# Magnetic field of a dipole in co ordinate free form

• Sylvester1
In summary, the speaker is trying to calculate the magnetic field at a distance of 1cm from a cylindrical magnet with specific dimensions. They have used various equations and tools, such as MATLAB and WolframAlpha, but are still unable to get a result that aligns with their sensor readings. They suspect that higher orders of the magnetic field may be affecting their calculations and plan to do more measurements and find a calibration formula to improve their results.
Sylvester1
Hello everybody!
I am writing this topi because i got stuck in this!I have a cylindrical magnet with 1,5mm Radius,2mm thickness and Br 1,38 Tesla! I want to calculate the magnetic field in a distance s = [0 0 0.01](in meters) ,that means in 1cm distance while my magnet's position is α = [0 0 0].

the vector form is [x y z] .

using the equation found at http://en.wikipedia.org/wiki/Dipole#Magnitudefor Vector Form i got a result of [0 0 46.5498] Tesla which is impossible!

for the magnetic moment calculation i used the type m =π*Br*d^2*l/(4 μ0) where d is the diameter of my magnet and l the thickness

Can not spot my mistake since i expect to have uT as a result!Any opinion aprreciated!Thanx in advance!

I get m=0.01553 Am^2 (WolframAlpha) and 3.11mT (WolframAlpha), using your formulas and values and assuming the magnet is aligned with the z-axis.

Sylvester1 said:
Can not spot my mistake

We can't spot it either, because we can't see the details of how you actually did your calculation. (hint, hint... )

i used MATLAB to do my calculations!more specific!

MagnetLoc = [0 0 0];

Sensors = [0
0
0.01]

R = magnetLoc - Sensors';

rH = R./norm(R);

theta = acos(R(3)/norm(R));

gamma = atan(R(2)/norm(R));

m1 = 0.0155171294871; % magnitude of magnetic moment m

m = [m1*sin(theta)*cos(gamma) m1*sin(theta)*sin(gamma) m1*cos(theta)];

M = 1.2566370614 * (10^-6); %vacuum perneability (μ0)

A = M/(4*pi*(norm(R)^5));

C = (3 * dot( rH , m) * rH)' - ((norm(R)^2)*m);Field = A*C

OK, I'll move this over to the Matlab forum and maybe someone there can check whether you've implemented the equation properly.

The first part (3 * dot( rH , m) * rH) should use R I think. Otherwise, you have an expression which grows (with R->0) with 1/R^5.

xmm!the equation says to use unit vector of R!if i use R i get even bigger magnitude!

What do you mean with your second recommendation?

Sylvester1 said:
if i use R i get even bigger magnitude!
Now that is very surprising, as |R| < |rH|

What do you mean with your second recommendation?
That was just an explanation why the current calculation has to be wrong.
(3 * dot( rH , m) * rH) does not depend on the magnitude of R. For a constant direction, your expression can be simplified to c/R^5 (neglecting the second term here). That is wrong, a dipole field is proportional to 1/R^3.

ok you are right!when i use R instead of rH i get 0.0031 Tesla!

What i want to do is find the position of a magnet in an area 1cm to 3cm(see it as a cube) away from my sensor! In order to do that i calculate the theoritical value of the magnetic field at a specific position (here at 1cm) and then i use least square algorithm for the relationship : (Btheoritical-Bexperiment) in order to minimize this and find the best solution!the problem is that even 3100uT is not even close to the value which my sensor gives to me at 1cm distance which is approximately 1000uT according to my sensor!

- 1.38 Tesla could be the magnetic field at some specific point, not everywhere in the magnet
- higher moments (quadruple, ...) might influence the value a bit

(Btheoritical-Bexperiment) in order to minimize this and find the best solution!
You can solve this (analytically), there is no need to use a minimization algorithm.

Do you get the same ratio measured/calculated for other distances?

no is completely differenT!it drives me crazy!can not find where i make the mistake..

Sylvester1 said:
xmm!

xmm?

(By the way, we have a rule against using text-message abbreviations here. Now you know.)

Can you give some examples for "different"? It might be possible to use a different formula to fit the data.

ok let me get some measurements again and i will post them as soon as i get them !

ok got the measurements!after calibration and removing the Earth magnetic field i got :

for 1cm : x = 173uT y = -74.4733 z = 2048,1 uT

for 2cm : x = -19.7439 y = 53,2893 z = 402,8459 uT

for 3cm : x = -3,4647 y = 4,2611 z = 141,9412 uT

2048/402.8=5.08 < 23, your field drops slower than a dipole field.
402.8/141.9=2.84 < 1.53=3.375, same here.

The last value fits to the theoretic dipole prediction (~115μT), which is a hint that higher orders of the field could be relevant for 1cm and 2cm.

I neglected the small x- and y-components.

mfb said:
is a hint that higher orders of the field could be relevant for 1cm and 2cm.

so you mean that i need to find scale factors? maybe the calibration is not correct!i mean that how can i calibrate a magnetometer that is still?

Your magnet is not a perfect, point-like dipole. Those deviations can be expressed as quadrupole moment, sextupole moment, ...
Alternatively, measure more points, and find some effective formula as calibration.

ok!thanx for your time!i will come back when i manage to make it work :)!

## What is a dipole moment?

A dipole moment is a measure of the separation of positive and negative charges in a system. It is represented by a vector pointing from the negative to the positive charge, with a magnitude equal to the product of the charge and the distance between them.

## What is a magnetic field?

A magnetic field is a region in space where a magnetic force is exerted on charged particles. It is produced by moving electric charges and is represented by lines of force that show the direction and strength of the field.

## How is the magnetic field of a dipole calculated?

The magnetic field of a dipole is calculated using the formula B = (μ0/4π) * (3(cosθ)^2 -1) * m/r^3, where μ0 is the permeability of free space, θ is the angle between the dipole axis and the point of interest, m is the dipole moment, and r is the distance from the dipole to the point of interest.

## What is coordinate-free form?

Coordinate-free form refers to a mathematical approach that does not rely on a specific coordinate system. In the context of the magnetic field of a dipole, it means that the formula for calculating the field does not depend on a specific set of coordinates, such as Cartesian or spherical coordinates.

## How does the magnetic field of a dipole change with distance?

The magnetic field of a dipole follows an inverse cube relationship with distance, meaning that as the distance from the dipole increases, the strength of the field decreases rapidly. This is because the magnetic field lines spread out as they move away from the dipole, resulting in a decrease in field strength.

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