electric and magnetic fields, forces

The separation of positive and negative electric charge leads to the build-up of an electric field between the charges. The electric field is characterized by arrows, which point from positive toward negative charges: The length and direction of the arrow indicates respectively the strength and direction of the force the field exerts on a particle of unit positive charge. From this definition it follows that the force an electric field exerts on a charged particle is equal to the product of the field strength and the charge, see box.

We can also draw lines in the volume filled with electric field such that the lines cross the electric field arrows everywhere at a right angle. In three dimensions, these lines become curved surfaces. If a charged particle moves along such a surface, called potential surface, it does not gain energy from or loses energy to the field, hence its energy remains constant. If the particle nevertheless travels between two surfaces, its energy changes, in proportion to its charge. The energy change per charge is called voltage, see box.

Moving charge, creating an electric current, produces a magnetic field, too. This field can be generated by a macroscopic current flowing in a wire, see Fig. 2, or by a microscopic one, emerging in excited atoms or molecules, or by the intrinsic magnetic dipoles of microscopic particles, such as electrons or protons. The magnetic field is defined through the force it exerts on a moving charged particle. It is defined such that this force is equal to the product of the particle’s charge, velocity and the magnetic field strength, and directed perpendicularly to the direction of the field and the motion of the particle, see box.

The electromagnetic forces govern all microscopic phenomena outside the atomic core. Life and technologies (except for nuclear technologies) rely on them.

Electric field, electric force, voltage.

According to the definition of the electric field strength, E, the electric force \vec{F}_q exerted on a particle of charge q is connected to the electric field strength \vec{E} by the simple equation

\vec{F}_q = q \times \vec{E},

where the arrows indicate a quantity that has a direction and a value. Such a quantity is called a vector and can be represented by an arrow with a length and direction, representing the strength and direction of the quantity, respectively. For static charges, the electric force is also referred to as Coulomb force, named after its first investigator.

By definition, the voltage between two potential surfaces S_2 and S_1 is given by

U= W / q,

where W is the work we had to do on the charge q when moving it from S_1 to S_2. Because the unit of energy (1 joule, 1 J) can be derived from the units of mass (kg), length (m) and time (s):

1\,{\rm J} = 1\,{\rm kg} \times {\rm m}^2/{\rm s}^2

and the unit of charge (1 coulomb, 1 C) can be derived from the units of current (A) and time (s):

1\,{\rm C} = 1\,{\rm A} \times {\rm s},

the unit of voltage can be derived from the above connection as 1\,{\rm J}/1\,{\rm C}, which is called 1 volt (1 V). The energy gained (or lost) by an electron between a potential difference of 1 volt is

W = |e| \times 1\,{\rm V} = 1\,{\rm electronvolt} = 1\,{\rm eV},

which is the most widely used unit of energy in the microscopic world. With |e|=1.6\times10^{-19}\,{\rm C}, we obtain

1\,{\rm eV} = 1.6 \times 10^{-19}\,{\rm CV} = 1.6 \times 10^{-19}\,{\rm J}.

The energy gained (or lost) by the particle upon travelling between S_2 and S_1 can also be calculated as the product of the force F_e =qE and the travel length l of the particle along the force: W=F_e\,l=qE\,l, see Fig. 1. By equating this with the above expression of the energy gain/loss leads to qU=qE\,l, from which we obtain E=U/l. This expression delivers the unit of the electric field: 1 V/m, as well as the force: 1 VC/m.

Magnetic field, magnetic force.

A magnetic field exerts a magnetic force, also called Lorentz-force, \vec{F}_m on a charged particle moving with velocity \vec{v}

\vec{F}_m=q\,\vec{v}\times\vec{B},

where the product of the arrows results in a new arrow directed perpendicularly to the directions of the magnetic field as well as velocity. This equation defines the magnetic field (force exerted on a unit charge moving with unit velocity) and leads to its SI unit: , which is called 1 Tesla (1 T).

Fig. 1. Electric field of two charged spheres. One with a positive charge, the other with the same amount of negative charge. The black arrows pointing from the red to the green sphere indicate the electric field. A testcharge q will feel the force F_q. Between two equipotential surfaces there is a voltage U. (© bf,ch)

Fig. 2. Magnetic field produced by an electric current. The circular magnetic field results from moving electric charges. The magnetic field goes clockwise, if you look in the current direction. (© ch)

Fig. 3. A magnetic field induces a force on a moving charge. The direction of the force is described by the right hand rule. If the thumb of your right hand points in the direction of the velocity of a positive charge and your forfinger in the direction of the magnetic field, your middlefinger will indicate the direction of the force on the positive charge. (© ch)