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When measuring the angle between earths magnetic field and the magnetic field around a wire with a current running through it, does the size of the compass affect the results? If the size of the compass does/doesn’t affect the results, why/why not?

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Assuming that you are not trying to block earth's magnetic field, it will be present when you are measuring the magnetic field around a wire and will have to be taken into account.

Regarding the effect of the compass size.

Let's assume that you are using an old fashioned compass with a magnetized needle.

If we are measuring the magnetic field of the earth, far away from large ferromagnetic objects, the results should not be affected by the size of the needle, since the magnetic field would be reasonably uniform.

It could be a different story when we deal with the current in a wire. The magnetic field around a wire runs in circles and therefore would be changing from point to point (won't be uniform).

Let's assume for a moment, that we have a straight vertical wire, so all magnetic lines (circles) lie in horizontal planes and we could measure the field holding the compass horizontally, which is how it is supposed to be used.

Let's now assume that the needle of the compass is symmetric. Let's also assume that, when we want to determine the direction of the field in a particular point, we place the center of the needle over that point.

For instance, on the picture below, which is a top view of a vertical wire, one of the compasses measures the field in point A on the magnetic line 1, therefore its center is aligned with line 1. Similarly, the second compass measures the field in point B of the magnetic line 2 and therefore its center is aligned with line 2.

enter image description here

Both needles point along the tangents to the corresponding magnetic line circles. Since a needle and a circle are symmetric relative to a point of measurement, the direction of the needle should not depend on the size of the needle or the radius of the circle.

So, for the case of a straight vertical wire, we can state that the measurements of the magnetic field in any horizontal plane won't be affected by the size of the compass.

Now, instead of a straight vertical wire, let's consider a vertical loop and attempt to determine the direction of the magnetic field in the horizontal plane dissecting the loop in the middle. The top view of the magnetic loop lines in the horizontal plane is shown below:

enter image description here

Judging by the density of the lines, we can tell that the field is not uniform.

If we try to determine the direction of the field with a small needle (on the left), the difference in field strength around its ends won't be significant and it'll still point more or less along the tangent to the magnetic line at the center of the needle.

However, if we try a large needle (on the right), its two ends will be exposed to different field strengths and therefore it won't be pointing along the tangent to the magnetic line going through its center.

So, with this simple examples we were able to demonstrate that for uniform and some symmetric fields measured in a certain way, the results of the measurements won't be affected by the size of the compass, while for non-uniform fields the results may be affected by the size of the compass.

Of course, if the size of a compass/needle is small in comparison with the radius of the lines, of a non-uniform magnetic field, at the point of the measurement, we could approximate this field as uniform and therefore the results will be the same for small and large (relatively speaking) compasses.

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  • $\begingroup$ Thank you for this well written explanation. I have marked it as accepted and hope others will be able to learn from it as well. $\endgroup$ – ProWolf Apr 3 '18 at 11:56

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