Liénard–Wiechert potential

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.

Equations

Definition of Liénard–Wiechert potentials

The retarded time is defined, in the context of distributions of charges and currents, as
:$t_r(\mathbf,\mathbf, t) = t - \frac, \mathbf - \mathbf_s, ,$
where $\mathbf$ is the observation point, and $\mathbf_s$ is the observed point subject to the variations of source charges and currents.
For a moving point charge $q$ whose given trajectory is $\mathbf(t)$, $\mathbf$
is no more fixed, but becomes a function of the retarded time itself. In other words, following the trajectory
of $q$ yields the implicit equation
:$t_r = t - \frac, \mathbf - \mathbf_s(t_r), ,$
which provides the retarded time $t_r$ as a function of the current time (and of the given trajectory):
:$t_r = t_r(\mathbf,t)$.

The Liénard–Wiechert potentials $\varphi$ (scalar potential field) and $\mathbf$ (vector potential field) are, for a source point charge $q$ at position $\mathbf_s$ traveling with velocity $\mathbf_s$:

:$\varphi(\mathbf, t) = \frac \left(\frac \right)_$

and

:$\mathbf(\mathbf,t) = \frac \left(\frac \right)_ = \frac \varphi(\mathbf, t)$

where:

* $\boldsymbol_s(t) = \frac$ is the velocity of the source expressed as a fraction of the speed of light;

* $$ is the distance from the source;

* $\mathbf_s = \frac$ is the unit vector pointing in the direction from the source and,

* The symbol $(\cdots)_$ means that the quantities inside the parenthesis should be evaluated at the retarded time $t_r = t_r(\mathbf,t)$.

This can also be written in a covariant way, where the electromagnetic four-potential at $X^=(t,x,y,z)$ is:
:$A^(X)= -\frac \left(\frac \right)_$
where $R^=X^-R_^$ and $R_^$ is the position of the source and $U^=dX^/d\tau$ is its four velocity.

Field computation

We can calculate the electric and magnetic fields directly from the potentials using the definitions:
$\mathbf = - \nabla \varphi - \dfrac$ and $\mathbf = \nabla \times \mathbf$

The calculation is nontrivial and requires a number of steps. The electric and magnetic fields are (in non-covariant form):
$\mathbf(\mathbf, t) = \frac \left(\frac + \frac \right)_$
and
$\mathbf(\mathbf, t) = \frac \left(\frac + \frac \right)_ = \frac \times \mathbf(\mathbf, t)$
where $\boldsymbol_s(t) = \frac$, $\mathbf_s(t) = \frac$ and $\gamma(t) = \frac$ (the Lorentz factor).

Note that the $\mathbf_s - \boldsymbol_s$ part of the first term updates the direction of the field toward the instantaneous position of the charge, if it continues to move with constant velocity $c \boldsymbol_s$. This term is connected with the "static" part of the electromagnetic field of the charge.

The second term, which is connected with electromagnetic radiation by the moving charge, requires charge acceleration $\dot_s$ and if this is zero, the value of this term is zero, and the charge does not radiate (emit electromagnetic radiation). This term requires additionally that a component of the charge acceleration be in a direction transverse to the line which connects the charge $q$ and the observer of the field $\mathbf(\mathbf, t)$. The direction of the field associated with this radiative term is toward the fully time-retarded position of the charge (i.e. where the charge was when it was accelerated).

Derivation

The $\varphi(\mathbf, t)$ scalar and $\mathbf(\mathbf, t)$ vector potentials satisfy the nonhomogeneous electromagnetic wave equation where the sources are expressed with the charge and current densities $\rho(\mathbf, t)$ and $\mathbf(\mathbf, t)$
$\nabla^2 \varphi + \left ( \nabla \cdot \mathbf \right ) = - \,,$
and the Ampère-Maxwell law is:
$\nabla^2 \mathbf - - \nabla \left ( + \nabla \cdot \mathbf \right ) = - \mu_0 \mathbf \,.$

