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In electromagnetism, electromagnetic waves in vacuum travel at the speed of light c, according to Maxwell's Equations. The retarded time is the time when the field began to propagate from the point where it was emitted to an observer. The term "retarded" is used in this context (and the literature) in the sense of propagation delays.
The calculation of the retarded time tr or t′ is nothing more than a simple "speed-distance-time" calculation for EM fields.
If the EM field is radiated at position vector r′ (within the source charge distribution), and an observer at position r measures the EM field at time t, the time delay for the field to travel from the charge distribution to the observer is |r − r′|/c, so subtracting this delay from the observer's time t gives the time when the field actually began to propagate - the retarded time, t′. [1] [2]
The retarded time is:
which can be rearranged to
showing how the positions and times correspond to source and observer.
Another related concept is the advanced timeta, which takes the same mathematical form as above, but with a “+” instead of a “−”:
and is so-called since this is the time the field will advance from the present time t. Corresponding to retarded and advanced times are retarded and advanced potentials. [3]
The retarded position can be obtained from the current position of a particle by subtracting the distance it has travelled in the lapse from the retarded time to the current time. For an inertial particle, this position can be obtained by solving this equation:
where rc is the current position of the source charge distribution and v its velocity.
Perhaps surprisingly - electromagnetic fields and forces acting on charges depend on their history, not their mutual separation. [4] The calculation of the electromagnetic fields at a present time includes integrals of charge density ρ(r', tr) and current density J(r', tr) using the retarded times and source positions. The quantity is prominent in electrodynamics, electromagnetic radiation theory, and in Wheeler–Feynman absorber theory, since the history of the charge distribution affects the fields at later times.
An electromagnetic field is a mathematical representation of the influences on and due to electric charges. The field at any point in space and time can be regarded as a combination of an electric field and a magnetic field. The way in which charges and currents interact with the electromagnetic field is described by Maxwell's equations and the Lorentz force law. Maxwell's equations detail how the electric field converges towards or diverges away from electric charges, how the magnetic field curls around electrical currents, and how changes in the electric and magnetic fields influence each other. The Lorentz force law states that a charge subject to an electric field feels a force along the direction of the field, and a charge moving through a magnetic field feels a force that is perpendicular both to the magnetic field and to its direction of motion. Because of the interrelationship between the fields, a disturbance in the electric field can create a disturbance in the magnetic field which in turn affects the electric field, leading to an oscillation that propagates through space, known as an electromagnetic wave.
In physics, the Lorentz force is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge q moving with a velocity v in an electric field E and a magnetic field B experiences a force of
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, electric and magnetic circuits. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar, etc. They describe how electric and magnetic fields are generated by charges, currents, and changes of the fields. The equations are named after the physicist and mathematician James Clerk Maxwell, who, in 1861 and 1862, published an early form of the equations that included the Lorentz force law. Maxwell first used the equations to propose that light is an electromagnetic phenomenon. The modern form of the equations in their most common formulation is credited to Oliver Heaviside.
A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to the magnetic field. A permanent magnet's magnetic field pulls on ferromagnetic materials such as iron, and attracts or repels other magnets. In addition, a nonuniform magnetic field exerts minuscule forces on "nonmagnetic" materials by three other magnetic effects: paramagnetism, diamagnetism, and antiferromagnetism, although these forces are usually so small they can only be detected by laboratory equipment. Magnetic fields surround magnetized materials, electric currents, and electric fields varying in time. Since both strength and direction of a magnetic field may vary with location, it is described mathematically by a function assigning a vector to each point of space, called a vector field.
An electric field is the physical field that surrounds electrically charged particles. Charged particles exert attractive forces on each other when their charges are opposite, and repulsion forces on each other when their charges are the same. Because these forces are exerted mutually, 2 charges must be present for the forces to take place. The electric field of a single charge describes their capacity to exert such forces on another charged object. These forces are described by Coulomb's Law, which says that the greater the magnitude of the charges, the greater the force, and the greater the distance between them, the weaker the force. Thus, we may informally say that the greater the charge of an object, the stronger its electric field. Similarly, the electric field is stronger nearer charged objects and weaker further away. Electric fields originate from electric charges and time-varying electric currents. Electric fields and magnetic fields are both manifestations of the electromagnetic field, one of the four fundamental forces of nature.
Electric potential is defined as the amount of work energy needed per unit of electric charge to move the charge from a reference point to a specific point in an electric field. More precisely, the electric potential is the energy per unit charge for a test charge that is so small that the disturbance of the field under consideration is negligible. The motion across the field is supposed to proceed with negligible acceleration, so as to avoid the test charge acquiring kinetic energy or producing radiation. By definition, the electric potential at the reference point is zero units. Typically, the reference point is earth or a point at infinity, although any point can be used.
In physics, specifically electromagnetism, the Biot–Savart law is an equation describing the magnetic field generated by a constant electric current. It relates the magnetic field to the magnitude, direction, length, and proximity of the electric current.
Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charges and currents using an extension of the classical Newtonian model. It is, therefore, a classical field theory. The theory provides a description of electromagnetic phenomena whenever the relevant length scales and field strengths are large enough that quantum mechanical effects are negligible. For small distances and low field strengths, such interactions are better described by quantum electrodynamics which is a quantum field theory.
An electromagnetic four-potential is a relativistic vector function from which the electromagnetic field can be derived. It combines both an electric scalar potential and a magnetic vector potential into a single four-vector.
In classical electromagnetism, magnetic vector potential is the vector quantity defined so that its curl is equal to the magnetic field: . Together with the electric potential φ, the magnetic vector potential can be used to specify the electric field E as well. Therefore, many equations of electromagnetism can be written either in terms of the fields E and B, or equivalently in terms of the potentials φ and A. In more advanced theories such as quantum mechanics, most equations use potentials rather than fields.
A classical field theory is a physical theory that predicts how one or more fields in physics interact with matter through field equations, without considering effects of quantization; theories that incorporate quantum mechanics are called quantum field theories. In most contexts, 'classical field theory' is specifically intended to describe electromagnetism and gravitation, two of the fundamental forces of nature.
In electromagnetism, the Lorenz gauge condition or Lorenz gauge is a partial gauge fixing of the electromagnetic vector potential by requiring The name is frequently confused with Hendrik Lorentz, who has given his name to many concepts in this field. The condition is Lorentz invariant. The Lorenz gauge condition does not completely determine the gauge: one can still make a gauge transformation where is the four-gradient and is any harmonic scalar function: that is, a scalar function obeying the equation of a massless scalar field.
In the physics of gauge theories, gauge fixing denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct configuration of the system as an equivalence class of detailed local field configurations. Any two detailed configurations in the same equivalence class are related by a gauge transformation, equivalent to a shear along unphysical axes in configuration space. Most of the quantitative physical predictions of a gauge theory can only be obtained under a coherent prescription for suppressing or ignoring these unphysical degrees of freedom.
The Wheeler–Feynman absorber theory, named after its originators, the physicists, Richard Feynman, and John Archibald Wheeler, is a theory of electrodynamics based on a relativistic correct extension of action at a distance electron particles. The theory postulates no independent field.
In electromagnetism, Jefimenko's equations give the electric field and magnetic field due to a distribution of electric charges and electric current in space, that takes into account the propagation delay of the fields due to the finite speed of light and relativistic effects. Therefore, they can be used for moving charges and currents. They are the particular solutions to Maxwell's equations for any arbitrary distribution of charges and currents.
There are various mathematical descriptions of the electromagnetic field that are used in the study of electromagnetism, one of the four fundamental interactions of nature. In this article, several approaches are discussed, although the equations are in terms of electric and magnetic fields, potentials, and charges with currents, generally speaking.
The Liénard–Wiechert potentials describe the classical electromagnetic effect of a moving electric point charge in terms of a vector potential and a scalar potential in the Lorenz gauge. Stemming directly from Maxwell's equations, these describe the complete, relativistically correct, time-varying electromagnetic field for a point charge in arbitrary motion, but are not corrected for quantum mechanical effects. Electromagnetic radiation in the form of waves can be obtained from these potentials. These expressions were developed in part by Alfred-Marie Liénard in 1898 and independently by Emil Wiechert in 1900.
In electrodynamics, the retarded potentials are the electromagnetic potentials for the electromagnetic field generated by time-varying electric current or charge distributions in the past. The fields propagate at the speed of light c, so the delay of the fields connecting cause and effect at earlier and later times is an important factor: the signal takes a finite time to propagate from a point in the charge or current distribution to another point in space, see figure below.
In physics, a field is a physical quantity, represented by a scalar, vector, or tensor, that has a value for each point in space and time. For example, on a weather map, the surface temperature is described by assigning a number to each point on the map; the temperature can be considered at a certain point in time or over some interval of time, to study the dynamics of temperature change. A surface wind map, assigning an arrow to each point on a map that describes the wind speed and direction at that point, is an example of a vector field, i.e. a 1-dimensional (rank-1) tensor field. Field theories, mathematical descriptions of how field values change in space and time, are ubiquitous in physics. For instance, the electric field is another rank-1 tensor field, while electrodynamics can be formulated in terms of two interacting vector fields at each point in spacetime, or as a single-rank 2-tensor field.
Magnetic current is, nominally, a current composed of fictitious moving magnetic monopoles. It has the unit volt. The usual symbol for magnetic current is , which is analogous to for electric current. Magnetic currents produce an electric field analogously to the production of a magnetic field by electric currents. Magnetic current density, which has the unit V/m2, is usually represented by the symbols and . The superscripts indicate total and impressed magnetic current density. The impressed currents are the energy sources. In many useful cases, a distribution of electric charge can be mathematically replaced by an equivalent distribution of magnetic current. This artifice can be used to simplify some electromagnetic field problems. It is possible to use both electric current densities and magnetic current densities in the same analysis.
