 Let us do a question, suppose there is a person whose head to eye distance is small h and eye to feet distance is capital H. You need to find minimum size of the mirror, you need to find minimum size of the mirror, so that this person can see himself completely. So, the hint is the light from the feet after reflection should reach the eye and light from the head after reflection should reach the eye, middle point will automatically it will reach, not right. So, light from the head if it has to reach eye, it should form an isosceles triangle or not. If you draw normal here, this angle will be equal to that angle, this will be a side bisector this one, this normal. Similarly, light from here has to reach here. So, if this is normal, this angle will be equal to that angle and this angle will be equal to this angle and this angle will be equal to that angle. So, you can make out actually that this angle will be equal to that angle and it will be an isosceles triangle again. So, you do not need this much of the mirror. So, you can do away with this, this much of the mirror is enough, is not it? So, you have to find out what is this length, this length is the minimum size of the mirror that is needed. How much is that? This is h by 2 plus small h by 2. So, this much is h by 2 and this much is small h by 2. So, sum of these two. You got it? Now, we will take up spherical mirrors. The first mirror is concave. Spherical mirror as the name suggests is a part of a sphere. So, getting it? So, every spherical mirror will have a radius of curvature or not. It will have a radius of curvature and it will have a center. So, if you take this, this is a part of a sphere. It is a part of a sphere. I am not drawing an entire sphere, but this is part of a sphere. If you silver this side, it becomes what? It becomes concave. So, you can remember like this. There is a cave like thing and looking inside and suppose this is center. See, center of a sphere for which this mirror is a part of. Can you draw a normal? How many normals you can draw? How many normals can you draw? Infinite. Any line that passes through center is a normal. But there is one special normal which is cutting the mirror into two equal parts. We call it principal axis. So, this is principal axis. This point is pole. It is just the name. It is a pole. Distance, we also represent pole as p. Distance between center and pole is radius of curvature. It is denoted as capital R. The size of the mirror, size of the mirror or lens, whatever it is, this is aperture. When you look the mirror like this, what kind of shape will you see? When you look the mirror like that, how you will see the mirror? What kind of shape it will be? If you look the mirror from this side, how the mirror will look like? This is sideways. When you look sideways, it looks like a line. When you look from the front, how it will look like? It will be circular. It is a part of a sphere which you have cut. When you look from that side, it is just a curved line. When you look from front, it is a circular thing. So, the diameter of the circle is actually the aperture. Now, before I proceed, I will tell you few assumptions that are valid throughout the chapter. Valid as in that are used. Assumption number 1, aperture is very small. Second, angle of incidence and reflection are very small. So, what are the implication of these assumptions? The implication of assumption number 1 is that if you drop a perpendicular from here on the principle axis, you can ignore this distance. You can ignore the distance, this one, the foot of the perpendicular and the pole distance. You can ignore. So, it is so small that it looks as if it is a straight line. You can think like that. Angle of incidence and reflection they are very small. What is the implication of this? The implication is that sign of incident angle or sign of incident or reflected angle. Let us say this is theta is approximately equal to tan of theta and it approximately equal to theta. So, we are going to use these approximation extensively throughout the chapter. At times, it is written that the observer is looking at the mirror or whatever it is very close to normal. It will be written. What does it mean? Angles are less. It automatically means that. So, observer if it is looking very close to normal, all the rays whose angles are more, will not be able to catch them. So, automatically these will be true and otherwise also if it is not written otherwise, then also we will consider these to be true. So, this chapter is full of approximation. So, this one is very common. We will keep on using this. Now, the thing is one thing is very clear that there can be multiple types of concave mirrors made up of different different radiuses of curvature. There can be multiple different types of this thing. So, it is one way to differentiate between the mirror in terms of radius of curvature or in terms of how they behave or how they affect the light. What do you think is more useful? Ultimately, you are using mirrors or lenses to affect the light only. So, usually we define the radius of curvature about mirrors or lenses with respect to what happens to the light which falls on it. So, radius of curvature, pole, aperture and all these, they are the geometrical parameters. They are not optical parameters. Optical parameters should be related to light, what it does to the light. There is one optical parameter which you will see which is common across is focal length. So, write down definition of focal length or let us say focus. We will define first focus. Focus is a point where parallel ray of light that is parallel to the principal axis meet the principal axis. In a simpler words, you can write down focus is a point where parallel rays of light that is parallel to the principal axis, parallel rays of light that are parallel to the principal axis meet the principal axis. Or in a simpler words, you can write down focus is a point where the image of an object which is at infinity falls. Suppose there is an object at infinity, the image will be fall falling at the focus. Because from infinity if lights are coming in, suppose there is an object light comes from that object, two rays are coming from the object. So, where these two rays are meeting, they start from the object and object is at infinity. They are meeting at infinity. So, which ray meet at infinity? Parallel rays. So, you can go by the definition also like that. So, now if you draw a ray diagram, for example, there is a parallel ray coming in like this. Can you draw a ray diagram and show where it meets the principal axis? Show angle of incidence and angle of reflection and show where the focus lie. So, this will be the normal passing through like this. This is angle of incidence and the light will reflect like this and this is angle of reflection. Let us say that is also I because they both are equal. So, I will just write I only. This is the point f that is focus. You draw any parallel ray, it will hit here. But if you take larger aperture, then light may not hit at the same point. Then that approximation for which you can prove that this is a single point is not valid. Angles become larger. So, if larger aperture you take, there will be spherical aberration. You cannot define focus very accurately. But our assumption is aperture is small. This angle will be what? It is 2i. Now, you have to find out this. Suppose this distance is f and this distance is r. What is the relation between f and r? Got it? What is the relation? This is the basic construction which you all the time see. You drop a perpendicular from the point where incident light hits the mirror. Let us say this is height h. Can I say tan of 2i is equal to h by f and this angle is what? I can say tan of i is equal to h by r. This is equal to 2i approximately and this is i because angles are small. So, I can say that 2 times of this is equal to that. So, 2h by r is equal to h by f. This will give you f equals to r by 2. So, in case of mirror f is always equal to r by 2. But same thing is not true for the lens. So, do not blindly use anywhere. So, what is focus? Focus is a point where parallel rays to the principal axis converge. Simple. And there is a principle of retraceability also. You should know right now itself. Write down principle of retraceability. It says that if a light has to go from point a to point b, there can be lenses, mirror and all that can be there to change its path. And suppose somehow it changes the path. There are lenses and mirrors and all that. This is how it reaches b, light. Now, if light has to go from b to a, this is the only path. So, it will retrace its path. If it reflects of normally, it will just go back same path. This is what the principle of retraceability is. So, if parallel rays converge at focus, then if light ray from focus hits the mirror, what will happen to that? It will go parallel. It will retrace its path back. So, any ray which passes through the focus will become parallel after hitting the mirror. And any ray that is parallel to the principal axis will hit the focus. So, with these two knowledge, you can have two rays. And you can, you know find out location of image using ray diagram.