The Lorentz force law simply adds the magnetic force to the electric force.  The electric aspect comes from equation [1]:

The magnetic aspect comes from equation [3]:

The sum of these forces describes the total electromagnetic force on a charged particle:

The previous equation, formulated by Hendrik Antoon Lorentz in 1895, embodies modern classical electromagnetism.  In essence, equation [6] is the definition of E and B.  They're the fields needed to account for the force F.

The Lorentz force law is the culmination of the shift from the idea of action at a distance—forces reaching out across empty space without a mechanism or speed limit—to the idea of a field.

A "field" is the medium or mechanism that transmits stress across a distance, from quantum to neighboring quantum at a finite speed.

In physics, electromagnetism was the first field theory.  The electromagnetic force is transmitted by virtual photons—quantum fluctuations in the electromagnetic field.