Dispersion

Transcript

0:00Introduction

Hello everyone! Here is a small puzzle. White light goes through a glass prism, and comes out bent, and split into colours. So, can glass bend white light and still keep the colours together? Most students say no. But two prisms of two different glasses can do it, and by the end of this video you will design such a pair yourself. Dispersive power and these prism pairs come up often in JEE questions. So, let's begin.

0:29Why a prism splits white light

So, look at the figure. White light, the grey ray, enters a glass prism, and out come red on top, then green, and violet at the bottom. They land on the black vertical line, where capital R marks red and capital V marks violet. The formulas written beside the figure, we will build one by one as we go. Okay? Here is the reason. The refractive index, n, depends on the wavelength, lambda, the Greek letter on screen. Roughly, n equals small a, plus small b over lambda squared, where small a and small b are constants. This is Cauchy's relation. Got it? Now let me ask you this. Red has a longer wavelength than violet. So in glass, which one has the bigger refractive index? Violet, yes, exactly! For transparent media, n decreases as the wavelength increases. So n V, that is n with a small V below for violet, is greater than n R, for red; that pointed sign means greater than. Right? And a bigger n means a smaller speed, since v, the speed in glass, equals c over n, with c the speed of light in vacuum. So violet is slower in glass, and it is deviated more than red, just as the figure shows. Clear? This splitting of white light into its colours is called dispersion, okay? Now a quick trap. In vacuum, which colour travels the fastest? If you said red, oh no, careful, that is true only in glass! In vacuum, all colours travel at the same speed, c, so there is no dispersion at all. With me?

2:11Thin prism: deviation and dispersive power

Now take a thin prism, with a small prism angle called capital A, and its mean deviation, the deviation of yellow, the middle colour. Delta, the Greek letter, equals n Y minus one, times capital A, with n Y the index for yellow. Okay? And n Y is roughly n V plus n R, over two; that wavy equals sign means approximately. Right? Now the spread. Violet is deviated by delta V, red by delta R, and the angle between them is the angular dispersion, theta. Theta equals delta V minus delta R, which is n V minus n R, times capital A, as in the figure. Got it? Divide the spread by the mean deviation, and you get the dispersive power, omega, the Greek letter like a round w. Omega equals theta over delta, which is n V minus n R, over n Y minus one. Clear? Now look at that fraction carefully. Does omega depend on the prism angle, capital A? No, it does not, very good! Capital A sits in theta and in delta, so it cancels. Omega depends only on the material, the kind of glass, okay?

3:24Two thin prisms, base to apex

Now, back to our puzzle. Take two thin prisms of different glasses, with angles capital A and A prime; the little dash, the prime, marks the second prism. Place them base to apex, so the second one bends the light back. With me? So tell me. We want the light bent, but the colours together. What must cancel? The angular dispersions, yes, exactly! n V minus n R, times capital A, equals n V prime minus n R prime, times A prime. This is deviation without dispersion, an achromatic pair. Right? But the deviations do not cancel here, since the two glasses have different dispersive powers. Net deviation equals n Y minus one, times capital A, minus n Y prime minus one, times A prime. So the light still bends, with no colours. Got it? Now flip it. Make the deviations cancel: n Y minus one, times capital A, equals n Y prime minus one, times A prime. Yellow goes through undeviated, but the colours still spread. This is dispersion without deviation, a direct-vision pair. With me? Its net dispersion is n V minus n R, times capital A, minus n V prime minus n R prime, times A prime. So, for no colours, equate the dispersions; for no bending, equate the deviations. Students often mix these up, okay?

4:53Example 1: dispersive power of crown glass

Now example one, an easy one. For crown glass, n V is one point five two three, n Y is one point five one nine, and n R is one point five one five. Find its dispersive power. Pause the video and try it first. Done? Okay, let's check. The top is n V minus n R, so one point five two three, minus one point five one five, gives zero point zero zero eight. With me? Now the bottom. We just divide by n Y, one point five one nine, right? No, no, careful, that is the trap! The bottom is n Y minus one, so one point five one nine, minus one, gives zero point five one nine. Clear? So omega equals zero point zero zero eight over zero point five one nine, which is about zero point zero one five four. Keep this crown glass in mind for example two, okay?

5:53Example 2: crown-flint achromatic pair, angle and deviation

Now example two, our puzzle with numbers. Here, a prism of angle six degrees, that small raised circle, made of the same crown glass, is combined with a flint glass prism. For flint, n V is one point six six two, n Y is one point six five three, and n R is one point six four four. For deviation without dispersion, find the angle of the flint prism, and the net deviation. Pause the video and try it first. Done? Okay, let's check. Do we equate the deviations, or the angular dispersions? The dispersions, exactly, since we want no spread of colours! For crown, n V minus n R is zero point zero zero eight. For flint, with primes, one point six six two minus one point six four four gives zero point zero one eight. Right? So, zero point zero zero eight times six degrees equals zero point zero one eight times A prime. Got it? Zero point zero zero eight times six is zero point zero four eight. Divide by zero point zero one eight, and A prime is about two point six seven degrees, the angle of the flint prism. Clear? Now, the colours cancel, so the deviations cancel too, and the net deviation is zero, right? No, no, wait, the deviations use n Y minus one, which differs for the two glasses! Net deviation equals zero point five one nine times six degrees, minus zero point six five three times two point six seven degrees. Here zero point six five three is n Y prime minus one, for flint. With me? That is three point one one four degrees, minus one point seven four four degrees, which gives about one point three seven degrees. So the light bends, and the colours stay together. That is our puzzle, solved with real numbers!

7:58Quick revision: dispersive power and two-prism pairs

Okay? So, quick revision. n V is greater than n R, so violet is deviated more, and in vacuum there is no dispersion. For a thin prism, theta equals n V minus n R, times capital A, and omega equals n V minus n R, over n Y minus one. And for two prisms: for no colours, equate the dispersions; for no bending, equate the deviations. Now try the practice problems on this page. All the best!