Since the potentials are not unique, but have gauge freedom, these equations can be simplified by gauge fixing. A common choice is the Lorenz gauge condition:
$+ \nabla \cdot \mathbf = 0$

Then the nonhomogeneous wave equations become uncoupled and symmetric in the potentials:
$\nabla^2 \varphi - = - \,,$
$\nabla^2 \mathbf - = - \mu_0 \mathbf \,.$

Generally, the retarded solutions for the scalar and vector potentials (SI units) are
$\varphi(\mathbf, t) = \frac\int \frac d^3\mathbf' + \varphi_0(\mathbf, t)$
and
$\mathbf(\mathbf, t) = \frac \int \frac d^3\mathbf' + \mathbf_0(\mathbf, t)$

where $t_r' = t - \frac , \mathbf - \mathbf',$ is the retarded time and $\varphi_0(\mathbf, t)$ and $\mathbf_0(\mathbf, t)$
satisfy the homogeneous wave equation with no sources and the boundary conditions. In the case that there are no boundaries surrounding the sources then
$\varphi_0(\mathbf, t) = 0$ and $\mathbf_0(\mathbf, t) = 0$.

For a moving point charge whose trajectory is given as a function of time by $\mathbf_s(t')$, the charge and current densities are as follows:

$\rho(\mathbf', t') = q \delta^3(\mathbf - \mathbf_s(t'))$
$\mathbf(\mathbf', t') = q\mathbf_s(t') \delta^3(\mathbf - \mathbf_s(t'))$

where $\delta^3$ is the three-dimensional Dirac delta function and $\mathbf_s(t')$ is the velocity of the point charge.

Substituting into the expressions for the potential gives
$\varphi(\mathbf, t) = \frac \int \frac d^3\mathbf'$
$\mathbf(\mathbf, t) = \frac \int \frac d^3\mathbf'$

These integrals are difficult to evaluate in their present form, so we will rewrite them by replacing $t_r'$ with $t'$ and integrating over the delta distribution $\delta(t' - t_r')$:
$\varphi(\mathbf, t) = \frac \iint \frac \delta(t' - t_r') \, dt' \, d^3\mathbf'$
$\mathbf(\mathbf, t) = \frac \iint \frac \delta(t' - t_r') \, dt' \, d^3\mathbf'$

We exchange the order of integration:
$\varphi(\mathbf, t) = \frac \iint \frac q\delta^3(\mathbf - \mathbf_s(t')) \, d^3\mathbf' dt'$
$\mathbf(\mathbf, t) = \frac \iint \frac q\mathbf_s(t') \delta^3(\mathbf - \mathbf_s(t')) \, d^3\mathbf' dt'$

The delta function picks out $\mathbf' = \mathbf_s(t')$ which allows us to perform the inner integration with ease. Note that $t_r'$ is a function of $\mathbf'$, so this integration also fixes $t_r' = t - \frac , \mathbf - \mathbf_s(t'),$.
$\varphi(\mathbf, t) = \frac \int q\frac dt'$
$\mathbf(\mathbf, t) = \frac \int q\mathbf_s(t') \frac \, dt'$

The retarded time $t_r'$ is a function of the field point $(\mathbf, t)$ and the source trajectory $\mathbf_s(t')$, and hence depends on $t'$. To evaluate this integral, therefore, we need the identity
$\delta(f(t')) = \sum_i \frac$
where each $t_i$ is a zero of $f$. Because there is only one retarded time $t_r$ for any given space-time coordinates $(\mathbf, t)$ and source trajectory $\mathbf_s(t')$, this reduces to:
$\begin\delta(t' - t_r') =& \frac = \frac \\ &= \frac\\ &= \frac\end$
where $\boldsymbol_s = \mathbf_s/c$ and $\mathbf_s$ are evaluated at the retarded time $t_r$, and we have used the identity $, \mathbf, ' = \hat \cdot \mathbf$ with $\mathbf = \mathbf'$. Notice that the retarded time $t_r$ is the solution of the equation $t_r = t - \frac , \mathbf - \mathbf_s(t_r),$. Finally, the delta function picks out $t' = t_r$, and
$\varphi(\mathbf, t) = \frac \left(\frac\right)_ = \frac \left(\frac\right)_$
$\mathbf(\mathbf, t) = \frac \left(\frac\right)_ = \frac \left(\frac\right)_$
which are the Liénard–Wiechert potentials.

