IIT-JEE Questions: Dispersion and Instruments

Transcript

0:00Introduction

Hello everyone! White light passes through two thin glass prisms placed together. On average it is not bent at all, and yet the colours still spread out. How can light split into colours without being bent? Hold that thought. Today we solve two past IITJEE questions on dispersion, and the second is exactly this puzzle. Let's begin.

0:26Dispersion: the index depends on colour

So, first the idea behind both questions. In the figure, white light, the grey ray, splits in a prism: red on top, then green, then violet, landing at R and V on the right. This happens because the refractive index, n, depends on the wavelength, lambda, the Greek letter on screen. Roughly, n equals a, plus b over lambda squared, where the letters a and b are constants for the glass. This is Cauchy's relation. Okay? Now let me ask you this. Lambda squared sits in the denominator. So when the wavelength gets longer, does n go up, or down? It goes down, yes, exactly! Longer wavelength, smaller index. So violet has the largest index and red the smallest: n V is greater than n R, where the small V and R below are labels for violet and red. That is why violet is deviated the most in the figure. Right?

1:26Critical angle for each colour

Now our second tool, total internal reflection, TIR for short. The source S sits in the denser medium, of index n one, below the rarer one, of index n two; the letters h and r belong to another problem, so ignore them. TIR needs light going from denser to rarer, and the angle of incidence i greater than theta c, the critical angle; theta is the Greek letter, with a small c below. Okay? At exactly theta c, the refracted ray just grazes the surface, the ray marked grazing. The figure gives sine theta c as n two over n one; going from glass to air, n two is one, so sine theta c is one over n. Beyond theta c, the light is totally reflected, as marked. Got it? Now tell me this. Violet has the bigger index, so does it have the bigger critical angle, or the smaller one? The smaller one, very good! Bigger n means a smaller sine theta c, so a smaller critical angle, and violet gets trapped more easily. Many students say bigger index, bigger critical angle; that is the trap. With me?

2:40Thin prisms, and pairs placed base to apex

Now, for a thin prism, with a small angle capital A and near normal incidence, the deviation is n minus one, times capital A, okay? Delta, the Greek letter, is the deviation; for the mean ray we use n Y, the index for yellow. Clear? And n Y is roughly the average of n V and n R; the wavy equals sign means approximately. Right? Violet and red deviate by different amounts, and the gap is the angular dispersion, theta, equal to n V minus n R, times capital A, okay? This theta, with no small c, is not the critical angle. The figure writes it as delta V minus delta R as well, and the omega there is the dispersive power, which we do not need today. With me? Now two thin prisms, base to apex, with angles capital A and capital A prime; the prime, the small dash, marks the second prism. For no dispersion, equate the dispersions: n V minus n R, times capital A, equals n V prime minus n R prime, times capital A prime. That is deviation without dispersion. Okay? And for no deviation, equate the mean deviations instead: n Y minus one, times capital A, equals n Y prime minus one, times capital A prime. That gives dispersion without deviation, our opening puzzle! Got it?

4:08Example 1: green at critical angle, which colours escape

Now example one, from IITJEE two thousand and four. White light, travelling in glass, hits the glass to air surface at such an angle that green light is just totally reflected. Which colours come out into the air? Pause the video and try it first. Done? Okay, let's check. Green is just totally reflected, so the angle of incidence, i, is exactly the critical angle for green. Okay? Now, green sits in the middle of the spectrum. So which side escapes: blue, indigo and violet, or yellow, orange and red? Yellow, orange and red, yes! They have longer wavelengths than green, so a smaller index, and so a larger critical angle. Right? For them, the same angle i is less than their own critical angle; that is the less than sign. So yellow, orange and red refract out into the air. Clear? And blue, indigo and violet have smaller critical angles, so for them i is greater than theta c, and they are totally reflected. If you picked the violet side, that was the trap: bigger index, smaller critical angle. Got it?

5:25Example 2: crown-flint pair, flint angle and net dispersion

Now example two, our opening puzzle, from IITJEE two thousand and one. Here, a thin crown glass prism of angle six degrees, the small circle meaning degrees, has index one point five one for blue and one point four nine for red. It is combined with a thin flint glass prism, one point seven seven for blue and one point seven three for red, so that there is no mean deviation. Find the flint prism's angle and the net dispersion. Pause the video and try it first. Done? Okay, let's check. Blue takes the place of violet here, and the mean index is the average of blue and red. For crown, one point five one plus one point four nine, over two, gives one point five zero. For flint, one point seven seven plus one point seven three, over two, gives one point seven five. Okay? Now, no mean deviation. So which two things do we set equal: the mean deviations, or the dispersions? The mean deviations, yes, exactly! Base to apex, the two prisms bend the mean ray equally, in opposite directions. So one point five zero minus one, times six degrees, equals one point seven five minus one, times capital A prime. Equating the dispersions here is a common trap. Right? Zero point five times six is three degrees, so three degrees equals zero point seven five times capital A prime. The double arrow means which gives: three over zero point seven five gives four, so the flint prism has an angle of four degrees. Clear? Now the net dispersion: each prism gives its index difference times its angle, and the two act against each other. So it is one point five one minus one point four nine, times six degrees, minus one point seven seven minus one point seven three, times four degrees. With me? Crown gives zero point zero two times six, which is zero point one two degrees. Flint gives zero point zero four times four, which is zero point one six degrees. So the net is minus zero point zero four degrees. Okay? The minus sign just tells us that the flint prism wins, so the net dispersion is zero point zero four degrees, in the sense of the flint prism. Puzzle solved: the mean ray is not bent at all, yet the colours spread by zero point zero four degrees. Got it?

7:56Quick revision: critical angle by colour, thin prism pairs

So, quick revision. Longer wavelength, smaller index, so violet has the smaller critical angle. For a thin prism, the mean deviation is n Y minus one, times capital A, and the dispersion is n V minus n R, times capital A, okay? And base to apex, equate dispersions for no dispersion, and mean deviations for no deviation. Now try the practice problems on this page. All the best!