Reflection of Light by Spherical Mirrors

When you look in the mirror have you noticed something interesting about you and your image in the mirror? Let us carry out a small activity. Stand in front of the mirror and move your right hand. Now lift your left hand. Did you notice that in the mirror the right appears left and vice versa? Let us study in detail about a spherical mirror.

Spherical Mirror

Suppose you are sitting at the dining table and you don’t like the food, you start playing with the spoon. You look yourself in the spoon and you notice that you look pretty funny. The moment you get the spoon closer you get a magnified image and when taken far, you see an inverted image.
Do you know what’s really happening?

To understand what is happening lets us talk about the special class of mirrors known as spherical mirrors.

Some Terms of Spherical Mirrors

The radius of Curvature (c): It is the distance between the Pole and the Center of curvature.
Center of Curvature (r): The Center of Curvature of a spherical mirror is the point in the center of the mirror which passes through the curve of the mirror and has the same tangent and curvature at that point.
Aperture: It is a point from which the reflection of light actually happens.
Pole (p): Pole is the midpoint of a mirror. It’s twice the focus.
Focus: It is any point, where light rays parallel to the principal axis, will converge after reflecting from the mirror.
Principal axis: An imaginary line passing through the optical center and the center of curvature of the spherical mirror.
Focal Length: It is on the axis of a mirror where rays of light are parallel to the axis converge after reflection or refraction.

Spherical mirrors are of two types

1.Convex Mirror

2.Concave Mirror

Convex mirror

The convex mirror has a reflective surface that curves outward. These mirrors are “always” form virtual, erect and diminished regardless of the distance between the object and mirror.

Concave Mirror

A concave mirror is also known as the converging mirror as in these type of mirrors light rays converge at a point after they strike and are getting reflecting back from the reflecting surface of the mirror.

When parallel rays of light strike the mirror, they are reflected in a way wherein they spread out or diverge. For this reason, a convex mirror is also a diverging mirror too. If these reflected rays are extended behind the mirror by dotted lines, they meet at a point.
This point is the focus of the convex mirror. The concave mirror is used in the vehicle so that the driver is aware of the vehicle coming from behind. They are also used in street light reflectors.

Reflection of Light

Any polished or shiny surface like that of water can act as a mirror. When a ray of light falls on such smooth or shiny object light from the object bounces back those rays of light to our eyes and this phenomenon is the Reflection of Light.

Laws of Reflection

In the diagram given above, the ray of light that approaches the mirror is the “Incident Ray”. The ray that leaves the mirror is the “Reflected Ray”. At the point of incidence where the incident ray strikes the mirror, a perpendicular line is drawn is the “Normal”. This normal is what divides the incident ray and the reflected ray equally and gives us the “Angle of Incidence” θ_i and “Angle of Reflection” θ_r.

The Laws of Reflection are :

1.The angle of incidence is equal to the angle of reflection.

2.The incident ray, the normal and the reflected ray, all lie in the same plane.

3.The reflected ray and the incident ray are on the opposite sides of the normal.

Types of Reflection

1.Regular Reflection

2.Diffused Reflection

Regular Reflection

It is a mirror-like reflection of rays of light. Here the rays of light which are reflected from a smooth and shiny object such as a mirror, are reflected at a definitive angle and each incident ray which is reflected along with the reflected ray has the same angle to the normal as the incident ray.

Diffused Reflection

This is a non-mirror-like reflection of light. In this type of reflection rays of light hit an irregular object with a rough surface, and reflects back in all directions. Here, the incident ray which is reflected along with reflected ray doesn’t have the same angle to the normal as the incident ray.

Reflection of Rays From a Spherical Mirror

Lab Activity :

Aim: Observing the types of images and measuring the object distance and image distance from the mirror.

Material required: A candle, paper, concave mirror (known focal length), V-stand, measuring tape or meter scale.

Procedure: Place the concave mirror on V-stand, arrange a candle and meter scale as shown in figure-6. Keep the candle at different distances from the mirror (10cm to 80cm) along the axis and by moving the paper (screen) find the position where you get the sharp image on paper. (Take care that flame is above the axis of mirror, paper is below the axis).

Group your observations based on the type of image you see (e.g. Image is bigger and inverted). It is possible you may not get any image at some positions, note down that too!

Since we know the focal point and centre of curvature, we can reclassify our above observations as shown in table-2. What do you infer from this table?.

At this point we suggest that you make one more observation. You have been trying to get the image on a paper when the object is at different positions. At the same time also look into the mirror and note your observations about how the image appears.

Is it inverted or erect, enlarged or diminished?

What do you infer from the table-2? Let us try to draw ray diagrams with concave mirrors and compare them with your inferences. Ray diagrams for concave mirror In activity-5 we saw the ray diagram of sunrays parallel to the concave mirror and the image of the sun was very small at the focal point (See figure-4). Now we shall develop a technique to draw ray diagrams when an object is placed anywhere on the axis of the mirror and validate the above observations. Here we will take at least two rays originating from the same point on the object but with different direction, see how they get reflected from the mirror and find out the point where they meet to form the image. Let us take an example. As shown in the figure-7, assume a concave mirror and a candle placed at some distance along the axis of the mirror.

The diagram shows two rays coming from the tip of the flame (object).   The reflected rays are constructed based on the laws of reflection.They meet at point A. The tip of the flame of the reflected image will be at the point of intersection, A.

Why only at point A?

