University Electromagnetism: Magnetic field (induction) of a rotating conducting charged sphere (with surface charges)

1. Magnetic Field of a Rotating Charged Conducting Sphere © Frits F.M. de Mul

2. B -field of a rotating charged conducting sphere Question: Calculate B -field in arbitrary points on the axis of rotation inside and outside the sphere Available: A charged conducting sphere (charge Q, radius R ), rotating with  rad/sec 

3. Analysis and Symmetry (1) Calculate B -field in point P inside or outside the sphere P P O  Assume Z-axis through O and P. z P Z Y X Coordinate systems: - X,Y, Z   r - r, 

4. Analysis and Symmetry (2) Conducting sphere , all charges at surface: surface density:   Q/( 4  R 2 ) [C/m 2 ] Rotating charges will establish a “surface current”  P P z P Y X Z   r O Surface current density j’ [A/m] will be a function of  j’

5. Analysis and Symmetry (3) dB, dl and e r mutual. perpendic.  P z P Y X Z   r O T Cylinder- symmetry around Z-axis: dB z Z-components only !! Direction of contributions dB: P O dB T  r e r dl Biot & Savart : r P dB

6. Approach (1): a long wire note: r and vector e r !! dB  dl and e r dB  AOP dB Biot & Savart : Z Y X P z I.dl in dz at z dl e r r P y P  A O

7. Approach (2): a volume current dB  dl and e r dB  AOP j : current density [A/m 2 ] dB Biot & Savart : Z Y P j.dA.dl = j.dv dl e r y P dA j A O r P

8. Approach (3): a surface current dB  dl and e r dB  AOP dB Biot & Savart : Z Y P dl e r y P dl j’ A O r P Current strip at surface: j’ : current density[A/m] j’.db.dl = j’.dA dl db

9. Approach (4) Conducting sphere, surface density:  Q/( 4  R 2 ) surface element: dA = ( R.d   R. sin  d   Z d  R   d  R sin  R.d  . R. sin  d  Surface element:

11. Conducting sphere (2) Z R  d  d  R sin  R.d  R. sin  d   Full rotation over 2  R sin  in 2   s. Charge on ring with radius R. sin  and width db is:  . 2  R. sin  db current: dI =  . 2  R. sin  db / ( 2   ) =   R sin  db current density: j’ =   R sin  [A/m]  dA = db.dl dl = R. sin  d   db= Rd 

12. Conducting sphere (3) R  d  d  R sin  R.d  R. sin  d  P z P j’ e r r P dA = R.d  . R. sin  d  j’  e r : => | j’ x e r | = j’.e r = j’ j’ =  R sin   

13. Conducting sphere (4) R  d  d  R sin  P z P j’ e r r P dA = Rd  R. sin  d  j’ =  R sin    Z-components only !! dB z  Cylinder- symmetry: P O dB  R r P z P  e r

14. Conducting sphere (5) R  d  d  R sin  P z P j’ e r r P dA = Rd  .R. sin  d  P O dB dB z   R r P z P  r P 2 = ( R. sin  ) 2 + ( z P - R. cos  ) 2 j’ =  R sin   

15. Conducting sphere (6) R  d  d  R sin  P z P j’ e r r P with r P 2 = ( R. sin  ) 2 + ( z P - R. cos  ) 2 Integration: 0<  <   

16. Conducting sphere (7) this result holds for z P >R ; for -R< z P <R the result is: and for z P <-R :  P P z P Y X Z   R O

17. Conducting sphere (8) inside sphere: constant field !!  P P z P Y X Z   r O result for | z P |>R : result for | z P |<R : B directed along + e z for all points everywhere on Z-axis !!

18. Conducting sphere (9) With surface density:  Q/ (4  R 2 ) : result for | z P | > R : result for | z P | < R :

19. Conducting sphere (10) Plot of B for: Q = 1  0 = 1  = 1 (in SI-units) z P / R

20. Conclusions (1) Homogeneously charged sphere (see other presentation) | z P | < R | z P | > R Conducting sphere | z P | > R | z P | < R

21. Conclusions (2) Plot of B for: Q = 1  0 = 1  = 1 (in SI-units) z P / R Homogeneously charged sphere Conducting sphere The end !!