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Along the same geographic latitude does magnetic field change or not, and the same for the gravitational field, does it change along a geographic latitude? Please explain.

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They can both change along any direction.

Here's Earth's magnetic field strength in early 2014, from ESA via NASA.

Earth's magnetic field, early 2014

And here's Earth's gravitational field strength courtesy of NASA's GRACE satellite. Note that the anomalies are quite exaggerated; the gravitational field is very close to uniform.

Earth's gravitational field strength

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