For a single charge e with position vector R, the charge density ρ and and current density j are given by:
ρ(r,t)=eδ3(r−R(t)),
j(r,t)=edRdtδ3(r−R(t)).
Suppose we want to check the equation of continuity
∂ρ∂t+∇⋅j=0.
How to do it? How to deal with the derivatives of a delta function?
Answer
It is almost no trouble to generalize to a finite number of point charges qi at positions ri(t). Then the charge density is
ρ(r,t) = ∑iqiδ3(r−ri(t)),
and the current density
j(r,t) = ∑iqi˙ri(t)δ3(r−ri(t)).
For clarity let us write ∇≡∂∂r. The chain rule then yields the continuity equation
−∂ρ(r,t)∂t = −∑iqi∂∂tδ3(r−ri(t)) = ∑iqi˙ri(t)⋅∂∂rδ3(r−ri(t)) = ∂∂r⋅∑iqi˙ri(t)δ3(r−ri(t)) = ∂∂r⋅j(r,t).
The same calculation can be repeated more carefully with the help of test functions.
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