Lorenz gauge, electric and magnetic fields

In order to calculate the derivatives of $\varphi$ and $\mathbf$ it is convenient to first compute the derivatives of the retarded time. Taking the derivatives of both sides of its defining equation (remembering that $\mathbf = \mathbf(t_r)$):
$t_r + \frac, \mathbf-\mathbf, = t$
Differentiating with respect to t,
$\frac + \frac\frac\frac= 1$

$\frac \left(1 - \mathbf_s\cdot_s\right) = 1$

$\frac = \frac$

Similarly, taking the gradient with respect to $\mathbf$ and using the multivariable chain rule gives

$t_r + \frac , \mathbf-\mathbf, = 0$

$t_r + \frac \left( t_r \frac + \mathbf_s\right) = 0$

$t_r \left(1 - \mathbf_s\cdot_s\right) = -\mathbf_s/c$

$t_r = -\frac$

It follows that

$\frac = \frac\frac = \frac$

$, \mathbf-\mathbf, = t_r \frac + \mathbf_s = \frac$

These can be used in calculating the derivatives of the vector potential and the resulting expressions are

\begin
\frac =&
-\frac\frac\frac\left \mathbf-\mathbf, (1-\mathbf_s\cdot_s)\right\
=& -\frac\frac\frac\left _\mathbf_s\cdot_s_+_^2_-_(\mathbf-\mathbf)\cdot_\dot__s_/c_\rightend

\begin\cdot\mathbf_=&
-\frac\frac\big(_\left \mathbf-\mathbf, -(\mathbf-\mathbf)\cdot_s\right)\rightcdot_s_-_\left \mathbf-\mathbf, -(\mathbf-\mathbf)\cdot_s\right)\rightcdot_s\big)\\
=&_-_\frac\frac\cdot\\
&\left \mathbf-\mathbf, -(\mathbf-\mathbf)\cdot_s\big)(\mathbf_s\cdot_\dot__s/c)\right\\=&\frac\frac\left beta_s^2_-_\mathbf_s\cdot_s_-_(\mathbf-\mathbf)\cdot_\dot__s/c\rightend

These_show_that_the_Lorenz_gauge_is_satisfied,_namely_that_\frac_+_c^2_\cdot\mathbf_=_0_.

Similarly_one_calculates:

\varphi_=_-\frac\frac\left mathbf_s\left(1-^2_+_(\mathbf-\mathbf)\cdot_\dot__s/c\right)_-__s(1-\mathbf_s\cdot_s)\right/math>

\frac_=_\frac\frac\left \mathbf-\mathbf, \dot__s_(1-\mathbf_s\cdot_s)/c\right/math>

By_noting_that_for_any_vectors_$\mathbf$,_$\mathbf$,_$\mathbf$:
\mathbf\times(\mathbf\times\mathbf)_=_(\mathbf\cdot\mathbf)\mathbf-_(\mathbf\cdot_\mathbf)\mathbf
The_expression_for_the_electric_field_mentioned_above_becomes
\begin\mathbf(\mathbf,_t)_=&_\frac_\frac\cdot__\\
&\left \mathbf_-_\mathbf_s, (\mathbf_s_\cdot_\dot_s/c)_(\mathbf_s_-__s)_-_, \mathbf_-_\mathbf_s, \big(\mathbf_s_\cdot_(\mathbf_s_-__s)\big)_\dot_s/c_\rightend

