Plane Angle and Solid Angle

Transcript

0:00Introduction

Hello everyone! So, one small question before we start. Look up at the full Moon. It subtends about half a degree at your eye. That tells you how wide it looks, okay? But tell me: how much of the sky does it actually cover? The Moon is a whole patch of sky, not just a line across it, right? So a flat, plane angle is not enough. We need a new kind of angle: the solid angle. By the end of this video, you will calculate it for the Moon yourself. And remember, radians come up again and again in JEE physics. So, let's begin.

0:37Plane angle: the radian

So, first the plane angle, drawn on a flat page. Look at the figure: the circle on its left side. Take an arc, and divide its length by the radius. That ratio is the angle, in radian. Theta equals arc over radius, that is, s over r, okay? Theta is the Greek letter you see on screen, and the small letters r a d after it stand for radian. Clear? Now think. What if the arc is exactly as long as the radius? Yes, arc over radius is one! So that angle is one radian. See the figure: the coloured arc s is exactly equal to the radius. That is the definition of one radian. Alright, now go all the way round. How many radians in one full circle? The whole arc is now the circumference, two pi r, okay? Divide by r, and you get two pi. So a full circle is two pi radian. As simple as that.

1:36Solid angle: the steradian

Now, for the Moon, we need a sphere. Look at the sphere, on the right side of the figure: from the centre, a cone cuts out a patch on its surface, the shaded part. The solid angle is the area of that patch, divided by r squared. Capital omega, the horseshoe shaped letter on screen, equals area over r squared, and the letters s r stand for steradian. In the figure, this area is labelled capital A, okay? But why r squared, and not just r? Because area is length times length, right? Area over area: a pure ratio, just like the radian. So, if the patch has an area of exactly r squared, the solid angle is one steradian. Same idea as the radian, just one dimension up. Got it? Now you tell me: how many steradians in the whole sphere? The total surface area is four pi r squared. Divide by r squared, and you get four pi! So a full circle is two pi radian, and a full sphere is four pi steradian. Remember this pair, it is really important.

2:43Degrees, minutes, seconds and radians

Now, let's connect radians to the degrees you already know. Half a circle is one hundred eighty degrees, and in radian it is half of two pi, that is, pi. So, pi radian equals one hundred eighty degrees. This one line does every conversion. Divide both sides by pi: one radian equals one hundred eighty by pi degrees, about fifty seven point three degrees. Divide by one hundred eighty instead: one degree is pi by one hundred eighty, about zero point zero one seven four five radian. So one radian is a big angle. Don't think it is tiny! Smaller angles are measured in minutes and seconds. One degree is sixty minutes, that is the single dash on screen, and one minute is sixty seconds, the double dash, okay? So, one minute is one sixtieth of a degree: pi by one hundred eighty, divided by sixty. That is about two point nine one into ten to the power minus four radian. And one second is pi by one hundred eighty, divided by three thousand six hundred: about four point eight five into ten to the power minus six radian. Very small angles, right? Keep these handy.

3:57JEE trap: no dimensions, but still units

Now, the famous JEE trap. Radian is length over length, and steradian is area over area. Both are pure ratios, so they have no dimensions. So, are they even units? Can we just forget about them? Hmm, no, no, careful! They have no dimensions, but they are still units. The unit tells you how the angle was measured. See, one radian and one degree are very different angles, aren't they? The unit matters! And so, the formula s equals r theta works only with theta in radian. If the question gives you degrees, convert first, and only then put it in. Many students lose easy marks here. Got it?

4:41Small angles and cones

Now two tools for real problems. First: for a small angle, the arc and the chord are almost equal. So a distant object of size d, at a distance capital D, subtends an angle theta approximately equal to d over capital D, okay? Second: a cone with half-angle alpha, measured from its axis to its side. It cuts out a solid angle omega equal to two pi into one minus cos alpha. Note this formula down carefully. Here is a topper's habit: check every new formula on a case you already know. What solid angle should a hemisphere have? Half of four pi, that is two pi! Now check: a hemisphere is a cone of half-angle ninety degrees. Cos ninety is zero, so omega is two pi into one minus zero: two pi steradian. Perfect match! And when alpha is small, the formula becomes simpler: omega is approximately pi alpha squared. But alpha must be in radian here, right? Remember our trap. We will need this one for the Moon.

5:46Example 1: angle of a 15 cm arc on a 60 cm circle

Let's do example one. An arc of fifteen centimetres lies on a circle of radius sixty centimetres. Find the angle it subtends at the centre, in radian and in degree. Pause the video, and try it first. Done? Okay, let's check. Theta equals s over r: fifteen over sixty. Do we need metres here? No! Both lengths are in centimetres, so the units cancel. Fifteen over sixty is one fourth: zero point two five radian. Now in degrees. One radian is about fifty seven point three degrees, so multiply: zero point two five into fifty seven point three, about fourteen point three degrees. Two lines, and done. That's it!

6:34Example 2: solid angle of the full Moon

Now, the question from the start. JEE level. The full Moon subtends about half a degree at the eye. Estimate the solid angle it subtends. Pause and try. Hint: treat the Moon as a thin cone from your eye, and use the small cone formula. Alright, so alpha equals zero point five degrees, right? No, no, careful! This is the first trap. Alpha is the half-angle, measured from the axis of the cone. Half a degree is the full angle across the Moon. So, the half-angle alpha is half of zero point five degrees: zero point two five degrees. Second trap: can we put degrees straight into pi alpha squared? Never! Convert first. Zero point two five into pi by one hundred eighty: about four point three six into ten to the power minus three radian. Now the formula: omega is approximately pi alpha squared, that is, pi into four point three six into ten to the power minus three, whole squared. Work it out: about six point zero into ten to the power minus five steradian. So that answers our puzzle: the full Moon covers only about six into ten to the power minus five steradian, compared to four pi for all directions around you. Tiny, isn't it? And see how both traps were waiting for you?

8:02Quick revision: radian, steradian, convert degrees first

So, quick revision. Radian is arc divided by radius, and a full circle is two pi radian. Steradian is area divided by r squared, and a full sphere is four pi steradian. Both have no dimensions, but they are still units, so convert degrees to radian before you use s equals r theta, or pi alpha squared. Now go and try the practice problems on this page. You will find them easy, I promise. All the best!