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In electromagnetism, an electromagnetic wave (light) in vacuum travels at a finite speed (the speed of light c). The retarded time is the propagation delay between emission and observation, since it takes time for information to travel between emitter and observer. This arises due to causality.

Retarded and advanced times

Retarded time t_r or t′ is calculated with a "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. Subtracting this delay from the observer's time t then gives the time when the field began to propagate, i.e. the retarded time t′.^[1]^[2]

The retarded time is: $t'=t-{\frac {|\mathbf {r} -(\mathbf {r} ')|}{c}}$

(which can be rearranged to $c=|\mathbf {r} -\mathbf {r} '|/(t-t')$ , showing how the positions and times of source and observer are causally linked).

A related concept is the advanced time t_a, which takes the same mathematical form as above, but with a “+” instead of a “−”:

t_{a}=t+{\frac {|\mathbf {r} -\mathbf {r} '|}{c}}

This is the time it takes for a field to propagate from originating at the present time t to a distance $|\mathbf {r} -\mathbf {r} '|$ . Corresponding to retarded and advanced times are retarded and advanced potentials.^[3]

Retarded position

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:

\mathbf {r} -\mathbf {r'} =\mathbf {r} -\mathbf {r_{c}} +{\frac {|\mathbf {r} -\mathbf {r'} |}{c}}\mathbf {v}

,

where r_c is the current position of the source charge distribution and v its velocity.

Application

A moving source emit a signal at periodic intervals. As the signal propagates at a finite speed, a detector will only see the signal after a retarded time has passed.

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', t_r) and current density J(r', t_r) 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.

References

^ Electromagnetism (2nd Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008, ISBN 978-0471-927129
^ Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, ISBN 81-7758-293-3
^ McGraw Hill Encyclopaedia of Physics (2nd Edition), C.B. Parker, 1994, ISBN 0-07-051400-3
^ Classical Mechanics, T.W.B. Kibble, European Physics Series, McGraw-Hill (UK), 1973, ISBN 007-084018-0

[1] Electromagnetism (2nd Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008, ISBN 978-0471-927129

[2] Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, ISBN 81-7758-293-3

[3] McGraw Hill Encyclopaedia of Physics (2nd Edition), C.B. Parker, 1994, ISBN 0-07-051400-3

[4] Classical Mechanics, T.W.B. Kibble, European Physics Series, McGraw-Hill (UK), 1973, ISBN 007-084018-0

[1]

[2]

[3]

[4]

