Directions of vec E, vec B and Propagation

Transcript

0:00Introduction

Hello everyone! Let us start with a small puzzle. An electromagnetic wave travels along plus z, and its electric field points along plus x. We know the magnetic field is perpendicular to both of them, so it lies along the y axis. But which way, plus y or minus y? Hmm, both look perpendicular, right? Today we will learn one simple rule that settles every such question. Questions on these directions come up often in JEE, so let's begin.

0:34The right-hand rule

So, look at the figure first. This wave travels along the x axis, the electric field swings along y, and the magnetic field swings along z. The little arrow on top of E and of B means a vector. And the figure says it too: E cross B points along the direction of travel. Okay? Now some notation. The small cap on top of a letter, the hat, means a unit vector: a vector of length one that only gives a direction. So n hat is the unit vector along the direction in which the wave travels. Clear? Here is the rule. E hat cross B hat equals n hat, where the cross is the vector cross product. In words: E, then B, then the direction of travel, in exactly that order, form a right-handed set, just like x, y and z. Right? Let me ask you this quick one from the course. At some instant, the electric field is along plus y and the magnetic field is along plus z. In which direction is the wave travelling? Along plus x, yes, exactly! E hat cross B hat is the plus y direction crossed into the plus z direction, and that gives plus x. Just like our figure. With me? Now, the full vectors. B equals one over c, times n hat cross E, where c is the speed of light. So to find B, cross the direction of travel into E, and then divide by c. That one over c also fixes the size: the magnitude of B is the magnitude of E over c. Got it? Now, many students turn this around and write E equals c times n hat cross B, with n hat first. Looks fine, right? Is it correct? No, no, it is not, and this is the classic trap! Swap the order of a cross product, and the sign flips. The correct line is E equals c times B cross n hat: B first, then n hat, the same cyclic order E, B, n. Write n hat cross B, and your answer points exactly the wrong way. Okay? And one more form. The propagation vector, k with an arrow, equals the wave number k times n hat, and c equals omega over k, where omega is the angular frequency. Then B equals k cross E, over omega. This is handy when the wave equation gives you k directly. Clear?

3:02Topper's trick: the cyclic order

Now a trick that saves a lot of time. Remember the cyclic order: x goes to y, y goes to z, and z goes back to x, as the arrows on screen show. Any two neighbours in this order, crossed, give the next one. Right? So, a wave travels along plus x. If E is along y, then B is along z, simple. But what if E is along z instead? Where does B point now? Along minus y, very good! Check it: z cross minus y gives plus x, the direction of travel. So for travel along plus x, the pair E, B is y and z, or z and minus y. Okay? Same pattern for travel along plus y: the pair is z and x, or x and minus z. Got it? And for travel along plus z: the pair is x and y, or y and minus x. Now remember our puzzle? Travel along plus z, with E along plus x. So B is along plus y, not minus y. Puzzle solved, in one line!

4:11Example 1: wave along -x, B along +z, find direction of E

Now our first example. Read it with me. Here, a wave travels along minus x. At a point, the magnetic field is along plus z. Find the direction of the electric field. Pause the video and try it first. Done? Okay, let's check. From the trick, the pair E, B is y and z. B is along plus z, so E is along plus y, right? Oh no, careful, that pair was for travel along plus x! We write i hat, j hat and k hat for the unit vectors along x, y and z. Here the wave travels along minus x, so n hat is minus i hat. The two vertical lines on screen mean parallel to: E is along B cross n hat. That is k hat cross minus i hat. Now k hat cross i hat is j hat, so with the minus sign, we get minus j hat. With me? Always check with the rule E hat cross B hat. Minus j hat cross k hat gives minus i hat, and that is exactly the direction of travel. Perfect! So the electric field is along minus y. Reversing the direction of travel flipped the answer. Never forget the sign of the travel direction. Clear?

5:30Example 2: find B and its amplitude for E along 3i + 4j

Now a harder one. For a wave in vacuum, E equals three i hat plus four j hat, times sine of, omega t minus k z, in volts per metre. The capital V and the m to the power minus one on screen mean volts per metre. Find B, and its amplitude. Pause the video and try it first. Done? Okay, let's check. First, the direction of travel. There is a minus sign before k z, so the wave travels along minus z, right? No, wait, that is another trap! What matters is whether omega t and k z have the same sign or opposite signs. Here they have opposite signs, so the wave travels along plus z, and n hat is k hat. Okay? Now cross n hat into the electric field. k hat cross i hat is j hat, so k hat cross three i hat gives three j hat. And k hat cross j hat is minus i hat, so k hat cross four j hat gives minus four i hat. Together: three j hat minus four i hat. Right? So B equals one over c, times minus four i hat plus three j hat, times the same sine of, omega t minus k z. Same bracket, so E and B stay in phase. Got it? Now the amplitude, B naught, the B with a small zero. The size of the vector is the square root of three squared plus four squared, which is five. So B naught is five, over three times ten to the power eight. That gives about one point six seven times ten to the power minus eight tesla; the capital T on screen is tesla. If you got this one, very good!

7:16Quick revision: E x B along n, B = (n x E)/c, sign first

So, quick revision. E hat cross B hat equals n hat: E, B and the direction of travel form a right-handed set. B equals one over c times n hat cross E, and E equals c times B cross n hat; keep the order. And always find the direction of travel first, with its sign. Now go and try the practice problems on this page. You can do this. All the best!