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

B-field of a rotating homogeneously charged 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 sphere (charge Q, radius R), rotating with  rad/sec Available: A charged sphere (charge Q, radius R), rotating with  rad/sec 

B-field of a rotating homogeneously charged 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 homogeneously charged sphere4 Analysis and Symmetry (2) Homogeneously charged sphere, volume charge density:  Q/(4  R 3 /3) [C/m 3 ] Homogeneously charged sphere, volume charge density:  Q/(4  R 3 /3) [C/m 3 ]  P P zPzP Y X Z   r O Rotating charges will establish a “volume current” Volume current density j [A/m 2 ] will be a function of (r,  ) j

B-field of a rotating homogeneously charged 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 Biot & Savart : erer dl rPrP dB dB, dl and e r mutual. perpendic!

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

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

B-field of a rotating homogeneously charged sphere8 Approach (3) Homogeneously charged sphere, volume density:  Q/(4  R 3 /3) Homogeneously charged sphere, volume density:  Q/(4  R 3 /3) volume element: dv = (dr).(r.d  (r.sin  d  volume element: dv = (dr).(r.d  (r.sin  d  Z r   dd dd r.sin  r.d  r.sin  d  dr `volume element:

B-field of a rotating homogeneously charged sphere9 Homogeneously charged sphere (1) Charge .dv  in dv will rotate with  dv = dA.dl dl = r.sin  d  dA=dr.rd  Needed: j, e r, r P Needed: j, e r, r P Z r   dd dd r sin  r.d  r.sin  d  dr  P zPzP j erer rPrP

B-field of a rotating homogeneously charged sphere10 Homogeneously charged sphere (2) dv = dA.dl dl = r.sin  d  dA=dr.rd  Full rotation over 2  r.sin  in 2  s. charge in torus with radius r.sin  and cross section dA is:  2  r.sin  dA current: dI = .2  r.sin  dA / (2  ) =  r sin  dA current density: j =  r sin  m 2 ] r.sin  d  Z r  dd dd r sin  r.d  dr  

B-field of a rotating homogeneously charged sphere11 Homogeneously charged sphere (3) dv = dr. 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  r  dd dd r sin  r.d  r.sin  d  dr P zPzP j erer rPrP  

B-field of a rotating homogeneously charged sphere12 Homogeneously charged sphere (4) Cylinder- symmetry: dv = dr. r.d . r.sin  d  Z-components only !! dB z  r  dd dd r sin  P zPzP j erer rPrP P O dB  r rPrP zPzP  erer  

B-field of a rotating homogeneously charged sphere13 Homogeneously charged sphere (5) r  dd dd r sin  P zPzP j erer rPrP dv = dr. r.d . r.sin  d  dv = dr. r.d . r.sin  d  P O dB dB z   r rPrP zPzP  r P 2 = (r.sin  ) 2 + (z P - r.cos  ) 2  

B-field of a rotating homogeneously charged sphere14 Homogeneously charged 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<r<R ; 0<  <  Integration: 0<r<R ; 0<  <   

B-field of a rotating homogeneously charged sphere15 Homogeneously charged sphere (7) Z  P P zPzP Y X   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 homogeneously charged sphere16 Homogeneously charged sphere (8)  P P zPzP Y X Z   r O result for |z P | > R : result for |z P | < R : B points along +e z for all points everywhere on Z-axis !!

B-field of a rotating homogeneously charged sphere17 Homogeneously charged sphere (9) result for |z P | > R : result for |z P | < R : With volume density:  Q/(4  R 3 /3) :

B-field of a rotating homogeneously charged sphere18 Homogeneously charged 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 homogeneously charged sphere19 Conclusions (1) Conducting sphere (see other presentation) Homogeneously charged sphere |z P | < R |z P | > R |z P | < R

B-field of a rotating homogeneously charged sphere20 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 !!