@@ Line 1: / Line 1: @@
+{{distinguish|Retardation time}}
-Merlin is retarded. {{electromagnetism}}
+{{Short description|Propagation delay of EM radiation (light)}}
-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 a point in a [[charge distribution]] to an observer. The term "retarded" is used in this context (and the literature) in the sense of [[propagation delay]]s.
+{{electromagnetism}}
+In [[electromagnetism]], an [[electromagnetic wave]] (light) in [[vacuum]] travels at a finite speed (the [[speed of light]] ''c''). The '''retarded time''' is the [[propagation delay]] between emission and observation, since it takes time for information to travel between emitter and observer. This arises due to [[Causality (physics)|causality]].
-==Retarded and advanced times==
-[[File:Universal charge distribution.svg|250px|right|thumb|Position vectors '''r''' and '''r&prime;''' used in the calculation.]]
-The calculation of the retarded time ''t<sub>r</sub>'' or ''t''<big>′</big> is nothing more than a simple "[[speed|speed-distance-time]]" calculation for EM fields.
+==Retarded and advanced times==
-If the EM field is radiated at [[position vector]] '''r'''<big>′</big> (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'''&nbsp;−&nbsp;'''r'''<big>′</big>|/''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''<big>′</big>.<ref>Electromagnetism (2nd Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008, ISBN 978-0471-927129</ref><ref>Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, ISBN 81-7758-293-3</ref>
+[[File:Universal charge distribution.svg|250px|right|thumb|Position vectors '''r''' and '''r&prime;''' used in the calculation]]
+Retarded time ''t<sub>r</sub>'' or ''t''<big>′</big> is calculated with a "[[speed|speed-distance-time]]" calculation for EM fields.
-:<math>t' = t - \frac{|\mathbf{r}-\mathbf{r}'|}{c}</math>
+If the EM field is radiated at [[position vector]] '''r'''<big>′</big> (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'''&nbsp;−&nbsp;'''r'''<big>′</big>|/''c''. Subtracting this delay from the observer's time ''t'' then gives the time when the field began to propagate, i.e. the retarded time ''t''<big>′</big>.<ref>Electromagnetism (2nd Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008, {{ISBN|978-0471-927129}}</ref><ref>Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, {{ISBN|81-7758-293-3}}</ref>
-which can be rearranged to
+The retarded time is:
-:<math>c = \frac{|\mathbf{r}-\mathbf{r}'|}{t - t'}</math>
+<math>t' = t - \frac{|\mathbf{r}-(\mathbf{r}') |}{c}</math>
+(which can be rearranged to
-showing how the positions and times correspond to source and observer.
+<math>c = |\mathbf{r}-\mathbf{r}'| / (t - t')</math>, showing how the positions and times of source and observer are causally linked).
-Another related concept is the '''advanced time''' ''t<sub>a</sub>'', which takes the same mathematical form as above, but with a “+” instead of a “−”:
+A related concept is the '''advanced time''' ''t<sub>a</sub>'', which takes the same mathematical form as above, but with a “+” instead of a “−”:
 :<math> t_a = t + \frac{|\mathbf r - \mathbf r'|}{ c}</math>
-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 potential]]s.<ref>McGraw Hill Encyclopaedia of Physics (2nd Edition), C.B. Parker, 1994, ISBN 0-07-051400-3</ref>
+This is the time it takes for a field to propagate from originating at the present time ''t'' to a distance <math>|\mathbf{r}-\mathbf{r}'|</math>. Corresponding to retarded and advanced times are [[retarded and advanced potential]]s.<ref>McGraw Hill Encyclopaedia of Physics (2nd Edition), C.B. Parker, 1994, {{ISBN|0-07-051400-3}}</ref>
 == Retarded position ==
-The retarded position can be obtained from the current position of a particle by subtracting the distance that has travelled in the lapse from the retarded time to the current time.
+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:
-:<math>\mathbf{r}-\mathbf{r'} = \mathbf{r}-\mathbf{r_c}+\frac{|\mathbf{r}-\mathbf{r'}|}{c}\mathbf{v}</math>
+:<math>\mathbf{r}-\mathbf{r'} = \mathbf{r}-\mathbf{r_c}+\frac{|\mathbf{r}-\mathbf{r'}|}{c}\mathbf{v}</math>,
-Where '''r'''<sub>c</sub>'' is the current position of the source charge distribution and '''v''' its velocity.
+where '''r'''<sub>c</sub>'' is the current position of the source charge distribution and '''v''' its velocity.
 ==Application==
+[[File:Retarded time.webm|400px|thumb|alt=Left panel: a yellow point, representing a source, moving on a thin grey Lissajous curve, and emitting circles at regular time intervals, whose radius grows linearly with time. The colour of the circle changes gradually from red for the first circle to blue for the last one. A white dot just above the Lissajous curve represents a detector. Right panel: a plot of time of emission vs time of detection, with each point appearing when one of the coloured circles hit the detector. The dots form a wavy monotone curve.|A moving source emit a signal at periodic intervals. As the signal propagates at a finite speed, a detector will only see the signal after a retarded time has passed.]]
-Perhaps surprisingly - electromagnetic fields and forces acting on charges depend on their history, not their mutual separation.<ref>Classical Mechanics, T.W.B. Kibble, European Physics Series, McGraw-Hill (UK), 1973, ISBN 007-084018-0</ref> The calculation of the electromagnetic fields at a present time includes integrals of [[charge density]] ρ('''r'''', ''t<sub>r</sub>'') and [[current density]] '''J'''('''r'''', ''t<sub>r</sub>'') 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.
+Perhaps surprisingly - electromagnetic fields and forces acting on charges depend on their history, not their mutual separation.<ref>Classical Mechanics, T.W.B. Kibble, European Physics Series, McGraw-Hill (UK), 1973, {{ISBN|007-084018-0}}</ref> The calculation of the electromagnetic fields at a present time includes integrals of [[charge density]] ρ('''r'''', ''t<sub>r</sub>'') and [[current density]] '''J'''('''r'''', ''t<sub>r</sub>'') 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.
 ==See also==