Maxwell's Equations

Transcript

0:00Introduction

Hello everyone! Here is a small puzzle. Take two constants you already know: epsilon naught from electricity, and mu naught from magnetism. Multiply them, take the square root, and then take one over that. Out comes three times ten to the power eight metres per second, the speed of light! So how can electricity and magnetism know the speed of light? The answer is hidden in four equations, Maxwell's equations, and questions on them come up often in JEE, so let's begin.

0:34In vacuum, integral form: the two Gauss's laws

So, here are Maxwell's four equations, in vacuum, in integral form. Number one is Gauss's law for electricity: the closed integral of E dot d A equals q over epsilon naught, where E is the electric field. The circle on the integral sign means a closed surface, and d A is a small piece of it. The arrows on top mean vectors, the dot is the dot product, and epsilon naught, the Greek letter epsilon with a small zero below it, is the permittivity of free space. Clear? Now, q is the charge inside the closed surface. So if there is no charge inside, what does the left side become? It becomes zero, exactly! No charge inside means zero flux, so charges are the sources of electric field. Got it? Now number two, Gauss's law for magnetism. Here we take the closed integral of B dot d A, with B, the magnetic field, in place of E, and the right side is simply zero, always. Why do you think the magnetic one is always zero? Because there are no magnetic monopoles, no isolated north pole or south pole on its own. So magnetic field lines are closed. Questions on this come up often, so remember it. Okay?

1:55Equations 3 and 4: changing fields

Now the other two equations, which are about closed loops. Number three is Faraday's law: the closed integral of E dot d l equals minus d phi B by d t. Here d l is a small piece of the loop, and capital phi with a small B below it is the magnetic flux through the loop, and d phi B by d t is its rate of change. So tell me, if that flux changes, what appears around the loop? An electric field, correct! So, a changing magnetic field produces an electric field. With me? Number four is the Ampère Maxwell law. The closed integral of B dot d l equals mu naught times i c, plus mu naught epsilon naught, d phi E by d t. Mu naught, the Greek letter mu with a small zero, is the permeability of free space. The small i with a little c below it is the conduction current, and capital phi with a small E below it is the electric flux. Okay? Now a question that catches many students. Suppose there is no current at all through the loop. Can there still be a magnetic field around it? Many students say no, but careful, the answer is yes! If the electric flux is changing, the second term is not zero, even with no current. So a current, or a changing electric field, produces a magnetic field. Right? And this second term, mu naught epsilon naught, d phi E by d t, is exactly what Maxwell added to Ampère's law. It is the displacement current term, and it is non zero only where the electric flux is changing. Clear?

3:34Topper's trick: read the pattern

Now a topper's trick to never mix these up. Equations one and two have d A, so they are about a closed surface, and they tell you the sources: charges for the electric field, and no monopoles for the magnetic one. Okay? Equations three and four have d l, so they are about a closed loop, with the changing flux. Now, in Faraday's law, the loop of E has which flux on the right, electric or magnetic? It has the magnetic flux, phi B, yes! And the loop of B has the electric flux, phi E, so the two fields are crossed. Writing phi E in Faraday's law is a common slip, so watch out. Got it?

4:15Why waves follow

Now this crossed pattern gives something beautiful. Equations three and four show a symmetry: a changing B creates E, and a changing E creates B, okay? Now, in empty space, with no charges and no currents anywhere, can a disturbance in these fields keep going? Yes, it can keep going! The changing E makes a changing B, which makes a changing E again, and so on. Each field regenerates the other, so the disturbance travels through empty space without any charges. With me? And Maxwell showed that its speed in vacuum is one over the square root of mu naught epsilon naught. That is exactly the measured speed of light, the speed from our puzzle! Right? So the conclusion is huge: light is an electromagnetic wave. That is how electricity and magnetism know the speed of light.

5:07Example 1: equations with time derivatives, static case

Clear? Alright, let's do Example one. Which of Maxwell's equations contain a time derivative, and what do they reduce to for static fields? Pause the video and try it first. Done? Okay, let's check. Use our trick: the loop equations carry the changing flux. So the time derivatives sit in equation three, Faraday's law, and equation four, the Ampère Maxwell law. Right? Now, static fields means nothing is changing. So d phi B by d t is zero, and d phi E by d t is zero too. Then what is left of Faraday's law? Only a zero is left on the right side, yes! The closed integral of E dot d l equals zero, which means the electrostatic field is conservative. Got it? And in equation four, only the current term survives: the closed integral of B dot d l equals mu naught times i c. That is the old Ampère's law you already know. Nice, isn't it?

6:12Example 2: speed of light from mu naught and epsilon naught

Now let us solve our opening puzzle, Example two. Mu naught is four pi times ten to the power minus seven tesla metre per ampere; the raised minus one means per. And one over four pi epsilon naught is nine times ten to the power nine newton metre squared per coulomb squared. Find one over the square root of mu naught epsilon naught. Pause the video and try it first. Done? Okay, first we need epsilon naught. The question gives nine times ten to the power nine, so epsilon naught is nine times ten to the power nine, right? Oh no, careful, that is exactly the trap many students fall into! Nine times ten to the power nine is one over four pi epsilon naught. So epsilon naught equals one over, four pi times nine times ten to the power nine. Okay? Now multiply by mu naught. Four pi on top and four pi at the bottom cancel, beautiful! Ten to the power minus seven, over nine times ten to the power nine, gives one over, nine times ten to the power sixteen. Clear? Finally, one over the square root of that is the square root of nine times ten to the power sixteen, which gives three times ten to the power eight metres per second. There it is, the speed of light! And do not forget the square root at the end, okay?

7:38Quick revision: sources, changing fields, light as EM wave

So, quick revision. Gauss's laws: charges are the sources of the electric field, and there are no magnetic monopoles. Faraday and Ampère Maxwell: a changing field produces the other one. And in vacuum, the speed is one over the square root of mu naught epsilon naught, three times ten to the power eight metres per second, so light is an electromagnetic wave. Now go and try the practice problems on this page. All the best!