The Earth's Magnetism

Transcript

0:00Introduction

Hello everyone! Hold a compass, and the needle turns until its north end points north. But like poles repel, and unlike poles attract, right? So what kind of magnetic pole is sitting up there, near the geographic north? Keep that question in mind. Today it is the Earth's magnetism, so let's begin.

0:22The Earth behaves like a giant bar magnet

So, near its surface, the Earth's field is roughly that of a magnetic dipole, a short bar magnet, at the Earth's centre. The wavy equals sign on screen means roughly equal to. Okay? Now back to our puzzle. Unlike poles attract, and the north end of a compass needle, marked with the letter N, points north. So which pole of the Earth's dipole must face geographic north? Its south pole, yes, exactly! The dipole's south pole, the letter S on screen, points towards geographic north, and it pulls the north end of the needle that way. The names are the opposite way round, so be careful. Right?

1:02Three magnetic elements fix the field

Now, how do we describe the Earth's field at one place? With three numbers, called the magnetic elements. Number one is the declination, written with a capital D, okay? In the top view, one arrow points to geographic north and the other to magnetic north; D is the angle between the geographic and the magnetic meridians. Clear? Number two is the dip, or inclination, written delta, the Greek letter. In the left figure, the arrow B E, that is B with a small E below it, is the Earth's field, in the vertical plane of the magnetic meridian. The dip is the angle between B E and the horizontal. Okay? Now let me ask you this, and the small circle in the question is the degree sign. At which places on the Earth is the dip zero, and where is it ninety degrees? At the magnetic equator the field is horizontal, so the dip is zero degrees. At the magnetic poles the field is vertical, so the dip is ninety degrees. With me? And number three is the horizontal component of the field, B H, with a small H below the B, okay? These three together fix the field at a place. Got it?

2:14Split the Earth's field into BH and BV

Now look at the dashed green arrows in the figure. The horizontal part, B H, equals B E times cos delta, where cos is the cosine. Cos, because delta is measured from the horizontal. Okay? And the vertical part, B V, with a small V below the B, equals B E times sine delta. Careful: measure the angle from the vertical by mistake, and the cos and the sine swap. Right? Now divide the two parts. What do you get for B V over B H? Sine delta over cos delta, so tan delta, yes! Tan delta equals B V over B H; you can see this line in the figure too. Write it down. Clear?

2:58Apparent dip: the needle in another plane

Now, the dip needle is a magnet that swings in a vertical plane. But suppose that plane makes an angle alpha, another Greek letter, with the magnetic meridian. With me? In that plane, the vertical part B V acts fully. But only the share of B H along the plane acts, and that is B H times cos alpha. Okay? So the needle settles at the apparent dip, delta prime, which is delta with a small dash. Tan delta prime equals B V, over B H times cos alpha, which is tan delta over cos alpha. Right? Now turn the plane until alpha is ninety degrees, at right angles to the meridian. What will the needle do? Cos ninety is zero, so tan delta prime goes to infinity; the arrow means tends to, and the sideways eight is infinity. The needle stands vertical, because only B V acts in that plane. So careful: a vertical needle does not always mean a magnetic pole. Got it? Now a lovely result: take readings delta one and delta two in two perpendicular vertical planes; the small one and two just number them. Then cot squared delta one, plus cot squared delta two, equals cot squared delta, the true dip. Cot is the cotangent, one over tan. Clear?

4:19Neutral points and the magnetometer

Now two more tools, and then the examples. At a neutral point, a bar magnet's field exactly cancels B H, the horizontal part. With the north pole pointing north, they lie on the equatorial line, where B H equals mu naught over four pi, times m over r cubed. Mu naught, the Greek mu with a small zero, is the permeability of free space, pi the circle constant, m the magnetic moment, and r the distance from the centre. Okay? Turn the magnet round, north pole pointing south, and the neutral points move to the axis, where the magnet's field has an extra factor of two. So there, mu naught over four pi, times two m over r cubed, equals B H, right? In the oscillation magnetometer, a magnet oscillates with time period T, equal to two pi times the square root of I, over m times B H, okay? The tick-shaped sign is the square root, and I is the moment of inertia. With me? So if you measure T at a place, how does B H depend on it? B H is proportional to one over T squared, and so to f squared, where f is the frequency; the curly sign means proportional to. So more oscillations per minute means a stronger B H, okay?

5:38Example 1: BH and BV for a 0.5 G field at 60 degree dip

Here is Example one. At a place, the Earth's field is zero point five gauss, capital G for gauss, and the dip is sixty degrees. Find B H and B V, the two parts of the field. Pause the video and try it first. Done? Okay, let's check. B H is zero point five times cos sixty, and cos sixty is a half; the cross sign means times. So B H is zero point two five gauss. Clear? And B V is zero point five times sine sixty, which is root three over two. That gives about zero point four three gauss. So B V is bigger than B H here, which makes sense, since the dip is above forty five degrees, right?

6:24Example 2: true dip from apparent dips of 45 and 30 degrees

Now a harder one, Example two. Here, a dip needle shows apparent dips of forty five degrees and thirty degrees, in two perpendicular vertical planes. Find the true dip. Pause the video and try it first. Done? Okay, let's check. Two perpendicular planes, so cot squared delta equals cot squared forty five, plus cot squared thirty. Cot forty five is one, and cot thirty is root three, whose square is three. So one plus three gives four. With me? So cot delta is two. The double arrow means so, and tan delta is a half. Right? Now let me ask you this: is the true dip somewhere between thirty and forty five degrees, the two readings? Actually, it is not, and this is the trap. Tan inverse of a half is about twenty six point six degrees; the small minus one means inverse, not a power. So the true dip is smaller than both readings. Interesting, isn't it? And it has to be so: in tan delta prime equals tan delta over cos alpha, the cos alpha is never more than one. So the apparent dip is always greater than or equal to the true dip; that is what these two signs say. Got it?

7:42Quick revision: BH, BV, apparent dip, cot squared rule

So, quick revision. The dip is measured from the horizontal, so B H is B E cos delta, and B V is B E sine delta. Tan delta is B V over B H, and tan delta prime is tan delta over cos alpha. For two perpendicular planes, cot squared delta one plus cot squared delta two equals cot squared delta. Now go and try the practice problems on this page. All the best!