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# Class_23new - PHYS809 Class 23 Notes Magnetostatics Steady...

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1 PHYS809Class23Notes Magnetostatics Steady currents give rise to steady magnetic fields. Assuming that charges are not created nor destroyed (e.g. by beta decay processes), steady localized currents can flow only in closed loops. Biot & Savart experimentally determined that the magnetic field (in Gaussian units) produced by a current I flowing in a loop is ( ) ( ) 0 3 . 4 loop I d μ π - = × - x x B x x x x (23.1) This can be written in a general form as ( ) ( ) ( ) 3 0 3 , 4 allspace d x μ π - = × - x x B x J x x x (23.2) where ( ) J x is the current density. Using the identities ( ) 3 1 , - = -∇ - - x x x x x x (23.3) and ( ) , ψ ψ ψ ∇× = ∇ × + ∇× A A A (23.4) equation (23.2) becomes ( ) ( ) 3 0 . 4 allspace d x μ π = ∇× - J x B x x x (23.5) From this we see that 0. ∇⋅ = B (23.6) Also, on taking the curl, and using the identity ( ) ( ) 2 , ∇× ∇× = ∇ ∇ -∇ A A A i (23.7) equation (23.5) gives

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2 ( ) ( ) ( ) 3 0 3 2 3 0 0 4 1 1 . 4 4 allspace allspace allspace d x d x d x μ π μ μ π π ∇× = ∇× ∇× - = ⋅∇ - - - J x B x x J x J x x x x x (23.8) Since ( ) 2 1 4 , πδ = - - - x x x x (23.9) and 1 1 ,
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