I am following along well several textbooks (Geophysics) that helps me understand the in-depth physics behind the magnetic field of a dipole magnet. I understand that the basic magnetic potential (when observing a dipole magnet where the one of the poles is far enough to have negligible effect on its partner is:
$$W= -\int_r^{\infty}B \space dr = \frac{u_op}{4 \pi r}$$
Then to find the magnetic potential at point P due to both poles are:
$$W(\theta, r)=\frac{u_om\space \cos \theta}{4 \pi r^2}$$
Now to find the magnetic field strength at point P, I know its the vector addition of both $B_r$ and $B_\theta$ (radial and tangential respectively), and to find both I need to differentiate the potential with respect to r and $\theta$
Now here is finally where I get to ask my question. To find $B_r$ is easy enough by:
$$B_r=\frac{\partial W}{\partial r} = -\frac{2 u_o m\space \cos\theta}{4 \pi r^3}$$
But when I take the potential and differentiate with respect to $\theta$ I should get:
$$B_\theta = \frac{\partial W}{\partial \theta} = -\frac{u_o m\space \sin\theta}{4 \pi r^2}$$
but in ALL the textbooks they multiply $\frac{1}{r}$ to get:
$$B_\theta = \frac{1}{r}\frac{\partial W}{\partial \theta} = -\frac{u_o m\space \sin\theta}{4 \pi r^3}$$
Which is needed to find the total magnetic strength field ($B_r + B_\theta$)
Can anyone please explain where they got that extra $\frac{1}{r}?$
I have a feeling it has something to do with a unit vector, but I can't seem to connect the dots.
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