University Electromagnetism: Magnetic field (induction) of a long thin wire supporting a current

1. Magnetic Field of a Long Wire © Frits F.M. de Mul

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

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  

8. Conclusions | B  ~ 1/ y P cylinder symmetry y P I P B

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 