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What Is A Field?



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 it has both a value and a direction, force is a vector.  The mathematical symbol for a force vector is:

Force vector

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):


E = F/q [1]


Like force, E is a vector.  And since the unit of charge is the coulomb (C), the unit of electric field strength (E) 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 one meter apart.  One V/m is exactly one N/C.

1 V/m = 1 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

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 (FE) between two charges (q1 & q2) that are separated by a distance r is:

Coulomb's Law [2]


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 or 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.





Magnetic Field

Magnetic fields arise from moving charge whether microscopic (like spinning electrons) or macroscopic (like electron current through a wire).

The magnetic field lines surrounding a bar magnet are depicted here.  The field is due to aligned electron spins in the bar.

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

Note the magnetic compass needles in the drawing ‒ opposite poles attract and like poles repel.  The Earth's geographic north pole is a magnetic south pole.


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.

Magnetic Field

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

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.





Moving charge not only creates a magnetic field, it also reacts to one, including its own!  Electricity and magnetism are coupled aspects of a single, electromagnetic force.

The electromagnetic force tugs on a moving charge perpendicular to both its velocity (v) and the surrounding magnetic field (B).

The force (F) on a positive electric charge moving through a magnetic field is illustrated below.

Magnetic Force

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)


Flux (symbol Phi Φ) is the flow of field lines through a unit of area perpendicular to the flow.  Flux density is the net amount of flux leaving or entering a closed surface.

The magnetic flux density (B) at any point in a magnetic field is simply defined as the magnetic force F that would be felt there by a charge q moving at velocity v… per unit of charge and velocity, of course.


B = F/qv


F = qvB

But there's one more variable ‒ the angle between the charge's velocity and the direction of the magnetic field.  The force is greatest on a charge moving perpendicular to the magnetic flux and zero on a charge moving parallel to it.

A vector cross product is therefore needed in our definition:

F = q(v x B) [3]

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

1 T = 10 kG

From B=F/qv, we see that we could also call a tesla a newton per ampere-meter.



Magnetic Field Strength (H)


In materials like iron, electron spins and their magnetic fields will line up with any external magnetic field, adding to the total flux density.  Such materials are said to be magnetically permeable.

Magnetic field strength (symbol H) does not include the effect of a region's permeability.  To calculate flux density (B) from field strength (H) in free space, the permeability of free space (μ0) must be factored in:


B = μ0H


The factor μ0 is a "dimensional" constant—that is, it has units.  So, while the unit of flux density (B) is the tesla, the unit of field strength (H) is the ampere per meter (A/m).

One A/m is the magnetic strength of a coiled wire passing one ampere of current per meter of coil length (not wire length).  It relates to flux density as follows:

1 A/m = 1.2567 microteslas = 0.012567 gauss

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 (μr) of the material:

B = μrμ0H

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:

F = qE [1b]

The magnetic aspect comes from equation 3:

F = q(v x B) [3]


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


F = q(E + v x B) [4]



Field Theory


The previous equation, formulated by Hendrik Antoon Lorentz in 1895, embodies modern classical electromagnetism.  In essence, equation 4 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.




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