Loops Moving Through a Field

Transcript

0:00Introduction

Hello everyone! Let's begin with a puzzle. Push a square loop of wire at a steady speed into a uniform magnetic field, until it is completely inside. Its sides are still moving through the field. So, does a current flow in the loop now? Many students say yes. By the end of this video, you will know the real answer, and why. This idea comes up often in JEE, so let's begin.

0:26A square loop entering the field

So, take a square loop of side l, small l, with its plane perpendicular to a uniform field, B, the arrow showing it is a vector. The upside-down T sign means perpendicular. The loop moves into the field with speed v. Okay? Now, the loop has four sides. Which of them have an emf right now? The front side is inside the field, and the side, its velocity and the field are all mutually perpendicular. So, just like a moving rod, its emf, the curly E on screen, is B times l times v. Right? The top and bottom sides move along their own length, so they have no emf at all. Clear? And the back side is still outside the field, so it has no emf either. Got it? So the net emf of the loop is B times l times v. Let's check with Faraday's law. If x is the length of the loop inside the field, the flux, capital phi, is B times l times x. So d phi over d t is B times l, times d x over d t, which is B times l times v, the same answer. With me?

1:35The rule: only the boundary matters

So here is the general rule. While a loop is crossing the edge of a uniform field, only the part along the boundary matters. Curly E equals B times v, times the length of the loop cut by the boundary. For our square, that cut length is just l, the side. Okay? Now, back to our puzzle. The loop moves on, and now it is wholly inside the field. Is there a current now? There is no current, and that surprises many students! Inside a uniform field, the flux through the loop stays constant. The double arrow means which gives, so the current, capital I, is zero. The same is true when the loop is wholly outside. Clear? But wait, the sides are still moving through the field, right? Yes, and the front and back sides each have an emf of B times l times v. But around the loop, these two emfs oppose each other and cancel. So each side may have an emf, and yet no current flows. That solves our puzzle. With me?

2:38Crossing a field region of width 3l

Now let's follow one loop all the way through. The field points into the page, in a region of width three times l, and our square loop of side l crosses it at a constant speed v. Okay? First, it enters, for a time l over v, while it moves its own length. Curly E is plus B times l times v. And the direction? The flux into the page is increasing, so by Lenz's law the current opposes this change. It makes a field out of the page inside the loop, so it flows anticlockwise. Right? Next, the loop is wholly inside. For how long? The region is three times l wide, so is it three times l, over v? No, careful, that is the trap, because it is only two times l, over v. The loop is wholly inside from the moment its back side enters until its front side reaches the far edge, and in that time the front side travels just two times l. All that time the flux is constant, so the emf is zero. Clear? Finally, the loop leaves, again for a time l over v. Now the flux into the page is decreasing, so the current tries to keep it up, and flows clockwise. So curly E is minus B times l times v, the opposite sign. Plus, then zero, then minus: remember this pattern. Got it?

4:02Example 1: current and force as a square loop enters a field

Now our first example. Here is a square loop of side ten centimetres, written c m, with a resistance of one ohm, the Greek capital letter omega. It enters a field of zero point five tesla, the capital T, at two metres per second, the m s to the power minus one, with its plane normal to the field. Find the current and the force needed while it is entering, and when it is fully inside. Pause the video and try it first. Done? Okay, let's check. First, the formula needs metres, so ten centimetres becomes zero point one metres. Right? While it enters, only the front side is inside the field, so curly E is B times l times v. Zero point five times zero point one times two gives zero point one volts, capital V for volt. Okay? The current is curly E over R, the resistance, so zero point one over one gives zero point one amperes, capital A for ampere. Clear? Now, the loop moves at a constant speed, so why do we need a force at all? Yes, exactly, it is because of Lenz's law! The current in the front side feels a magnetic force, B times I times l, and it opposes the motion. So we must push with an equal force, zero point five times zero point one times zero point one, which gives five times ten to the power minus three newtons. That is five millinewtons, the m N on screen. Got it? And once the loop is fully inside, the emfs of the front and back sides cancel. So the current is zero, and no force is needed, just like in our puzzle. With me?

5:47Example 2: emf of a circular loop crossing a field boundary

Now a harder one, with a round loop. Here is a circular loop of radius r, moving with velocity v into a uniform field, B, that has a straight boundary. Find the emf when its centre is at a distance, x, from the boundary, and its greatest value. The bars around x mean mod x, the modulus, and mod x less than r means the loop is crossing the boundary. Pause the video and try it first. Done? Okay, let's check. Our rule says B times v, times the length cut by the boundary. So, for a circle, which length is that? It is the chord along the boundary, yes, exactly! Many students take the curved arc inside the field instead, and that is the trap. Right? Now, the centre is a distance, x, from this chord, and the radius r reaches the end of the chord. So by Pythagoras, half the chord is the square root of r squared minus x squared, with the tick-shaped root sign on screen. Okay? The full chord is twice that. So curly E equals two times B times v, times the root of r squared minus x squared. Clear? Now, for which value of x is this emf the greatest? At x equal to zero, that is right! Then the centre is on the boundary, and the chord is the full diameter, two times r. So curly E max, with max written below it, equals two times B times r times v. Got it?

7:22Quick revision: cut length, no current inside, Lenz's law

So, quick revision. First, while a loop crosses the edge of a uniform field, curly E equals B times v, times the length of the loop cut by the boundary. Second, wholly inside or wholly outside a uniform field, the flux is constant, so there is no current. Third, entering and leaving give opposite emfs and opposite currents, by Lenz's law. Now go and try the practice problems on this page. All the best!