University Electromagnetism: Gauss's Law for cylindrical symmetry (charges and currents)

1. Gauss’ Law for Cylinder Symmetry © Frits F.M. de Mul

2. Gauss’ Law for Cylinder Symmetry Question: Calculate E- field in arbitrary points inside and outside cilinder Two cases: A: homogeneously charged B: charged at surface walls only Available: Cilinder, radius R, infinitely long, carrying charge density  [C/m]

4. Analysis and Symmetry (1) 2. Charge distribution:   C/m] ; homogeneous. 4. Cylinder symmetry: 1. Cylinder : infinitely long, radius R 3. Coordinate axes: Z-axis = symm. axis Y X Z e z e r all points at equal r are equivalent, even if at different z or  

5. Analysis and Symmetry (2) 5. Consequences : a point charge will not move tangentially. Y X Z 4. Cylinder symmetry: e z e r all points at equal r are equivalent, even if at different z or   E directed radially everywhere. all planes z = const. are equivalent.

7. Calculations (1) top and bottom lids do not contribute ( E  dA ) wall contributes: E. 2  rL charge enclosed:  L result: E ( r ) =  2   r  Gauss’ Law: Z E L pill box: radius r>R ; height L

8. Calculations (2) result: E ( r ) =  2   r  but wait !! this holds for r>R ! A: homogeneously charged B: charged at surface only Gauss’ Law: Z L L for r<R :

9. Conclusions (1) field strength dependent of distance to cylinder => no homogeneous field A: homogeneously charged B: charged at surface only for infinite cylinder:

10. Conclusions (2) for infinite cylinder: A: homogeneously charged; B: surface charge only the end R 2 R r 0 E ( r ) A+B A B