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

In summary, the derivation of the electric field at any point around a dipole involves considering 2a as a vector from the negative dipole to the positive one, and using the vector "r" from the midpoint of "a" to the point of observation. By rewriting the vectors in terms of "r" and "a" and using the relation r=(r.r)^1/2, one can obtain terms for both 1/r+ and 1/r-, which can then be substituted into the standard expression for electric potential. Finally, taking the negative gradient of the potential in cylindrical polars will give the electric field in the requested terms of charge, charge separation, angle, and the constant k.
  • #1
Ashu2912
107
1
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!
 
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  • #2
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.
 
  • #3
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.
 

What is an electric dipole?

An electric dipole is a pair of equal and opposite charges separated by a small distance. This results in a non-uniform distribution of electric charge, creating an electric field.

How is the electric field of a dipole calculated?

The electric field of a dipole is calculated using the equation E = (1/4πε0) * (p/r3) * (3cosθr - p), where E is the electric field, ε0 is the permittivity of free space, p is the dipole moment, r is the distance from the dipole, and θ is the angle between the dipole moment and the direction of the electric field.

What is the direction of the electric field of a dipole?

The direction of the electric field of a dipole is always from the positive charge towards the negative charge.

How does the distance from a dipole affect the strength of the electric field?

The strength of the electric field of a dipole decreases as the distance from the dipole increases. This is because the inverse-square law states that the strength of the electric field is inversely proportional to the square of the distance from the source.

What is the significance of the electric dipole moment?

The electric dipole moment is a measure of the separation of the positive and negative charges in a dipole. It is a useful concept in electromagnetism and is used to calculate the strength of the electric field and the potential energy of a dipole in an electric field.

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