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I am trying to follow along with a derivation from Cullity, Introduction to magnetic materials 2011. I am getting stuck at a very early stage when computing the components of the magnetic field at a point P due to a magnetic dipole at the origin, where P is at an angle $\theta$ from the direction of the dipole's magnetic moment (in a 2D plane). See diagram:

I compute

$H_r = \frac{m \rm{cos}(\theta)}{r^3}$.

However, they compute

$H_r = \frac{2m \rm{cos}(\theta)}{r^3}$.

See bottom of diagram for the equations in the textbook. Or see the book chapter here

https://onlinelibrary.wiley.com/doi/pdf/10.1002/9780470386323.app1.

Why do the have $2 \times$ what I calculate for $H_r$? Are we not simply taking the cosine of the field, to find the field along from the line from the centre of the dipole to P?

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the magnetic field on the axis of the dipole is $2KM/r^3$ here dipole is $M$ in the general case, which you are talking about, the axial dipole is $Mcos(\theta)$ so the magnetic field on the axis of $Mcos(\theta)$ dipole will be $2KMcos[\theta]/r^3$ just replaced $M$ with $Mcos[\theta]$.

here 2 comes because of the formula of magnetic field on the axis of dipole moment, the derivation of the formula is attached in the image file, I have done the derivation in analogy to the electrostatics. enter image description here hope my answer helped you in any way. thank you, this is my first answer on this app.

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    $\begingroup$ Welcome to PSE. Making a tour in the site you'll realize that answers scanned images of bad handwritten math equations are not acceptable. $\endgroup$
    – Frobenius
    Apr 18, 2022 at 12:01

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