Thursday, October 11, 2018

homework and exercises - Derivatives of Dirac delta function and equation of continuity for a single charge


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(rR(t)),


j(r,t)=edRdtδ3(rR(t)).


Suppose we want to check the equation of continuity


ρt+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(rri(t)),


and the current density


j(r,t) = iqi˙ri(t)δ3(rri(t)).


For clarity let us write r. The chain rule then yields the continuity equation


ρ(r,t)t = iqitδ3(rri(t)) = iqi˙ri(t)rδ3(rri(t))  = riqi˙ri(t)δ3(rri(t)) = rj(r,t).


The same calculation can be repeated more carefully with the help of test functions.


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