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Concept
Retarded time
Summary
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′.
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 time ta, 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.
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. 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.
Official source
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Related concepts (5)
Jefimenko's equations
In electromagnetism, Jefimenko's equations (named after Oleg D. Jefimenko) 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 (retarded time) 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.
Charge density
In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in coulombs per cubic meter (C⋅m−3), at any point in a volume. Surface charge density (σ) is the quantity of charge per unit area, measured in coulombs per square meter (C⋅m−2), at any point on a surface charge distribution on a two dimensional surface.
Magnetic vector potential
In classical electromagnetism, magnetic vector potential (often called A) 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.