Could a magnetic field with a direction opposite the Earth's repel the Earth, or would it have to be of equal force?
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3$\begingroup$ Sounds like an experiment you could do at home with a permanent magnet and a kitchen scale. $\endgroup$– Solomon SlowCommented Jul 21, 2015 at 20:52
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3 Answers
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The situation is best described by two magnetic dipoles. Since opposite magnetic pols attract each another and the same magnetic poles repel each other, one could in principle obtain a mere repulsive force. However, if you take two permanent magnets and place them on a table, you will not be able to push one magnet using the second one -- if the friction of the table is "low" and the magnet is free to move in 2D. The reason is that the total energy is minimised, if one of the magnets rotates by 180° -- so that their poles have the same orientation (magnetic moments are parallel). Hence, the configuration is an unstable equilibrium state. It's like trying to place a metal ball on top of a second ball.
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A magnet could repel the Earth and the force between the magnet and the Earth depends on the strength of the magnet seeing as how the Earth field is constant.
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Firstly, we have to clarify what we mean by "a magnetic field with a direction opposite the Earth's". The Earth has dynamic magnetic poles that approximate but do not exactly coincide with the terrestrial poles, which are the poles that define the axis of rotation of the Earth.
Generally, the 'direction' of any magnet is defined by its magnetic field, which is taken (arbitrarily) to travel from the north pole to the south. In three dimensions, the field leaves the magnetic north pole and curves around toward the south pole in a toroidal loop. Of course, the field has no single position or path; rather, it is continuous at all points. Field lines that we draw to represent the field of any magnet are only used to assist our visualisation of the orientation of its field.
So, if we were to position a magnet above Earth's north magnetic pole and orientate it the opposite way (with the same direction but opposite sense), the Earth and the magnet would repel each other with some force proportional to their respective field strengths and inversely proportional to an exponent of their distance.
Alternatively, if we were to instead position our magnet above the equator, it would again be oriented parallel to the Earth's poles, but with the same sense, since the magnetic field would be acting 'downward' toward the south pole. This is how a compass works: the needle of a compass is a magnetic dipole which is repelled by the Earth's magnetic field at one end and attracted to it at the other, automatically aligning itself to be parallel. The end of the needle that points "north" is, in fact, its own south pole.
