# Calculating magnetic declination for a non-axial dipole field

I'm trying to answer the following question:

Compute the inclination and declination of the magnetic field that would be observed in Boulder, Colorado (40°N, 105°W) if the Earth’s field corresponded to a perfect geocentric dipole whose axis penetrates the Earth’s surface at 80°N, 72°W.

Source: question 4, exercise set 5.10, on p. 360 of Lowrie (2007).

I found the inclination using $\arctan(2\tan(40))$, but am unsure of how to get the declination.

Reference:

Lowrie, W. (2007). Fundamentals of Geophysics, 2nd ed. Cambridge University Press.

## 1 Answer

Inclination is given by

$$I=\arctan(2\cot\theta)$$ where theta is equal to 90° minus the observer's latitude (in the Northern Hemisphere), therefore

$$I=\arctan(2\cot(90-40))=59,21^\circ$$

To get declination the following figure helps:

Declination is the angle between geographic north and magnetic north. (I suppose 80°N, 72°W are the coordinates of the North Geomagnetic Pole.) Let's do the math!

$$D=\arctan\left({{{33}\over{40}}}\right)=39,52^\circ$$