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If we write the Biot Savart law as

where

Then take the curl, making use of curl(

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

and

Now according to my book, this reduces to

because only the value of

However, we're integrating over (x',y',z') and so varying

What is it I am misunderstanding? Thanks.

Edit: If the answer is something along the lines of 'we may aswell 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.

**B**(**r**)=μ_{0}/4π∫(**J**(**r'**)x**n**/n^{2})dV'where

**B**is the magnetic field which depends on**r**=(x,y,z), a fixed point,**J**is the volume current density depending on**r'**=(x',y',z'), and**n**=**r**-**r'**, a vector from the volume element dV' at**r'**to the point**r**. Note we integrate over the primed coordinates as**J**, the source of the current varies with these.Then take the curl, making use of curl(

**A**x**B**)=**A**(**∇**.**B**)+(**B**.**∇**)**A**-(**A**.**∇**)**B**-**B**(**∇**.**A**), and noting**∇**.**J**=0 (it depends on the primed coordinates) and that (**n**/n^{2}.**∇**)**J**=0 for the same reason, we get**∇**x**B**=μ_{0}/4π∫**J**(**∇**.**n**/n^{2})dV'-μ_{0}/4π∫(**J**.**∇**)**n**/n^{2}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

**∇**x**B**=μ_{0}/4π∫**J**(**∇**.**n**/n^{2})dV'and

**∇**.**n**/n^{2}=4πδ^{3}(**r**-**r'**). So**∇**x**B**=μ_{0}/4π∫4π**J**(**r'**)δ^{3}(**r**-**r'**)dV'**∇**x**B**=μ_{0}∫**J**(**r'**)δ^{3}(**r**-**r'**)dV'Now according to my book, this reduces to

**∇**x**B**=μ_{0}J(**r**), which makes sense in a way because we can say**∇**x**B**=μ_{0}∫**J**(**r**)δ^{3}(**r**-**r'**)dV'because only the value of

**J**at the 'spike' is actually useful anyway. This is constant so**∇**x**B**=μ_{0}**J**(**r**)∫δ^{3}(**r**-**r'**)dV' and then this integral is simply one giving the desired result.However, we're integrating over (x',y',z') and so varying

**r'**, keeping**r**fixed. As our integral only need cover all of our current, and**r**could be outside of our current distribution, (according to my brain) we need not even integrate over the 'spike' of the delta function at**r'**=**r**, which makes the integral zero.What is it I am misunderstanding? Thanks.

Edit: If the answer is something along the lines of 'we may aswell 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.

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