The Lorentz Force

Transcript

0:00Introduction

Hello everyone! Okay, a small puzzle before we start. Here a charged particle moves for five seconds in a uniform magnetic field, and a force of ten to the power minus twelve newton acts on it all the time. So the field must be doing some work on it, right? Hmm, let us not be so sure. Today we meet the Lorentz force, the force on a moving charge, and by the end you will know the answer. This force starts the whole chapter and comes up often in JEE problems. So, let's begin.

0:34The Lorentz force

So, a charge q moves with velocity v through an electric field E and a magnetic field B, okay? The force on it is F equals q E plus q v cross B, the Lorentz force. The small arrow on top of each letter means a vector, and the cross sign means the cross product of the velocity and the field. With me? Look at the two parts. q E is the electric part, the force from the electric field. And q v cross B is the magnetic part, our focus today. Got it? Now its size. The vertical bars mean the magnitude, so the magnitude of q v cross B equals q v B sine theta. Theta, the Greek letter, is the angle between the velocity and the magnetic field. Right?

1:22Direction of the magnetic force

Now the direction of the magnetic part. It is perpendicular to both v and B; the upside down T symbol on screen means perpendicular. So this force always sits at right angles to the velocity and to the field. Clear? Now tell me, when does this magnetic force vanish completely? Yes, exactly, in two cases. When the charge is at rest, v is zero, so there is no force at all. And when it moves along the field, or exactly opposite to it, theta is zero or one eighty degrees, and the sine is zero, so again there is no force. Got it? And there is one more rule. For a negative charge, like an electron, the force is opposite to v cross B, okay? This is exactly where students lose marks. Let us test it with a course example. Take i hat east, j hat north and k hat up; the little caps on top mean unit vectors. Now an electron moves east, and the magnetic field points vertically up. Which way is the force on it? The velocity is along i hat, the field is along k hat, and i hat cross k hat is minus j hat, which points south for plus q, a positive charge. So the force on our electron points south, right? Oh no, careful, south is the answer only for a positive charge. The electron is negative, so we reverse it, and the force points north. Forget this flip, and you get the opposite answer. With me?

2:58Does the magnetic force do work?

Now the most important result of today. The magnetic force is always perpendicular to the velocity. So how much work does a force do, if it always pushes at right angles to the motion? It does no work at all, absolutely zero. Since F is perpendicular to v, the magnetic force does no work; the double arrow on screen means implies. Right? And no work means no change in kinetic energy. So when only the magnetic field acts, the speed and the kinetic energy stay constant. An electric field, of course, can still change the speed. Okay? So this force only changes the direction of motion. It steers the particle, but never speeds it up or slows it down. Clear? Now back to our puzzle. Ten to the power minus twelve newton for five seconds; capital N is newton and small s is seconds. The work done is zero, because the force is always perpendicular to the velocity, however long it acts. Got it?

3:59Fleming's left-hand rule: the same direction

Now, a hand rule gives the same direction: Fleming's left-hand rule. Use your left hand, not the right. Point your forefinger along the magnetic field. With me? Then point your middle finger along the current, which means the direction of motion of a positive charge. Okay? And your thumb points along the force, the direction q v cross B gives. Right?

4:22Example 1: electron at 30 degrees to a 1 mT field, find F

Alright, example one. An electron moves at two times ten to the power seven metres per second, at thirty degrees to a magnetic field of ten to the power minus three tesla. Here m s to the power minus one means metres per second, the small circle means degrees, and capital T is tesla. Find the force, and pause the video to try it first. Done? Okay, let's check. Many students write e v B straight away and get three point two times ten to the power minus fifteen newton. Is that correct? No, no, wait, the velocity is at thirty degrees to the field, not ninety, so we need the sine. So F equals e v B sine thirty degrees, where e is the size of the electron's charge. Clear? Now put in the numbers: one point six times ten to the power minus nineteen, times two times ten to the power seven, times ten to the power minus three, times zero point five, since sine thirty degrees is one half. Got it? Multiply the numbers first: one point six times two times zero point five gives one point six. The powers of ten give ten to the power minus fifteen. So the force is one point six times ten to the power minus fifteen newton. Okay?

5:44Example 2: v cross B by determinant for a 2 microC charge

Now a harder one, with vectors. Here a charge of two micro coulomb moves with velocity two i hat plus three j hat metres per second, in a magnetic field of four j hat minus k hat tesla. The Greek letter mu before capital C means micro, so the charge is two times ten to the power minus six coulomb. Find the force, and pause the video to try it first. Done? Okay, let's check. We write v cross B as a determinant, between the vertical bars on screen. The top row is i hat, j hat, k hat. Then the components of v, two, three and zero, and the components of B, zero, four and minus one. With me? Now we expand along the top row. Before you write anything, tell me, which sign goes in front of the j hat term? Yes, the middle term always takes a minus sign. For i hat, three times minus one, minus zero times four. For j hat, a minus outside the square brackets, then two times minus one, minus zero times zero. For k hat, two times four, minus three times zero. Right? So v cross B is minus three i hat, plus two j hat, plus eight k hat. Forget that minus sign and you get minus two j hat, and the whole answer goes wrong. Clear? Now F equals q times v cross B, so multiply by the charge, two times ten to the power minus six. Okay? That gives minus six i hat, plus four j hat, plus sixteen k hat, all times ten to the power minus six newton. Got it? And here is a quick check toppers use. The force must be perpendicular to v, so v dot, v cross B, must be zero; the dot is the dot product. Two times minus three, plus three times two, plus zero times eight, gives zero, so the answer passes. Right?

7:49Quick revision: qvB sin theta, no work, flip for electrons

So, quick revision. The Lorentz force is q E plus q v cross B, and the magnetic part has size q v B sine theta. The magnetic force is perpendicular to the velocity, so it does no work: with only a magnetic field, the speed stays constant and only the direction changes. And for a negative charge, reverse v cross B to get the force. Now go and try the practice problems on this page. All the best!