which_is_easily_seen_to_be_equal_to_$-\varphi_-_\frac$

Similarly_$\times\mathbf$_gives_the_expression_of_the_magnetic_field_mentioned_above:
\begin_=&_\times\mathbf_=
-\frac\frac\big(_\left \mathbf-\mathbf, -(\mathbf-\mathbf)\cdot_s\right)\righttimes_s_-_\left \mathbf-\mathbf, -(\mathbf-\mathbf)\cdot_s\right)\righttimes_s\big)\\
=&_-_\frac\frac\cdot\\
&\left \mathbf-\mathbf, -(\mathbf-\mathbf)\cdot_s\big)(\mathbf_s\times_\dot__s/c)\right\\=&
-\frac_\frac\cdot__\\
&\left \mathbf_-_\mathbf_s, (\mathbf_s_\cdot_\dot_s/c)_(\mathbf_s\times__s)_+_, \mathbf_-_\mathbf_s, \big(\mathbf_s_\cdot_(\mathbf_s_-__s)\big)_\mathbf_s\times\dot_s/c_\right=_\frac\times\mathbf
\end
The_source_terms_$\mathbf_s$,_$\mathbf_s$,_and_$\mathbf_s$_are_to_be_evaluated_at_the_retarded_time.

_Implications

The_study_of_classical_electrodynamics_was_instrumental_in_Albert_Einstein's_development_of_the_theory_of_relativity.__Analysis_of_the_motion_and_propagation_of_electromagnetic_waves_led_to_the_special_relativity_description_of_space_and_time.__The_Liénard–Wiechert_formulation_is_an_important_launchpad_into_a_deeper_analysis_of_relativistic_moving_particles.

The_Liénard–Wiechert_description_is_accurate_for_a_large,_independently_moving_particle_(i.e._the_treatment_is_"classical"_and_the_acceleration_of_the_charge_is_due_to_a_force_independent_of_the_electromagnetic_field)._The_Liénard–Wiechert_formulation_always_provides__two_sets_of_solutions:_Advanced_fields_are_absorbed_by_the_charges_and_retarded_fields_are_emitted._Schwarzschild_and_Fokker_considered_the_advanced_field_of_a_system_of_moving_charges,_and_the_retarded_field_of_a_system_of_charges_having_the_same_geometry_and_opposite_charges._Linearity_of_Maxwell's_equations_in_vacuum_allows_one_to_add_both_systems,_so_that_the_charges_disappear:_This_trick_allows_Maxwell's_equations_to_become_linear_in_matter.
Multiplying_electric_parameters_of_both_problems_by_arbitrary_real_constants_produces_a_coherent_interaction_of_light_with_matter_which_generalizes_Einstein's_theory_which_is_now_considered_as_founding_theory_of_lasers:_it_is_not_necessary_to_study_a_large_set_of_identical_molecules_to_get_coherent_amplification_in_the_mode_obtained_by_arbitrary_multiplications_of_advanced_and_retarded_fields.
To_compute_energy,_it_is_necessary_to_use_the_absolute_fields_which_includes_the_zero_point_field;_otherwise,_an_error_appears,_for_instance_in_photon_counting.

It_is_important_to_take_into_account_the_zero_point_field_discovered_by_Planck._It_replaces_Einstein's_"A"_coefficient_and_explains_that_the_classical_electron_is_stable_on_Rydberg's_classical_orbits._Moreover,_introducing_the_fluctuations_of_the_zero_point_field_produces_Willis_E._Lamb's_correction_of_levels_of_H_atom.

Quantum_electrodynamics_helped_bring_together_the_radiative_behavior_with_the_quantum_constraints._It_introduces_quantization_of_normal_modes_of_the_electromagnetic_field_in_assumed_perfect_optical_resonators.

