Lorentz Force for a
Continuous Charge∗
David N. Williams
Randall Laboratory of Physics
University of Michigan
Ann Arbor, MI 48109 –1040
February 10, 2013
Contents
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2
3
4
Classical electrodynamics . . . . . .
Retarded and advanced kinematics
Liénard-Wiechert potentials . . . .
Lorentz force body density . . . . .
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1
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This note describes our notation and conventions for the Lorentz force exerted by a continuous charge distribution on itself.
1 Classical electrodynamics
We use Jackson’s special relativity and Gaussian unit conventions:
sig = (+ − − −)
xµ = (ct, x)
j µ = (cρ, j)
∂ ·F =
4π
j
c
µ = 0, 1, 2, 3
∂
∂µ =
,∇
∂ct
Aµ = (φ, A)
∂ ·j = 0
(1.2)
(1.3)
∂ ·F D = 0
Fµν = ∂µ Aν − ∂µ Aµ ≡ [∂A]µν
∗
(1.1)
∂ ·A = 0 (Lorentz gauge)
This document was started on August 6, 2012.
1
(1.4)
(1.5)
4π
A ≡ ∂ ·∂A =
j
c
1
1
Θ=
F ·F + g F : F
4π
4
(1.6)
1
∂ ·Θ = − F ·j
c
1
f = F ·j (Lorentz force density)
c
Z
Z
dPQ
3
= f (x) d x (Lorentz force)
Q = ρ(x) d3 x
dt
(1.7)
(1.8)
(1.9)
In Eq. (1.9), P is the total four-momentum of the charge distribution.
2 Retarded and advanced kinematics
Retarded and advanced sources for field point x.
Four-vector spacetime points on the world lines of charge elements are parametrized by proper time τ plus three, invariant spatial coordinates α or
β, corresponding to three-positions in a reference body. We also call α
and β body coordinates. We use α for spacetime points x = y(τ, α) which
experience the electromagnetic field, also called field points, and β for retarded or advanced spacetime source points y(τR,A , β). We use R for the
displacement from a source to a field point.
x = [ct, x(t, α)] = y(τ, α)
(2.1)
R = x − y(τ 0, β) = y(τ, α) − y(τ 0, β)
(2.2)
2
Here are the two solutions of the lightcone condition R · R = 0 for τ 0
and R, along with the corresponding retarded and advanced four-velocity
u, four-acceleration a, and four-lurch l:
τR,A = τR,A (x, β) = τR,A [y(τ, α), β]
(2.3)
RR,A = RR,A (x, β) = y(τ, α) − y(τR,A , β)
(2.4)
∂y
(τR,A , β)
∂τ
∂u
= aR,A (x, β) =
(τR,A , β)
∂τ
∂a
= lR,A (x, β) =
(τR,A , β)
∂τ
uR,A = uR,A (x, β) =
(2.5)
aR,A
(2.6)
lR,A
(2.7)
We use the term lurch instead of jerk for ȧ because we want to use j for
current.
Let J be the Jacobian of the transformation α → x(t, α) at time t.
Then
d3 x = J d3 α
(2.8)
dq = ρ(x) d3 x = ρ0 (α) d3 α
j(x) =
ρ0
u(τ, α)
Jγ
ρ(x) =
ρ0
J
Z
Q=
γ ≡ u0 (τ, α) = γ(x)
dq
(2.9)
(2.10)
Note that point charge formulas for derivatives with respect to x of τR,A ,
RR,A , uR,A , etc., also hold for continuous charge, with the understanding
that ∂/∂x is taken at constant source reference point β.
3 Liénard-Wiechert potentials
In the following, blackboard bold symbols are used for reference body volume densities.
AR,A = AR,A (x) = AR,A [y(τ, α)]
Z
Z
uR,A
3
= d β ρ0 (β)
= d3 β ρ0 (β) AR,A
|RR,A ·uR,A |
3
(3.1)
(3.2)
AR,A = AR,A (x, β) (vector potential body density)
Z
FR,A = FR,A (x) = d3 β ρ0 (β) FR,A
(3.3)
FR,A = FR,A (x, β) = [∂AR,A ] (x, β)
(3.5)
(3.4)
4 Lorentz force body density
Now we call f (x) the Lorentz force field, and write it in terms of a Lorentz
force body density, f(x, β).
Z
fR,A (x) = d3 β ρ0 (β) fR,A
(4.1)
fR,A = fR,A (x, β) =
ρ0 (α)
fR,A (x) =
Jγc
Z
FR,A (x, β)·j(x)
FR,A (x, β)·ρ0 (α) u(τ, α)
=
c
Jγc
(4.2)
d3 β ρ0 (β) FR,A (x, β)·u(τ, α)
4
(4.3)