E-field of a dipole1 The Electric Field of a Dipole © Frits F.M. de Mul

E-field of a dipole2 The Electric Field of a Dipole Given: Electric Dipole, situated in O, with dipole moment p [Cm] P P O p p Question: Calculate E-field in arbitrary points P around the dipole

E-field of a dipole3 The Electric Field of a Dipole Analysis and symmetry Approach to solution Calculations Conclusions Field lines in XY-plane

E-field of a dipole4 Analysis and Symmetry P P O p p Coordinate axes: X,Y, Z Y-axis = dipolar axis X Z Y Symmetry: cylindrical around Y-axis Cylindrical coordinates: r,  and  perpendicular’ Cylindrical coordinates: r,  and  perpendicular’ r  erer erer ee ee erer erer  ee ee All points at equal r and  are equivalent, even if at different 

E-field of a dipole5 Approach to solution Several approaches possible: 1. E with “Coulomb”; 2. E from V with: E = - grad V r  P P O erer erer p p ee ee erer erer X Y Z  ee ee Choose option 2. Necessary: calculate V(r,  ) first !

E-field of a dipole6 Intermezzo: V(r,  ) -calculation P -Q+Q+Q p=Qa a Calculate: Potential in P : (reference in inf.) r+r+ r-r-   If a << r +,r -, then P -Q+Q+Q r-r- r+r+  r a a cos  Contributions to V from +Q and –Q :

E-field of a dipole7 Intermezzo: V(r,  ) -calculation P -Q+Q+Q r-r- r+r+  erer r a a cos  This approximation is called: Far-field approximation

E-field of a dipole8 Calculation of E (1) Calculate E from V using : E = - grad V X Y Z In general : UV W with s u,s v and s w line elements

E-field of a dipole9 Calculation of E (2) r  P P O p p ee ee erer erer X Y Z  ee ee here u=r, v= ,w= 

E-field of a dipole10 Calculation of E (3) here u=r, v= ,w=  r  P P O p p ee ee erer erer X Y Z  ee ee

E-field of a dipole11 Calculation of E (4) r  O p p ee ee erer erer X Y Z  ee ee here u=r, v= ,w= 

E-field of a dipole12 Calculation of E (5) r  O erer erer p p ee ee erer erer X Y Z ee ee  ErEr ErEr EE EE EE

E-field of a dipole13 Conclusions (1) r  O erer erer p p ee ee erer erer X Y Z  ee ee ErEr ErEr EE EE EE ErEr ErEr

E-field of a dipole14 Conclusions (2)  r erer ee erer ee erer ee erer ee I I II III IV I II III IV cos sin E r and E  -comp. E E E -directions in the 4 quadrants:

E-field of a dipole15 Field lines in XY-plane (1) r  O p p ee ee erer erer X Y Field lines in XY-plane :

E-field of a dipole16 Field lines in XY-plane (2) Parameter: distance a between -Q and +Q a = 20 a = 10 a = 1 a = 0.1 far field

E-field of a dipole17 Field lines in XY-plane (3) Parameter: distance a between -Q and +Q a = 20 a = 10 a = 1 a = 0.1 far field

E-field of a dipole18 Field lines in XY-plane (4) Parameter: distance a between -Q and +Q a = 20 a = 10 a = 1 a = 0.1 far field

E-field of a dipole19 Field lines in XY-plane (5) Parameter: distance a between -Q and +Q a = 20 a = 10 a = 1 a = 0.1 far field

E-field of a dipole20 Field lines in XY-plane (6) Parameter: distance a between -Q and +Q a = 20 a = 10 a = 1 a = 0.1 far field