University Electromagnetism: Electric field calculated from potential distribution

1. The Electric Field of a Dipole © Frits F.M. de Mul

2. The Electric Field of a Dipole Given: Electric Dipole, situated in O, with dipole moment p [Cm] Question: Calculate E- field in arbitrary points P around the dipole P O p

4. Analysis and Symmetry All points at equal r and   are equivalent, even if at different  P O p Coordinate axes: X,Y, Z Y-axis = dipolar axis X Z Y Symmetry: cylindrical around Y-axis Cylindrical coordinates: r,  and   perpendicular’ r  e r e  e r  e 

5. Approach to solution Several approaches possible: 1. E with “Coulomb”; 2. E from V with: E = - grad V Choose option 2. Necessary: calculate V(r,  ) first ! r  P O e r p e  e r X Y Z  e 

6. Intermezzo: V(r,  ) -calculation Calculate : Potential in P : (reference in inf.) If a << r + ,r - , then Contributions to V from +Q and – Q : P -Q + Q p=Qa a r + r -     P -Q + Q r - r +   r a a cos 

7. Intermezzo: V(r,  ) -calculation This approximation is called: Far-field approximation P -Q + Q r - r +   e r r a a cos 

8. Calculation of E (1) Calculate E from V using : E = - grad V In general : X Y Z U V W with s u ,s v and s w line elements

9. Calculation of E (2) here u=r, v=  ,w=  r  P O p e  e r X Y Z  e 

10. Calculation of E (3) here u=r, v=  ,w=  r  P O p e  e r X Y Z  e 

11. Calculation of E (4) r  O p e  e r X Y Z  e  here u=r, v=  ,w= 

12. Calculation of E (5) r  O e r p e  e r X Y Z e   E r E  E

13. Conclusions (1) r  O e r p e  e r X Y Z  e  E r E  E E r

14. Conclusions (2) E - directions in the 4 quadrants:  r e r e  e r e  e r e  e r e  I II III IV I II III IV cos + - - + sin + + - - E r and E  -comp. E

15. Field lines in XY-plane (1) Field lines in XY-plane : r  O p e  e r X Y

16. Field lines in XY-plane (2) Parameter: distance a between -Q and +Q a = 20 a = 10 a = 1 a = 0.1 far field

17. Field lines in XY-plane (3) Parameter: distance a between -Q and +Q a = 20 a = 10 a = 1 a = 0.1 far field

18. Field lines in XY-plane (4) Parameter: distance a between -Q and +Q a = 20 a = 10 a = 1 a = 0.1 far field

19. Field lines in XY-plane (5) Parameter: distance a between -Q and +Q a = 20 a = 10 a = 1 a = 0.1 far field

20. Field lines in XY-plane (6) Parameter: distance a between -Q and +Q a = 20 a = 10 a = 1 a = 0.1 far field