Background: I would like to model the Earth's magnetic field with the magnetic dipole approximation. This is a low-level approximation with the following magnetic vector field in spherical coordinates relative to the geomagnetic north pole.
$$\vec{B}_m = \begin{bmatrix} -2 B_0 \left(\frac{R_E}{r}\right)^3 \cos \theta \\ - B_0 \left(\frac{R_E}{r}\right)^3 \sin \theta \\ 0 \end{bmatrix}$$ where $\theta$ is defined as the colatitude from the geomagnetic north pole, and $R_E$ is the mean Earth radius.
I would like to find a representation of this in the true geographic coordinate system.
Progress: Obviously, we can convert to a cartesian system oriented relative to the geomagnetic north pole through the following: $$ \begin{bmatrix} x \\ y \\ z \end{bmatrix}_m = \begin{bmatrix} r \sin \theta \cos \phi \\ r \sin \theta \sin \phi \\ r \cos \theta \end{bmatrix}_m $$ Additionally, after some geometry, we can obtain: $$\begin{bmatrix} x \\ y \\ z \end{bmatrix}_t = \begin{bmatrix} \cos \alpha \cos \beta & -\cos \beta & \sin \alpha \cos \beta \\ \cos \alpha \sin \beta & \cos \beta & \sin \alpha \sin \beta \\ -\sin \alpha & 0 & \cos \alpha \end{bmatrix} \cdot \begin{bmatrix} x \\ y \\ z \end{bmatrix}_m $$ where $\alpha$ is the azimuthal angle rotation, and $\beta$ is the angle between $y_t$ and $y_m$ ($y_m$ remains in the $x_t, y_t$ plane after rotation). More specifically, since the geomagnetic north pole is located at roughly 80.7°N, 72.7°W → $\alpha = 9.3°$, $\beta = -72.7°$.
I'd appreciate any help you might have.