If we hold the screen at any point before or beyond point A (for example at point B), we see that the rays will meet the screen at different points. Therefore, the image of the tip of the flame will be formed at different points due to these rays. If we draw more rays emanating from the same tip we will see that at point A they will meet but at point B they won’t. So, the image of the tip of the flame will be sharp if we hold the screen at point A and will become blurred (due to overlapping of multiple images) when we move the paper slightly in any direction (forward or backward). Is this not something that you observed during the previous experiment with sun rays? However, it is not going to be easy to evaluate the angle of reflection for any arbitrary ray, every time we will have to find the normal, measure the incident angle and construct a ray with equal angle on the other side. This would be a tedious task, can we find any other simpler method? Yes, there are a few. Based on our discussion so far, we can identify some appropriate rays which we can take as representative rays to find the point ‘A’.

We have seen that all rays that are parallel to the axis get reflected such that they pass through the focal point of the mirror. So, for drawing any diagram the most suitable ray to draw will be the one that comes from the object and goes parallel to the axis of the mirror. The reflected ray will be the line drawn from the point of incidence on the mirror and passes through

the focal point of the mirror. To make it more convenient we will always take rays that come from the tip of the object. See the ray R1 in figure-8. The converse situation of previous one is also true; that is, a ray that passes through the focal point of the mirror will travel parallel to the axis after reflection.

We know that a line drawn from the centre of curvature to the mirror is perpendicular to the tangent at the point where the line meets the curve. So if we draw a ray coming from the tip of the object going through the centre of curvature to meet the mirror, it will get reflected along the same line. This ray is shown as R3 in the figure-10. In general, a ray travelling along normal retraces its path. Along with these three rays ‘the ray which comes from the object and reaches the pole of the mirror’ is also useful in drawing ray diagrams. For this ray, the principal axis is the normal.

we can draw the ray diagram to get the point of intersection A, of any two rays coming from the top of the object and point of intersection B, of any two rays coming from the bottom of the object. We notice that point B isexactly at the same distance from mirror as point A.  Hence the image is vertical and inverted.

Where is the base of the candle expected to be in the image when the candle is placed on the axis of the mirror? Since any ray coming from any point on the axis and travelling along the axis will get reflected on the axis itself, we can conclude that the base of the image is going to be on the axis. Using the knowledge, that if the object is placed vertically on the axis, the image is going to be vertical, all that we need to do, is to draw a perpendicular from point A to the axis. The intersection point is the point where the base of the image of the candle is likely to be formed.

Does this conclusion match with your observations? (Lab Activity) Draw similar diagrams for other cases and verify that they match with your observations.

During the experiment, did you get any positions where you could not get an image on the screen? Consider the case shown in the figure-13. The candle object (O) is placed at a distance less than the focal length of the mirror.

The first ray (R1) will start from tip of the object and run parallel to axis to get reflected so as to pass through the focal point. This one is easy to draw. The second ray that we chose for earlier ray diagrams is the ray coming from the tip of the object and going through the focal point but it is not possible as such a ray will not meet the mirror. So we must use the third ray, a ray coming from the tip of the object and going through the centre of curvature.

We notice that thetwo reflected rays (figure-13) diverge and will not meet. While doing the experiments for a case such as this we were unable to find any place where we get a sharp image on the screen. This ray diagram tells us that since the reflected rays are diverging we will not get an image anywhere. So even if we had moved the screen much away from the mirror, we would not have found an image.

Ray diagrams for convex mirrors

One can draw ray diagrams for a convex mirror too. The ‘easy’ rays that we identified earlier can be used in this case with small modification. Here there are three rules which describe these rays. The procedure for drawing the diagram is similar and is not repeated here.

Rule 1: A ray parallel to the axis, on meeting the convex mirror will get reflected so as to appear as if it is coming from the focal point. See figure-17.

Rule 2: This is converse of Rule 1. A ray travelling in the direction of the focal point, after reflection, will become parallel to the axis. See figure-18.

Rule 3:  A ray travelling in the direction of the centre of curvature will, on reflection, travel in the opposite direction and appears to be coming from the centre of curvature. See figure-19.

Now let us use these rules to show the formation of images when the object is placed at different places infront of the convex mirror.

AB is the object placed at any point on the principal axis infront of the convex mirror.  Using Rule (1) and Rule (3), we get an erected, diminished, and virtual image between P and F on the back side of the mirror.  This image can not be caught on screen and visible only in the mirror.  Hence this is a virtual image. Verify this with an experiment.

Sign convention for the parameters related to the mirror equation

1.All distances should be measured from the pole.

2.The distances measured in the direction of incident light, to be taken positive and those measured in the direction opposite to incident light to be taken negative.

3.Height of object (ho) and height of image (hi) are positive if measured upwards from the axis and negative if measured downwards. Now let us understand magnification, i.e. the relation between the size of the object and the size of the image.

Making of solar cooker

You might have heard the story of Archimedes burning ships using mirrors. Can we at least heat up a vessel using a mirror? Let us try: We have already learnt that a concave mirror focuses parallel sun rays at the focal point of the mirror. So with a small concave mirror we can heat up and burn paper as shown in the figure-23. (Try this with convex mirror also. What do you observed?)

In the same way make a big concave mirror to heat up a vessel. You might have observed the TV dish antenna. Make a wooden/ iron frame in the shape of TV dish. Cut acrylic mirror sheets in to 8 or 12 pieces in the shape of isosceles triangles with a height equal to the radius of your dish antenna.

Your solar heater/cooker is ready. Arrange it so that concave part faces sun. Find its focal point and place a vessel at that point. The vessel gets heated enough to cook rice. In practical applications (like in car-headlights), concave mirrors are of parabolic shape.