# Derivation of ideal magnetic dipole field strength

• RedDeer44
In summary, the book derives the ideal magnetic dipole field strength using the vector potential and simplifies it to have a single component in phi. The phi value is defined when theta is equal to 0 and the moment is (0,0,m).
RedDeer44
For reference, this is from Griffiths, introduction to quantum mechanics electrodynamics, p253-255

When deriving the ideal magnetic dipole field strength, if we put the moment m at origin and make it parallel to the z-axis,
the book went from the vector potential A

$$A= \frac{\mu_0}{4\pi}\frac{\vec{m}\times \hat{r}}{r^2}$$
to
$$A = \frac{\mu_0}{4\pi}\frac{m\sin{\theta}}{r^2}\hat{\phi}$$

Can someone explain how the single ##\phi## component come about? This to me seems to indicate ##r## has non-zero ##\theta## component and zero ##\phi## component. But I thought ##r## is any point?

Also, for a point in spherical coordinates, is ##\phi## value defined when ##\theta = 0##? Or when ##r=0##?

Last edited:
You have ##\vec{m}=(0,0,m)## and thus
$$\vec{m} \times \hat{r} = \begin{pmatrix} 0\\0\\m \end{pmatrix} \times \begin{pmatrix} \cos \varphi \sin \vartheta \\ \sin \varphi \sin \vartheta \\ \cos \vartheta \end{pmatrix} =m \sin \vartheta \begin{pmatrix} -\sin \varphi \\ \cos \varphi \\ 0 \end{pmatrix} = m \sin \vartheta \hat{\varphi}.$$

Dale and RedDeer44

## 1. What is the ideal magnetic dipole field strength?

The ideal magnetic dipole field strength is a theoretical concept that represents the strength of a magnetic dipole, which is a magnet with a north and south pole, in a vacuum. It is used to describe the strength of a magnetic field at a specific distance from the center of the dipole.

## 2. How is the ideal magnetic dipole field strength derived?

The ideal magnetic dipole field strength is derived using the equation B = μ0m/(4πr^3), where B is the magnetic field strength, μ0 is the permeability of free space, m is the magnetic dipole moment, and r is the distance from the center of the dipole. This equation is based on the principles of electromagnetism and can be derived using vector calculus.

## 3. What is the significance of the ideal magnetic dipole field strength?

The ideal magnetic dipole field strength is significant because it allows scientists to calculate the magnetic field strength at any point in space around a magnetic dipole. This is important for understanding the behavior of magnets and their interactions with other objects.

## 4. How does the ideal magnetic dipole field strength differ from the actual field strength of a magnet?

The ideal magnetic dipole field strength is a theoretical concept and does not take into account factors such as the shape and size of the magnet, as well as external influences such as other magnetic fields. The actual field strength of a magnet may vary depending on these factors.

## 5. Can the ideal magnetic dipole field strength be applied to real-world situations?

While the ideal magnetic dipole field strength is a useful theoretical concept, it is not directly applicable to real-world situations. However, it can be used as a starting point for understanding and approximating the behavior of magnetic fields in practical applications.

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