_Universal_speed_limit

The_force_on_a_particle_at_a_given_location__and_time__depends_in_a_complicated_way_on_the_position_of_the_source_particles_at_an_earlier_time__due_to_the_ finite_speed,_c,_at_which_electromagnetic_information_travels._A_particle_on_Earth_'sees'_a_charged_particle_accelerate_on_the_Moon_as_this_acceleration_happened_1.5_seconds_ago,_and_a_charged_particle's_acceleration_on_the_Sun_as_happened_500_seconds_ago._This_earlier_time_in_which_an_event_happens_such_that_a_particle_at_location__'sees'_this_event_at_a_later_time__is_called_the_ retarded_time,_.__The_retarded_time_varies_with_position;_for_example_the_retarded_time_at_the_Moon_is_1.5_seconds_before_the_current_time_and_the_retarded_time_on_the_Sun_is_500_s_before_the_current_time_on_the_Earth.__The_retarded_time_''t_r''=''t_r''(''r'',''t'')_is_defined_implicitly_by

:$t_r=t-\frac$

where_$R(t_r)$_is_the_distance_of_the_particle_from_the_source_at_the_retarded_time._Only_electromagnetic_wave_effects_depend_fully_on_the_retarded_time.

A_novel_feature_in_the_Liénard–Wiechert_potential_is_seen_in_the_breakup_of_its_terms_into_two_types_of_field_terms_(see_below),_only_one_of_which_depends_fully_on_the_retarded_time._The_first_of_these_is_the_static_electric_(or_magnetic)_field_term_that_depends_only_on_the_distance_to_the_moving_charge,_and_does_not_depend_on_the_retarded_time_at_all,_if_the_velocity_of_the_source_is_constant._The_other_term_is_dynamic,_in_that_it_requires_that_the_moving_charge_be_''accelerating''_with_a_component_perpendicular_to_the_line_connecting_the_charge_and_the_observer_and_does_not_appear_unless_the_source_changes_velocity._This_second_term_is_connected_with_electromagnetic_radiation.

The_first_term_describes_ near_field_effects_from_the_charge,_and_its_direction_in_space_is_updated_with_a_term_that_corrects_for_any_constant-velocity_motion_of_the_charge_on_its_distant_static_field,_so_that_the_distant_static_field_appears_at_distance_from_the_charge,_with_no_aberration_of_light_or_ light-time_correction._This_term,_which_corrects_for_time-retardation_delays_in_the_direction_of_the_static_field,_is_required_by_Lorentz_invariance._A_charge_moving_with_a_constant_velocity_must_appear_to_a_distant_observer_in_exactly_the_same_way_as_a_static_charge_appears_to_a_moving_observer,_and_in_the_latter_case,_the_direction_of_the_static_field_must_change_instantaneously,_with_no_time-delay._Thus,_static_fields_(the_first_term)_point_exactly_at_the_true_instantaneous_(non-retarded)_position_of_the_charged_object_if_its_velocity_has_not_changed_over_the_retarded_time_delay._This_is_true_over_any_distance_separating_objects.

The_second_term,_however,_which_contains_information_about_the_acceleration_and_other_unique_behavior_of_the_charge_that_cannot_be_removed_by_changing_the_Lorentz_frame_(inertial_reference_frame_of_the_observer),_is_fully_dependent_for_direction_on_the_time-retarded_position_of_the_source._Thus,_electromagnetic_radiation_(described_by_the_second_term)_always_appears_to_come_from_the_direction_of_the_position_of_the_emitting_charge_at_the_retarded_time._Only_this_second_term_describes_information_transfer_about_the_behavior_of_the_charge,_which_transfer_occurs_(radiates_from_the_charge)_at_the_speed_of_light._At_"far"_distances_(longer_than_several_wavelengths_of_radiation),_the_1/R_dependence_of_this_term_makes_electromagnetic_field_effects_(the_value_of_this_field_term)_more_powerful_than_"static"_field_effects,_which_are_described_by_the_1/R²_field_of_the_first_(static)_term_and_thus_decay_more_rapidly_with_distance_from_the_charge.

