# Calculating distance (from magnet) to an arbitrary point in 3D space

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• egoodchild
In summary, the individual is an engineer seeking assistance with a project involving calculating the distance from a permanent magnet to an arbitrary point in 3D space. They have identified the use of the magnetic dipole equation and need help in deriving it in vector form. The responder suggests taking the magnitude of both sides of the equation and solving for the distance 'r'. However, they also point out that because the field does not have spherical symmetry, there may be multiple locations with the same magnitude of B, making it difficult to determine a unique distance. The individual acknowledges this but mentions that for their specific application, the field is mostly spherical and therefore they believe it is still possible to calculate the distance.
egoodchild
TL;DR Summary
Looking to learn how to calculate the distance from the center of a magnetic dipole to an arbitrary point in 3D space assuming we know the B-field at that arbitrary point as well as the characteristics of the dipole in question.
So I'm looking for some advice on a problem that I am trying to solve for a project. I'm an engineer but my background in vector mathematics is very minimal so I'm looking for some assistance.

Lets assume we have a permanent magnet. We know the characteristics of this magnet, such as it's dimensions and residual flux density, etc. Now let's say we pick some arbitrary point away from this magnet in 3D space and we define the distance from that point to the center of the magnet as 'r'. Additionally we measure the B-field at this arbitrary point and know the field strength in all 3 dimensions.

I would like to be able to calculate the distance 'r' from the center of magnet to the arbitrary point in meters.

Now from what I have gathered so far I think the way to do this is by using the equation for a magnetic dipole:

Above image from K&J Magnetics

Where m is:

B is known, m is known and we are looking to calculate 'r'.

Now where I get a bit lost is finding the reciprocal of this equation in vector form. As mentioned my end goal is an equation that has an input of B and output of r.

Any help on how to derive this or a resource to take a look at to help would be great.

You don't need to get a reciprocal in vector form. Just take the magnitude of both sides. Then you have
$$\|B\| =B = \frac{\mu_0}{4\pi} \frac{\|3\hat r(\hat r\cdot \vec m) - \vec m\|}{r^3} =\frac{\mu_0}{4\pi} \frac{\|(3 m \cos\theta)\hat r - \vec m\|}{r^3}$$
where ##\theta## is the angle between ##\hat r## (the unit direction vector pointing at the point you're interested in) and ##\vec m##.

The symbols ##\|...\|## indicate taking the magnitude of a vector.

This is a scalar equation that you can solve for ##r## in the usual way. The only vector bit is working out the vector ##(3 m \cos\theta)\hat r - \vec m## and getting its magnitude. You can do that with a diagram, or by calculation if you have the components.

Thanks so much for the reply! Solving a scaler equation in the usual way is where I'm stuck. I don't know how to go about solving a scaler equation. This level of math is not by strongest subject.

But you may have the same magnitude of B at different distances. The relationship does not have to be one-to-one. The field does not have spherical symmetry, does it?

You are correct, there are going to be multiple locations where B (and r) will have the same absolute magnitude. For my application this is fine, I'm only interested in knowing the total distance my arbitrary point is away from center of the magnet.

No, B will have the same magnitude for different values of r, not for the same r. So you cannot tell just from the magnitude of B what is the distance. There is no unique answer. It's not just the positon being undetermined but the distance itself. Look at a map of lines of equal B for a magnet. You will see that they are not circles (spheres).

For my application, size/shape of magnet my B-field is mostly spherical. This can be seen in a magnetics simulation:

The contours of the field do change direction as you move around a circle from the origin at a fixed radius, however the absolute value of B will be mostly constant. This can be observed above as the color contour being circular in nature.

So from a basic perspective, I know what I'm trying to achieve is possible (although be it only in constrained conditions).

## 1. How is distance calculated in 3D space?

In 3D space, distance is calculated using the Pythagorean theorem, which states that the distance between two points is equal to the square root of the sum of the squares of the differences between their coordinates. This can be represented as d = √((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2), where (x1, y1, z1) and (x2, y2, z2) are the coordinates of the two points.

## 2. What is the role of a magnet in calculating distance to an arbitrary point in 3D space?

A magnet is used as a reference point in calculating distance to an arbitrary point in 3D space. By knowing the distance between the magnet and the arbitrary point, we can use the Pythagorean theorem to calculate the distance between the arbitrary point and any other point in 3D space.

## 3. How do you determine the coordinates of the magnet in 3D space?

The coordinates of the magnet in 3D space can be determined by using a coordinate measuring device, such as a GPS or a compass. The device will provide the coordinates of the magnet, which can then be used in the distance calculation formula.

## 4. Can distance be calculated in 3D space without a magnet?

Yes, distance can still be calculated in 3D space without a magnet. However, a reference point is needed in order to determine the coordinates of the arbitrary point and calculate the distance. This reference point can be any known point with known coordinates.

## 5. Are there any limitations to calculating distance in 3D space?

One limitation to calculating distance in 3D space is the accuracy of the measuring device used to determine the coordinates of the magnet or reference point. Another limitation is the presence of obstacles or irregularities in the environment, which may affect the accuracy of the distance calculation. Additionally, the Pythagorean theorem assumes a straight line distance between two points, so it may not be accurate for calculating distance in curved spaces.

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