Problem : How is $\text{Magnetic Flux Density} = {\text{Magnetic Flux} \over A\sin(θ)}$ (this is the formula my text book has given me)

Where $θ$ is the angle between magnetic flux lines and plane and $A$ is the Area

I know that $\text{Magnetic Flux Density} = {\text{Magnetic Flux} \over A}$ but where does that $\sin(θ)$ come from ? Why is the formula multiplied by $1 \over \sin(θ)$ ?

thanks in advance

  • $\begingroup$ The element of flux is $|A| {\bf n}\cdot {\bf B}$ where $|A|$ is the area and ${\bf n}$ is the unit normal to the surface. This is $|A| \cos \theta$. For some reason your book uses $90-\theta$ hence the $\sin\theta$. $\endgroup$ – mike stone Sep 22 at 21:40
  • $\begingroup$ @mike stone how does the dot product of n.B gives us cos(θ) in the first place? $\endgroup$ – AmirWG Sep 22 at 21:49
  • 1
    $\begingroup$ Is is not just the definition of the dot product? ${\bf n}\cdot {\bf B}= |{\bf n}| |{\bf B}| \cos \theta$? Geometrically its just that it's only when the area is face-on that you see all of it. If it's at an angle you only see a fraction $\cos \theta$ of it. Edge on, there is no area for the field to penentrate. $\endgroup$ – mike stone Sep 22 at 23:32

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