The Curl of B

The Bio-Savart law for the volume current is
            B = μ₀/4π ∫J x r/ r₂  dτ                            … (1)

Talking the curl of both sides of Eq. (1),

             = x B = μ₀/4π ∫∇x(j x r/r²)dτ                        … (2)

But        ∇ x (J x r/²) = J (∇. r/r²) – (J.∇) r/r²                         … (3)

(Terms involving derivatives of J are dropped, because J does not depend on x, y, z). The second term integrates to zero.

Also,         ∇. r/r² = 4πδ³ (r)

. :        ∇ x B =     4π ∫J(r’)4πδ³ (r – r’)dτ’ = μ₀ J(r).


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