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Deriving Ampere's law from the Biot-Savart law
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[QUOTE="fayled, post: 4848564, member: 497826"] If we write the Biot Savart law as [B]B[/B]([B]r[/B])=μ[SUB]0[/SUB]/4π∫([B]J[/B]([B]r'[/B])x[B]n[/B]/n[SUP]2[/SUP])dV' where [B]B[/B] is the magnetic field which depends on [B]r[/B]=(x,y,z), a fixed point, [B]J[/B] is the volume current density depending on [B]r'[/B]=(x',y',z'), and [B]n[/B]=[B]r[/B]-[B]r'[/B], a vector from the volume element dV' at [B]r'[/B] to the point [B]r[/B]. Note we integrate over the primed coordinates as [B]J[/B], the source of the current varies with these. Then take the curl, making use of curl([B]A[/B]x[B]B[/B])=[B]A[/B]([B]∇[/B].[B]B[/B])+([B]B[/B].[B]∇[/B])[B]A[/B]-([B]A[/B].[B]∇[/B])[B]B[/B]-[B]B[/B]([B]∇[/B].[B]A[/B]), and noting [B]∇[/B].[B]J[/B]=0 (it depends on the primed coordinates) and that ([B]n[/B]/n[SUP]2[/SUP].[B]∇[/B])[B]J[/B]=0 for the same reason, we get [B]∇[/B]x[B]B[/B]=μ[SUB]0[/SUB]/4π∫[B]J[/B]([B]∇[/B].[B]n[/B]/n[SUP]2[/SUP])dV'-μ[SUB]0[/SUB]/4π∫([B]J[/B].[B]∇[/B])[B]n[/B]/n[SUP]2[/SUP]dV'. Now term two can be shown to integrate to zero, which I understand (incidentally, the book says this second term is integrated over a volume enclosing all current, as I suspected - this is related to my problem below), but I have a problem with term 1. We get [B]∇[/B]x[B]B[/B]=μ[SUB]0[/SUB]/4π∫[B]J[/B]([B]∇[/B].[B]n[/B]/n[SUP]2[/SUP])dV' and [B]∇[/B].[B]n[/B]/n[SUP]2[/SUP]=4πδ[SUP]3[/SUP]([B]r[/B]-[B]r'[/B]). So [B]∇[/B]x[B]B[/B]=μ[SUB]0[/SUB]/4π∫4π[B]J[/B]([B]r'[/B])δ[SUP]3[/SUP]([B]r[/B]-[B]r'[/B])dV' [B]∇[/B]x[B]B[/B]=μ[SUB]0[/SUB]∫[B]J[/B]([B]r'[/B])δ[SUP]3[/SUP]([B]r[/B]-[B]r'[/B])dV' Now according to my book, this reduces to [B]∇[/B]x[B]B[/B]=μ[SUB]0[/sub]J([B]r[/B]), which makes sense in a way because we can say [B]∇[/B]x[B]B[/B]=μ[SUB]0[/SUB]∫[B]J[/B]([B]r[/B])δ[SUP]3[/SUP]([B]r[/B]-[B]r'[/B])dV' because only the value of [B]J[/B] at the 'spike' is actually useful anyway. This is constant so [B]∇[/B]x[B]B[/B]=μ[SUB]0[/SUB][B]J[/B]([B]r[/B])∫δ[SUP]3[/SUP]([B]r[/B]-[B]r'[/B])dV' and then this integral is simply one giving the desired result. However, we're integrating over (x',y',z') and so varying [B]r'[/B], keeping [B]r[/B] fixed. As our integral only need cover all of our current, and [B]r[/B] could be outside of our current distribution, (according to my brain) we need not even integrate over the 'spike' of the delta function at [B]r'[/B]=[B]r[/B], which makes the integral zero. What is it I am misunderstanding? Thanks. Edit: If the answer is something along the lines of 'we may as well integrate over all space because there's no current anywhere else anyway', I ask, if there's no current anywhere else anyway, why do we get two different answers by integrating/not integrating over all space. [/QUOTE]
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Deriving Ampere's law from the Biot-Savart law
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