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If you shrunk the "magnetic part" from inside the Earth down to a fridge magnet size, how strong would it be in Gauss?

This is not a home work question. I just watched this video which included the statement: "A fridge magnet is 100x stronger than the Earth's magnetic field". I am trying to answer my friend's question of: "How come a compass usually points North instead of at the nearest fridge magnet?". I got into postulating some relationship based on the inverse cube of distance from the magnets centre and decided there might be a better approach (see above).

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I'm glad you posed your friend's original question of "How come a compass usually points North instead of at the nearest fridge magnet?" because that's a more straightforward and apt question than the question derived from it. The Earth's magnetic field at the surface of the Earth has an average magnitude of about 0.5 Gauss. The magnetic field strength right next to a refrigerator magnet is 100x greater or more.

However, you then have to consider how quickly the magnetic field strength from the magnet decreases as a function of distance from the magnet. If you were to model the magnetic field from a refrigerator magnet as simply the field from a magnetic monopole, the magnetic field strength would drop off as the inverse square of the distance from the magnet. In reality, though, the refrigerator magnet, like any bar magnet, has an 'N' pole and a 'S' pole with opposite polarities, and so if you're at a distance from the bar magnet there is significant 'cancellation' between the 'N' pole and the 'S' pole (visualize the 'N' pole of a magnet as being a source of magnetic flux lines, and the 'S' pole as being a sink of magnetic flux lines) and so the magnetic field strength actually drops off as about the inverse cube of the distance from the magnet.

I'll leave it to you as an exercise to figure out how what the approximate magnetic field strength due to a magnet having a size of 1 inch and a surface field of 500 Gauss is at a distance of, say, 3 feet from the refrigerator. You should be able to see that it doesn't take much distance to reduce even a very large magnetic field at the surface of a bar magnet down to much less than the Earth's magnetic field strength of about 0.5 Gauss.

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Magnetic intensity doesn't increase due to size. A neodymium magnet the size of the Earth with have the same pull on a given object as if it was a dime.

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