Does the magnetic north pole of a magnetic compass change direction through Earths hemispheres?

The easiest way to see the answer to your question is to go to a diagram of the earth's magnetic field that includes the field directional arrows. There is one at ref 1. The arrows show the direction in which magnetic north of a compass will point.

The tricky thing about the terminology is summed up in the first sentence of the article:

"Earth's magnetic field (and the surface magnetic field) is approximately a magnetic dipole, with the magnetic field South pole near the Earth's geographic north pole (see Magnetic North Pole) and the other magnetic field N pole near the Earth's geographic south pole (see Magnetic South Pole)."

In other words, the magnetic field South pole (labeled S in the diagram) IS the Magnetic North Pole, which is located near the geographic North Pole! Once you get used to this, you can use the diagram as follows:

You can see that, at the geographic South Pole, the spherical compass will point upwards--that is, outwards from the surface of the earth and perpendicular to it.

At the equator it will point towards magnetic North Pole, and tangent to the surface of the earth.

At magnetic North Pole (near geographic North Pole) it will point down towards the earth's center, perpendicular to the surface.

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Given that you are asking for both the vertical and horizontal aspects of the needle then first we look at the horizontal. The North seeking end of the magnet points to the appropriate pole of the earth which is a little away from the North pole itself.

For the vertical angles remember that the magnet will orient itself along a magnetic field line.

It doesn't point directly to a pole at all.

If you sketch the field lines around a bar magnet you can see that from the pole they are almost perpendicular to the surface. At the equator the lines are parallel to the surface.

Close to either pole but not at it then there is substantial declination. The magnet dips substantially toward the nearer pole.

Translated At the south pole ( north magnetic pole) the north seeking end of the magnet will point almost vertically up from the earth and strangely, away from the north pole.

At a latitude of around 70 degrees the North seeking end of the magnet would point north of course but it would point above the horizon by quite a bit.

As you get nearer to the equator the magnet would be flat and level.

Then getting closer to the North pole the South seeking end would be the one that points above the horizon.

Until finally at the north pole then the south seeking end would point straight up in the air.

All of the above can be easily estimated by simply drawing a series of concentric circles on a piece of paper, then putting a bar magnet on that paper.

Finally sprinkling iron filings on top to display the direction of field lines.

The angles of those field lines from the circle shows ( approximately) the vertical angle of the magnetic field of the earth at similar points.

Of course in practice there are variations caused by the amount of iron in the crust at different points but these are local variations on a more general theme.

To extend your slightly more simple view. Near the North pole ( south magnetic pole) the north end is attracted so it is pulled downwards.

Near the south pole ( north magnetic pole) the north end is repelled so it is pushed upwards.

I think this is the question that you are asking.