2. B -field of a long wire Question: Calculate B -field in arbitrary points around the wire I Available: A thin wire, infinitely long, carrying a current I
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4. Analysis and Symmetry Assume: thin wire = Current: ] = Coordinate axes: Z-axis // wire Z X Y = Symmetry: cylinder = Cylinder coordinates: r, z all perpendicular !! e r e e z
5. Analysis, field build-up 1. XYZ-axes Z Y X 2. Point P on Y-axis P z i z i at z i 3. all ( I. z i )’ s at z i contribute B i to B in P B i,y =B i,r B i,z B i,x 4. B i,x , B i,y , B i,z e r e z e 5. e i,r , e i,z , e i, B i, 6. B i,r , B i,z , B i,
6. Approach to solution Z Y X P z I.dl in dz at z dl current element I.dl note: r and vector e r !! dl x e r => - comp. only !! ( dB in XY-plane) e dB e r r y P Biot & Savart : dB in plane ┴ wire Z-axis
7. Calculations , r and e r are f(z Calc. vector product, then r, both as f(z) , and insert z Z Y X P I.dl in dz at z dB e r r dl y P e
9. Appendix: angular integration (1) dl x e r = dz . cos e z, dz, r and e r are f( Calc. vector product, then r, both as f( ) , and insert in dB z Z Y X P I.dl in dz at z dB e r r dl y P e
10. Appendix: angular integration (2) integration over from - to the end z Z Y X P I.dl in dz at z dB e r r dl y P e