# Electrodynamics

TheÂ **Electrodynamics**Â is the branch of physics that deals withÂ everything related to the movement of electrical charges.Â It describes the evolution in time of a set of N particles with mass and electric charge, whose initial position and velocity are known.

If it is a large set of particles withÂ smallÂ *momentum*Â , their motion and the interactions that take place between them are described macroscopically through classical electrodynamics, which makes use of Newton’s laws of motion and Maxwell’s laws.

And if the momentum of the particles is large and the number of particles is small, the relativistic and quantum effects must be taken into account.

Adding relativistic and quantum effects to the study of the system depends on the energy of the photons involved during the interaction.Â Photons are particles without charge or mass (for practical purposes) that are exchanged whenever there is an electrical attraction or repulsion.

If the momentum of the photons is small, compared to the momentum of the system, the classical description is sufficient to obtain its characterization.

**Brief History**

The laws that describe the dynamics of charged particles were discovered between the end of the 18th century and the middle of the 19th century, when the concept of electric current emerged, as a result of the experimental and theoretical work of many scientists.

The Italian physicist Alessandro Volta (1745-1827) made the first voltaic pile at the dawn of the 19th century.Â With it he obtained a direct current, the effects of which began to be studied immediately.

The link between electric charges in motion and magnetism was revealed with the experiments of the physicist Hans Christian Oersted (1777-1851) in 1820. In them it was observed that an electric current was capable of moving the compass needle of the same way as magnets.

It was AndrÃ© Marie Ampere (1775-1836) who established in mathematical form the link between current and magnetism, through the law that bears his name.

Simultaneously Georg Simon Ohm (1789-1854) quantitatively studied the way in which materials conduct electricity.Â He also developed the concept of electrical resistance and its relationship with voltage and current, through Ohm’s law for circuits.

Michael Faraday (1791-1867) found a way to generate a current through the relative motion between the source of the magnetic field and a closed circuit.

Later, the physicist James Clerk Maxwell (1831-1879) created a theory for electromagnetism that unified all the discovered laws, explaining the phenomena known until then.

Furthermore, through his equations, Maxwell predicted several effects that were later confirmed.Â For example, when Heinrich Hertz (1857-1894), discoverer of radio waves, verified that they moved at the speed of light.

With the advent of the theory of relativity, at the beginning of the 20th century, it was possible to explain the behavior of particles with speeds close to that of light, while quantum mechanics refined electrodynamics by introducing the concept of spin and thus explaining the origin of magnetism inÂ matterÂ .

**Fundamentals of Electrodynamics**

Electrodynamics is based on four laws, which are separately known as: Coulomb’s law, Gauss’s law, Ampere’s law, and Faraday’s law.

These four laws, plus the principle of conservation of charge, which is derived from them and the Lorentz force law, describe how electric charges interact from the classical point of view (without considering the photon as a mediator).

If the speed of the particles is close to that of light, their behavior is modified and it is necessary to add to the classical theory some relativistic corrections that are derived from Albert Einstein’s theory of relativity (Â *relativistic electrodynamics*Â ).

And when the scale of the phenomena to be studied is the atomic scale or smaller, the quantum effects acquire relevance, giving rise toÂ *quantum electrodynamics*Â .

**Mathematical foundations of electrodynamics**

The mathematics necessary for the study of electrodynamics are Vector Algebra and Vector Calculus, since electric and magnetic fields are entities of a vector nature.Â Scalar fields, such as electric potential and magnetic flux, also participate.

The mathematical operators for the derivatives of vector functions are:

- Gradient.
- Divergence.
- Rotational.
- Laplacian.

Coordinate systems are required for solving Maxwell’s equations.Â In addition to Cartesian coordinates, the use of cylindrical coordinates and spherical coordinates is common.

In integration appear the theorems of Green, Stokes and the divergence theorem.

Finally, there is a function calledÂ *Dirac delta*Â , which is defined through its properties and is very useful to express charge distributions confined to a certain dimension, for example a linear, surface distribution, a point or a plane.

**Electromagnetic waves**

The origin of electromagnetic waves is in electric charges whose movement is accelerated.Â A time-varying electric current produces an electric field, described by the vector functionÂ **E**Â (x, y, z, t) and in turn produces a magnetic fieldÂ **B**Â (x, y, z, t).

These fields combine to form the electromagnetic field, in which the electric field originates the magnetic field and vice versa.

**Laws of Electrodynamics**

When electric charges are static, there is electrostatic attraction or repulsion between them, while the magnetic interaction arises from the movement of the charges.

The four Maxwell equations relate each of the fields to its source, and together with the Lorentz force, they form the theoretical basis of electrodynamics.

**Gauss’s law**

The electric field flux that leaves aÂ volumeÂ enclosed by the closed surface S, is proportional to the net charge enclosed in it:

WhereÂ *dÂ A*Â is an area differential andÂ

*k*Â is the electrostatic constant.Â This law is a consequence of Coulomb’s law for the force between electric charges.

**Gauss’s law of magnetism**

The magnetic field flux through a volume delimited by a closed surface S is zero, because the magnetic monopoles do not exist.

Consequently, whenever a magnet is enclosed within a volume delimited by S, the number of field lines entering S is equal to the number of lines leaving:

**Faraday’s Law**

Michael Faraday discovered that the relative motion between a closed metallic loop C and a magnet generates an induced current.Â The induced voltage (electromotive force) ÎµÂ _{ind}Â , associated with said current, is proportional to the time derivative of the magnetic flux Î¦Â _{B}Â that crosses the area delimited by the loop:

The minus sign is Lenz’s law, which states that the induced voltage opposes the change in flux that produces it.Â But the induced electromotive force is the line integral of the electric field along the closed path C, therefore:

**Ampere-Maxwell Law**

The circulation of the magnetic field on a curve C is proportional to the total current that the curve encloses.Â There are two contributions to it: the conduction current I and the displacement current caused by the time variation of the electric flux Î¦Â _{E}Â :

Where Î¼Â _{o}Â and ÎµÂ _{o}Â are constants, the first is theÂ *permeability of the vacuum*Â and the second theÂ *electrical permittivity of the vacuum*Â .

**Lorentz’s Law**

Maxwell’s equations describe the relationship betweenÂ **E**Â ,Â **B**Â and their respective sources, but the dynamics of an electric charge is described by Lorentz’s law or Lorentz force.

She points out that the total force acting on a chargeÂ *q*Â moving with velocityÂ ** v**Â in the middle of an electric fieldÂ

**E**Â and a magnetic fieldÂ

**B**Â (not produced byÂ

*q*Â ) is given by:

**F**Â = qÂ **E**Â + qÂ **v**Â xÂ **B**

**Applications**

The charges in orderly movement constitute an electric current, which is capable of generating energy to do useful work: lighting light bulbs, moving motors, in short, starting numerous devices.

**Electric power distribution**

Electrodynamics makes possible the transmission of electrical energy, through alternating current, from distant places where energy is transformed and generated, to cities, industries and homes.

**Electronics**

By aiming at the study of charges in motion, electrodynamics is the physical foundation of electronics, which deals with designing devices that, through electronic circuits, make use of the flow of electrical charges to generate, transmit, receive and storing electromagnetic signals containing information.