University Electromagnetism: Electric field of a hollow sphere with surface charge

1. Electrical Field of a hollow Sphere © Frits F.M. de Mul

2. E -field of a hollow sphere Question: Calculate E -field in arbitrary points outside the sphere Available: A hollow sphere, radius R , with surface charge density [C/m 2 ]

4. Analysis and Symmetry 2. Coordinate axes: Z-axis = polar axis 3. Symmetry: spherical 4. Spherical coordinates: r,  1. Charge distribution:  (surface charge)   C/m 2 ] Z X Y R R R e r e   e  

5. Analysis, field build-up 4. E i,xy , E i,z 5. expect:  E i,xy = 0, to be checked !! 6. E = E z e z only ! 1. XYZ-axes X Y   R Z 2. Point P on Z-axis . P 3. all Q i ’ s at r i ,  i ,  i  contribute E i to E in P r Q i E i e r

6. Approach to solution dQ =  dA r and ( e r .e z ): see next page Z e r  r X Y . P e z  d  d  dQ at dA Distributed charge: dQ dE dA=(R.d  R. sin  d  R. sin 

7. Calculations (1) dA=(R.d  R. sin  d  r and ( e r .e z) : see next page Z X Y e r   . P r dQ at dA dE d  d  e z R. sin 

8. Calculations (2) r 2 = ( R. sin   + (z P - R.cos   ( e r .e z ) = (z P - R.cos  r Z X Y e r   . P r dQ at dA dE d  d  e z R. sin  r  R e r e z z P R.sin  R.sin  R.cos  z P - R.cos 

9. Calculations (3) dA=(R.d  R. sin  d  r 2 = ( R. sin   + (z P - R.cos   ( e r .e z ) = (z P - R.cos  r Z X Y e r   . P r dQ dE d  d  e z R. sin 

10. Calculations (4) result for E in P: z P < R : E = 0 Z X Y e r   . P r dQ dE d  d  e z R. sin  z P > R :

11. Conclusions for homogeneous charge distribution: total charge seems to be in center the end r < R : E = 0 E =0 r > R : E E