Derivation for Magnetic Field Due to Dipole

[boderam]

I am looking for a reference (derivation) for this exact formula given for "the magnetic field due to a dipole $$\mu$$ fixed at the origin" :

$$B=-\frac{\mu}{R^3}+\frac{3r(\mu\cdot r)}{R^5}$$

I don't really know anything about dipoles or how they are derived (I have only taken lower division E&M) so the $$\mu$$ has no meaning to me. It seems that this formula is derived from a vector potential $$A=\frac{\mu\times r}{r^3}$$ and I know that B = grad x A so it might help more to understand what this potential actually means as well as a good lesson on what is the dipole vector mu. Thanks.

note:i'm having trouble rendering the tex so here is the formula in plain text:

B=-u\R^3 + (3r(u dot r))\R^5 here i use small r for the position vector and R for the length of r and u stands for the greek letter mu for the dipole vector.

 

[jtbell]

I know it's in Griffiths's "Introduction to Electrodynamics." It's probably in other intermediate-level E&M textbooks such as Purcell or Lorrain & Corson.

Griffiths does it as part of the general "multipole expansion" of the magnetic vector potential from a general distribution of current. You might try a Google search for something like "magnetic vector potential multipole expansion."

 

[astrokreng]

Hello,

Thank you for your question. The formula you have provided is the expression for the magnetic field due to a magnetic dipole moment (represented by \mu) fixed at the origin. This formula can be derived from the vector potential A=\frac{\mu\times r}{r^3} using the relationship B = \nabla \times A, which is the curl of the vector potential.

The magnetic dipole moment, \mu, is a measure of the strength and orientation of a magnetic dipole. It is defined as the product of the current I and the area A of the loop of wire that forms the dipole, i.e. \mu = IA. The direction of \mu is perpendicular to the plane of the loop and follows the right-hand rule.

To derive the formula for the magnetic field due to a dipole, we can start with the expression for the vector potential A=\frac{\mu\times r}{r^3}. This expression represents the magnetic field at a point r due to a dipole located at the origin. We can then use the relationship B = \nabla \times A to obtain the expression for the magnetic field B.

Following the derivation process, we arrive at the formula B=-\frac{\mu}{R^3}+\frac{3r(\mu\cdot r)}{R^5}. The first term represents the contribution of the dipole itself to the magnetic field, while the second term represents the contribution of the dipole and its orientation. This second term is often referred to as the dipole term.

In summary, the formula you have provided is the expression for the magnetic field due to a dipole fixed at the origin. It can be derived from the vector potential A=\frac{\mu\times r}{r^3} using the relationship B = \nabla \times A. The dipole moment \mu is a measure of the strength and orientation of the dipole. I hope this helps to clarify the derivation and the meaning of the dipole vector.

Best regards,