University Electromagnetism: Electric field of a line of dipoles ( 3 directions)

1. Electrical dipoles on a thin long line © Frits F.M. de Mul

2. Electrical dipoles on a thin long line Question: Calculate E- field in arbitrary points around the line Available: Thin line, infinitely long, homogeneously filled with dipoles, each with dipole moment p [Cm]

4. Analysis and Symmetry (1) all points at equal r are equivalent, even if at different z or  1. Cylinder : infinitely long and thin 2. Distribution of dipoles: n dipoles / meter; homogeneous; all directions uniform each: dipole moment p [Cm] 3. Coordinate axes: X,Y,Z Z-axis = symm. axis Y X Z e z 4. Cylinder symmetry: e r 

5. Analysis and Symmetry (2) Assume: field components of dipole field at some distance in point P àre known. p Dipole :  r P e r e  E r E 

6. Analysis and Symmetry (3) Possible directions of the dipoles: X Z e z e r  Y X Y X Z e z e r  Y Y X Z e z e r  Z

7. Approach to solution Use symmetry: dp = n p dz n = dipole density (per meter) Z Y P y P Choose : Point P at distance y P from axis. r + r - Dipole elements at + z and - z will contribute symmetrically dp in dz at + z dp in dz at - z +z -z O

8. Calculations (1): Dipoles // X-axis P dp in dz at + z dp in dz at - z Z +z -z y P r + r - O Y X Case 1: all dipoles // +X-axis dE    d   ==> E // -X-axis E result  = 90 0 ==> e  + =e  

9. Calculations (2a): Dipoles // Y-axis resulting vector : ==> E // Y-axis Find directions of the field contributions dE dp in dz at - z P dp in dz at + z Z +z -z O Y X     Case 2: all dipoles // +Y-axis e  + e r+ e  - e r- dE r  cos  dE    sin  dipole + z - z >0 d  r  >0 d  r  dE   >0 d   >0 E result

10. Calculations (2b): Dipoles // Y-axis Y-components: P Z +z -z O Y X dE   d       e  - e  + e r- e r+ d  r  d  r  E result r    dp = n.p.dz ;      dE r+ and dE r- : each: dE  + and dE  - : each:

11. Calculations (3a): Dipoles // Z-axis resulting vector : ==> E // Z-axis P dp in dz at + z dp in dz at - z Z +z -z O Y X     Case 3: all dipoles // +Z-axis e  + e r+ e  - e r- dE r  cos  dE    sin  dipool + - <0 d  r  >0 d  r  dE   >0 d   >0 dE result

12. Calculations (3b) Z-components: P Z +z -z O Y X dE   d       e  - e  + e r- e r+ d  r  d  r  dE result r r      dp = n p dz       dE r+ and dE r- : each: dE  + and dE  - : each:

13. Conclusions dipole direction: dE - contributions:   and r are measured from dipole to P ) the end X Z e z e r  Y X Y Z