Orbital Current and Magnetism

Transcript

0:00Introduction

Hello everyone! Here is something strange. Take one hydrogen atom, with one proton and one electron, and no wire or battery. And yet there is a current inside it, and a magnetic field at its centre of about twelve point five tesla. So where does that current come from? Let us find out today. This idea does appear in JEE questions, and they are quick once you see it. So, let's begin.

0:27The orbiting electron is a current loop

So, picture the electron going round the nucleus, and stand at one point on its orbit. Many times a second, its charge passes you. So what do we call charge passing a point per second? Yes, exactly, it is a current. Here e is the size of the electron's charge, and f is the frequency, the number of rounds per second. So the current, small i, equals e times f. Okay? Now, one round is a distance of two pi r, where r is the radius and pi is the Greek letter you see on screen. At speed v, the rounds per second are v over two pi r. So i equals e times v, over two pi r. With me? Now Bohr's results: speed v is proportional to Z over n, and radius r to n squared over Z, okay? Here n is the principal quantum number and Z the atomic number; the stretched alpha sign means proportional to, and the small two means squared. So how does the current depend on n and Z? Look, e and two pi are constants, so i goes as v over r. Z over n, divided by n squared over Z, gives Z squared over n cubed; the small three means cubed. Clear? Now hydrogen in its ground state, n equal to one. The charge e, one point six times ten to the power minus nineteen, times the frequency, six point five eight times ten to the power fifteen rounds per second, gives about one point zero five milliampere; capital A is the ampere, and m A is the milliampere, okay?

2:08Magnetic moment and the Bohr magneton

Now, a current loop is a tiny magnet. Its magnetic moment, mu, the Greek letter like a u with a tail, is current times area, pi r squared. So put in our current, e times v over two pi r. What do you get? Very good, pi r squared over two pi r leaves r over two. So mu equals e times v times r, over two. Okay? Now, m times v times r, with m the mass of the electron, is the angular momentum, written capital L on screen. So mu equals e over two m, times the angular momentum. Got it? Now Bohr's rule: L equals n times h, over two pi, where h is Planck's constant. Put it in, and mu equals n times e h, over four pi m. Right? That package, e h over four pi m, is the Bohr magneton, written mu with a small B below. Its value is nine point two seven times ten to the power minus twenty four ampere metre squared, A m squared on screen. So in orbit n, the moment is n times the Bohr magneton. Clear? Now a direction question, and a common trap. The little arrow on top means a vector. Does the magnetic moment vector point the same way as the angular momentum vector? No, they point opposite ways. The electron is negative, so its current flows opposite to its motion. So mu vector equals minus e over two m, times L vector. With me?

3:42Magnetic field at the nucleus

Now the field at the nucleus, the centre of our loop. From magnetism, the field at the centre of a circular loop is B equals mu naught times i, over two r. Mu naught, mu with a small zero, is the permeability of free space. Okay? Now, the current goes as Z squared over n cubed. So the field also goes as Z squared over n cubed, right? Be careful, because that is exactly the trap. The r below also changes from orbit to orbit. So B goes as i over r: Z squared over n cubed, divided by n squared over Z, where each slash means divided by. That gives Z cubed over n to the power five. Got it? For hydrogen in the ground state, mu naught is four pi times ten to the power minus seven, the current is one point zero five times ten to the power minus three, and r is the Bohr radius, zero point five two nine times ten to the power minus ten. So B is about twelve point five T, where T stands for tesla. That is the field from our puzzle!

4:48Example 1: orbital current in the second orbit of He+

Now our first example. Find the orbital current in the second orbit of H e plus, the helium ion; the plus up top means it has lost one electron, leaving just one. Pause the video and try it first. Done? Okay, let's check. For helium, Z is two, and the second orbit means n equals two. Okay? Now notice that Z over n is two over two, which is one, so the speed equals the speed in hydrogen's ground state. Same speed, so the same current, one point zero five milliampere, right? Careful, because the speed is the same, but the orbit is not. The radius goes as n squared over Z, four over two, so this orbit is twice as big. So the electron passes a point only half as often. With me? Here i H, with the small H below, is the hydrogen ground-state current. So i over i H equals Z squared over n cubed, four over eight, which is one half. Clear? So the current is one point zero five over two milliampere, about zero point five three milliampere. The wavy equals sign means approximately, and we are done.

6:07Example 2: torque on ground-state hydrogen in a 2 T field

Now a harder one: a hydrogen atom in its ground state is placed in a magnetic field of two tesla, with the normal to the orbit at sixty degrees to the field; the small circle means degrees. Find the torque on the orbiting electron. Pause the video and try it, and here is a hint: find the magnetic moment first. Done? Okay, let's check. Ground state means n equals one, so the moment is one Bohr magneton, nine point two seven times ten to the power minus twenty four ampere metre squared. Okay? Now the angle. Some students say the field makes thirty degrees with the plane of the orbit, so they use thirty. So which angle goes into the torque, sixty or thirty? Sixty degrees is correct. The moment of a current loop points along the normal to its plane, so the angle between the mu vector and the field vector is sixty degrees. Right? From magnetism, the torque on a magnetic moment is mu times B times sine theta, where theta, the Greek letter like an o with a bar, is the angle between them. Torque is written tau, another Greek letter. So tau equals the Bohr magneton, times B, times sine sixty degrees. Got it? So, nine point two seven times ten to the power minus twenty four, times two, times sine sixty, which is zero point eight six six. That gives about one point six times ten to the power minus twenty three newton metre, N m on screen, and that's it!

7:44Quick revision: current, Bohr magneton, field at nucleus

So, quick revision. The current is e times v over two pi r, and goes as Z squared over n cubed. The moment is e over two m, times L, which is n times the Bohr magneton, opposite to the angular momentum. The field at the nucleus is mu naught times i, over two r, and goes as Z cubed over n to the power five. Now try the practice problems on this page, and all the best!