B-field of a rotating charged conducting sphere1 Magnetic Field of a Rotating Charged Conducting Sphere © Frits F.M. de Mul

B-field of a rotating charged conducting sphere2 Question: Calculate B-field in arbitrary points on the axis of rotation inside and outside the 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 Available: A charged conducting sphere (charge Q, radius R), rotating with  rad/sec 

B-field of a rotating charged conducting sphere3 Calculate B-field in point P inside or outside the sphere P P O  Analysis and Symmetry (1) Assume Z-axis through O and P. zPzP Z Y X Coordinate systems: - X,Y, Z Coordinate systems: - X,Y, Z   r - r, 

B-field of a rotating charged conducting sphere4 Analysis and Symmetry (2) Conducting sphere, all charges at surface: surface density:  Q/(4  R 2 ) [C/m 2 ] Conducting sphere, all charges at surface: surface density:  Q/(4  R 2 ) [C/m 2 ]  P P zPzP Y X Z   r O Rotating charges will establish a “surface current” Surface current density j’ [A/m] will be a function of  j’

B-field of a rotating charged conducting sphere5 Analysis and Symmetry (3)  P zPzP 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 erer dl Biot & Savart : rPrP dB dB, dl and e r mutual. perpendic.

B-field of a rotating charged conducting sphere6 Approach (1): a long wire dB Biot & Savart : note: r and vector e r !! note: r and vector e r !! dB  dl and e r dB  AOP Z Y X P z I.dl in dz at z dl erer rPrP yPyP  A O

B-field of a rotating charged conducting sphere7 Approach (2): a volume current dB Biot & Savart : dB  dl and e r dB  AOP j: current density [A/m 2 ] Z Y P j.dA.dl = j.dv dl erer yPyP dA j A O rPrP

B-field of a rotating charged conducting sphere8 Approach (3): a surface current dB Biot & Savart : dB  dl and e r dB  AOP Z Y P dl erer yPyP j’ A O rPrP Current strip at surface: j’: current density[A/m] j’.db.dl = j’.dA dl db

B-field of a rotating charged conducting sphere9 Approach (4) Z dd R   dd R sin  Conducting sphere, surface density:  Q/(4  R 2 ) Conducting sphere, surface density:  Q/(4  R 2 ) surface element: dA = (R.d  R.sin  d  surface element: dA = (R.d  R.sin  d  R.d . R.sin  d  Surface element:

B-field of a rotating charged conducting sphere10 Conducting sphere (1) dA = db.dl Surface charge .dA  on dA will rotate with  dl = R.sin  d  db= R d  Needed: j, e r, r P Needed: j, e r, r P with j’ in [A/m] R.sin  d  Z R  dd dd R sin  R.d   

B-field of a rotating charged conducting sphere11 Conducting sphere (2) Z R  dd dd R sin  R.d  R.sin  d   dA = db.dl dl = R.sin  d  db= Rd  Full rotation over 2  Rsin  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] 

B-field of a rotating charged conducting sphere12 Conducting sphere (3) R  dd dd R sin  R.d  R.sin  d  P zPzP j’ erer rPrP dA = R.d . R.sin  d  j’  e r : => | j’ x e r | = j’.e r = j’ j’  e r : => | j’ x e r | = j’.e r = j’ j’ =  R sin   

B-field of a rotating charged conducting sphere13 Conducting sphere (4) R  dd dd R sin  P zPzP j’ erer rPrP dA = Rd  R.sin  d  Z-components only !! dB z  Cylinder- symmetry: P O dB  R rPrP zPzP  erer j’ =  R sin   

B-field of a rotating charged conducting sphere14 Conducting sphere (5) R  dd dd R sin  P zPzP j’ erer rPrP dA = Rd .R.sin  d  P O dB dB z   R rPrP zPzP  r P 2 = (R.sin  ) 2 + (z P - R.cos  ) 2 j’ =  R sin   

B-field of a rotating charged conducting sphere15 Conducting sphere (6) R  dd dd R sin  P zPzP j’ erer rPrP with r P 2 = (R.sin  ) 2 + (z P - R.cos  ) 2 Integration: 0<  <  Integration: 0<  <   

B-field of a rotating charged conducting sphere16 Conducting sphere (7)  P P zPzP Y X Z   R O this result holds for z P >R ; for -R<z P <R the result is: and for z P <-R:

B-field of a rotating charged conducting sphere17 Conducting sphere (8) inside sphere: constant field !!  P P zPzP 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 !!

B-field of a rotating charged conducting sphere18 Conducting sphere (9) With surface density:  Q/(4  R 2 ) : result for |z P | > R : result for |z P | < R :

B-field of a rotating charged conducting sphere19 Conducting sphere (10) Plot of B for: Q = 1  0 = 1  = 1 (in SI-units) Plot of B for: Q = 1  0 = 1  = 1 (in SI-units) z P / R

B-field of a rotating charged conducting sphere20 Conclusions (1) Homogeneously charged sphere (see other presentation) |z P | < R |z P | > R Conducting sphere |z P | > R |z P | < R

B-field of a rotating charged conducting sphere21 Conclusions (2) Plot of B for: Q = 1  0 = 1  = 1 (in SI-units) Plot of B for: Q = 1  0 = 1  = 1 (in SI-units) z P / R Homogeneously charged sphere Conducting sphere The end !!