_Existence_and_uniqueness_of_the_retarded_time

_Existence

The_retarded_time_is_not_guaranteed_to_exist_in_general._For_example,_if,_in_a_given_frame_of_reference,_an_electron_has_just_been_created,_then_at_this_very_moment_another_electron_does_not_yet_feel_its_electromagnetic_force_at_all._However,_under_certain_conditions,_there_always_exists_a_retarded_time._For_example,_if_the_source_charge_has_existed_for_an_unlimited_amount_of_time,_during_which_it_has_always_travelled_at_a_speed_not_exceeding_$v_M_<_c$,_then_there_exists_a_valid_retarded_time_$t_r$._This_can_be_seen_by_considering_the_function_$f(t')_=_, \mathbf_-_\mathbf_s(t'), _-_c(t_-_t')$._At_the_present_time_$t'_=_t$;_$f(t')_=_, \mathbf_-_\mathbf_s(t'), _-_c(t_-_t')_=_, \mathbf_-_\mathbf_s(t'), _\geq_0$._The_derivative_$f'(t')$_is_given_by

:$f'(t')_=_\frac_\cdot_(-\mathbf_s(t'))_+_c_\geq_c_-_\left, \frac\_\,_, \mathbf_s(t'), _=_c_-_, \mathbf_s(t'), _\geq_c_-_v_M_>_0$

By_the_ mean_value_theorem,_$f(t_-_\Delta_t)_\leq_f(t)_-_f'(t)\Delta_t_\leq_f(t)_-_(c_-_v_M)\Delta_t$._By_making_$\Delta_t$_sufficiently_large,_this_can_become_negative,_''i.e.'',_at_some_point_in_the_past,_$f(t')_<_0$._By_the_ intermediate_value_theorem,_there_exists_an_intermediate_$t_r$_with_$f(t_r)_=_0$,_the_defining_equation_of_the_retarded_time._Intuitively,_as_the_source_charge_moves_back_in_time,_the_cross_section_of_its_light_cone_at_present_time_expands_faster_than_it_can_recede,_so_eventually_it_must_reach_the_point_$\mathbf$._This_is_not_necessarily_true_if_the_source_charge's_speed_is_allowed_to_be_arbitrarily_close_to_$c$,_''i.e.'',_if_for_any_given_speed_$v_<_c$_there_was_some_time_in_the_past_when_the_charge_was_moving_at_this_speed._In_this_case_the_cross_section_of_the_light_cone_at_present_time_approaches_the_point_$\mathbf$_as_the_observer_travels_back_in_time_but_does_not_necessarily_ever_reach_it.

_Uniqueness

For_a_given_point_$(\mathbf,_t)$_and_trajectory_of_the_point_source_$\mathbf_s(t')$,_there_is_at_most_one_value_of_the_retarded_time_$t_r$,_''i.e.'',_one_value_$t_r$_such_that_$, \mathbf_-_\mathbf_s(t_r), _=_c(t_-_t_r)$._This_can_be_realized_by_assuming_that_there_are_two_retarded_times_$t_1$_and_$t_2$,_with_$t_1_\leq_t_2$._Then,_$, \mathbf_-_\mathbf_s(t_1), _=_c(t_-_t_1)$_and_$, \mathbf_-_\mathbf_s(t_2), _=_c(t_-_t_2)$._Subtracting_gives__c(t_2_-_t_1)_=_, \mathbf_-_\mathbf_s(t_1), _-_, \mathbf_-_\mathbf_s(t_2), _\leq_, \mathbf_s(t_2)_-_\mathbf_s(t_1), _by_the_triangle_inequality._Unless_$t_2_=_t_1$,_this_then_implies_that_the_average_velocity_of_the_charge_between_$t_1$_and_$t_2$_is_$, \mathbf_s(t_2)_-_\mathbf_s(t_1), /(t_2_-_t_1)_\geq_c$,_which_is_impossible._The_intuitive_interpretation_is_that_one_can_only_ever_"see"_the_point_source_at_one_location/time_at_once_unless_it_travels_at_least_at_the_speed_of_light_to_another_location._As_the_source_moves_forward_in_time,_the_cross_section_of_its_light_cone_at_present_time_contracts_faster_than_the_source_can_approach,_so_it_can_never_intersect_the_point_$\mathbf$_again.

The_conclusion_is_that,_under_certain_conditions,_the_retarded_time_exists_and_is_unique.

_See_also

*Maxwell's_equations_

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,_and_electric_circuits._
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