# Derivation for Magnetic Field Due to Dipole

• boderam
In summary, the conversation discusses a search for a reference for a specific formula for the magnetic field due to a dipole fixed at the origin. The formula is derived from a vector potential and the conversation also touches on the concept of a dipole vector and suggests resources for further understanding.

#### 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.

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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."

## 1. What is a dipole?

A dipole is a pair of equal and opposite charges separated by a small distance. It is a fundamental concept in electromagnetism and is used to describe the behavior of magnetic and electric fields.

## 2. How is the magnetic field due to a dipole derived?

The magnetic field due to a dipole can be derived using the Biot-Savart law, which states that the magnetic field at a point is directly proportional to the current flowing through a wire and inversely proportional to the square of the distance from the wire. By considering the magnetic field produced by each element of the dipole, the total magnetic field can be calculated.

## 3. What is the direction of the magnetic field due to a dipole?

The direction of the magnetic field due to a dipole is perpendicular to the plane formed by the dipole and is determined by the right-hand rule. If the fingers of the right hand point in the direction of the current, the thumb will point in the direction of the magnetic field.

## 4. What is the formula for the magnetic field due to a dipole?

The formula for the magnetic field due to a dipole is given by B = μ0/(4πr^3) * (3(m⃗ · r̂)r̂ - m⃗), where μ0 is the permeability of free space, r is the distance from the dipole, m⃗ is the dipole moment, and r̂ is a unit vector in the direction of r.

## 5. What are some real-world applications of the magnetic field due to a dipole?

The magnetic field due to a dipole is used in a variety of technologies, such as electric motors, generators, and magnetic resonance imaging (MRI) machines. It is also important in understanding the Earth's magnetic field and its effects on compasses and navigation systems.