Page 344 - Intro to Tensor Calculus

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which is known as the Lorentz force law. The magnetic force due to a line charge density λ moving along
acurve C is the line integral
Z Z
~
~
~
~
~
∗
F mag = λ ds(V × B)= I × Bds. (2.6.52)
C C
Similarly, for a moving surface charge density moving on a surface
ZZ ZZ
~
~
~
~
~
F mag = µ dσ(V × B)= K × Bdσ (2.6.53)
∗
S S
and for a moving volume charge density
ZZZ ZZ
~
~
~
~
~
∗
F mag = ρ dτ(V × B)= J × Bdτ (2.6.54)
V V
∗ ~
~
~
∗ ~
~
∗ ~
where the quantities I = λ V , K = µ V and J = ρ V are respectively the current, the current per unit
length, and current per unit area.
A conductor is any material where the charge is free to move. The flow of charge is governed by Ohm’s
law. Ohm’s law states that the current density vector J i is a linear function of the electric intensity or
J i = σ im E m ,where σ im is the conductivity tensor of the material. For homogeneous, isotropic conductors
σ im = σδ im so that J i = σE i where σ is the conductivity and 1/σ is called the resistivity.
∗
Surround a charge density ρ with an arbitrary simple closed surface S having volume V and calculate
the flux of the current density across the surface. We find by the divergence theorem
ZZ ZZZ
~
~
J · ˆ n dσ = ∇· Jdτ. (2.6.55)
S V
If charge is to be conserved, the current flow out of the volume through the surface must equal the loss due
to the time rate of change of charge within the surface which implies
ZZ ZZZ d ZZZ ZZZ ∂ρ ∗
~
~
∗
J · ˆ n dσ = ∇· Jdτ = − ρ dτ = − dτ (2.6.56)
dt ∂t
S V V V
or
ZZZ ∂ρ ∗
~
∇· J + dτ =0. (2.6.57)
∂t
V
This implies that for an arbitrary volume we must have
∂ρ ∗
~
∇· J = − . (2.6.58)
∂t
Note that equation (2.6.58) has the same form as the continuity equation (2.3.73) for mass conservation and
so it is also called a continuity equation for charge conservation. For magnetostatics there exists steady line
~
currents or stationary current so ∂ρ ∗ = 0. This requires that ∇· J =0.
∂t