Deriving Electric Field of a Dipole in Cylindrical Polars: Can You Help?

[Ashu2912]

Can someone help me with the derivation of the electric field at any point around a dipole. I DO NOT want the dipole moment to be resolved into components and then the field found out. I want it in terms of charge q (+q and -q constitute the dipole), a (2a being the charge separation), theta (angle between the line joining the point to the center of the line joining the 2 charges and the line joining the 2 charges) and of course k (= 1/(4*pi*e0)). Please help!

 

[gareth182]

Hi,

Iv'e been struggling on the same kind of question but think I can do it up to the stage you're asking about.

It's useful to consider 2a as a vector going from the -ve dipole to the +ve one. Also take vector "r" to go from the half way point of "a" to the point of observation. You should now be able to rewrite the vectors from each end of the dipole to the point of observation in terms of "r" and "a".

Using the relation r=(r.r)^1/2 get terms for both 1/r+ and 1/r- then sub into the standard expression for electric potential.

Finally, take the negative gradient of the potential in cylindrical polars to get the electric field in the terms that you asked.

 

[gareth182]

Hi,

Iv'e been struggling on the same kind of question but think I can do it up to the stage you're asking about.

It's useful to consider 2a as a vector going from the -ve dipole to the +ve one. Also take vector "r" to go from the half way point of "a" to the point of observation. You should now be able to rewrite the vectors from each end of the dipole to the point of observation in terms of "r" and "a".

Using the relation r=(r.r)^1/2 get terms for both 1/r+ and 1/r- then sub into the standard expression for electric potential.

Finally, take the negative gradient of the potential in cylindrical polars to get the electric field in the terms that you asked.