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Hi all,

Consider one has a magnetic dipole, the field given by:

\begin{equation}

\vec{B} = \frac{\mu_0}{4\pi}\left(\frac{3(\vec{m}\cdot\vec{r})\vec{r}}{r^5}-\frac{\vec{m}}{r^3}\right)

\end{equation}

where we can take $$\vec{m} = m\hat{y}$$.

Let us say we have the a magnet vector which is theoretically somewhere in the dipole field. Is it possible to obtain the location where that magnet vector occurs?

To simplify things a bit, I looked at the fieldlines of a dipole, sliced through the XY plane:

If we have a magnet vector of B = a[1,1,0], the vector would lie somewhere along the line y = 2x (just a really rough approximation to get my point across). The magnitude of the vector should correspond to two points on this line due to the symmetry. However, the dipole equation becomes

\begin{equation}

a[1,1,0]^\intercal = \frac{\mu_0m}{4\pi}\left(\frac{3y\vec{r}}{r^5}-\frac{\hat{y}}{r^3}\right)

\end{equation}

Which remains quite a tough equation to solve.

What am I missing in my line of reasoning?

Consider one has a magnetic dipole, the field given by:

\begin{equation}

\vec{B} = \frac{\mu_0}{4\pi}\left(\frac{3(\vec{m}\cdot\vec{r})\vec{r}}{r^5}-\frac{\vec{m}}{r^3}\right)

\end{equation}

where we can take $$\vec{m} = m\hat{y}$$.

Let us say we have the a magnet vector which is theoretically somewhere in the dipole field. Is it possible to obtain the location where that magnet vector occurs?

To simplify things a bit, I looked at the fieldlines of a dipole, sliced through the XY plane:

If we have a magnet vector of B = a[1,1,0], the vector would lie somewhere along the line y = 2x (just a really rough approximation to get my point across). The magnitude of the vector should correspond to two points on this line due to the symmetry. However, the dipole equation becomes

\begin{equation}

a[1,1,0]^\intercal = \frac{\mu_0m}{4\pi}\left(\frac{3y\vec{r}}{r^5}-\frac{\hat{y}}{r^3}\right)

\end{equation}

Which remains quite a tough equation to solve.

What am I missing in my line of reasoning?

Last edited: