You saw in the section above that the image in a plane mirror has the same size as the object and forms at the same distance behind the mirror as the object is located in front of the mirror. A curved mirror, on the other hand, can form images that may be larger or smaller than the object and may form either in front of the mirror or behind it. Because of the variety of images possible, curved mirrors are exciting optical devices that find many applications.
Since it is much easier to manufacture spherically curved mirrors than other curved shapes, they are more common. Therefore, here, we will mostly study spherically curved mirrors. If the reflecting surface is the outer side of the spherical surface the mirror is called a convex mirror, and when the inside surface is the reflecting surface, it is called the concave mirror as illustrated in Figure 44.11.
Figure44.11.Reflecting sides of a spherical surface for convex and concave mirrors.
Only virtual images form in a convex mirror, while both virtual and real images form in a concave mirror if the object is within half the radius from the vertex. These are illustrated in Figure 44.12. In a concave mirror if the object is further away than half the radius from the vertex a real image is formed as illustrated in Figure 44.13. No real image is possible from a convex mirror.
Figure44.12.Virtual images by convex as well as concave mirror. There are no light rays passing the points on the virtual image since the image is behind the mirror and all the rays are in front of the mirror. However, we will see below that rays coming out after reflection from the mirror are diverging and when we extend them backwards, they meet behind the mirror where the image is.
Figure44.13.Real image by a concave mirror. A convex mirror cannot form a real image. To see a real image you need to place an opaque screen to see the point where the light rays converge.
Subsection44.2.1Image by Convex Mirror
A convex mirror is made by coating a spherical surface so that reflection occurs at the outside. The center of this sphere is called center (C) of the mirror and the tip of the mirror is called its vertex (V). The point midway between the center and the vertex is the focal point (F) of the mirror. The distance VF is called focal length of the mirror.
Figure44.14.
As we saw in the case of image formation by plane mirrors, we can use any two rays from a point object to find the image of that point. It turns out that there are three easy-to-draw special or helper rays in the case of convex mirors shown in Figure 44.15 that simlify our task. You should memorize these special helper rays. You will similar helper rays for concave mirrors and lenses.
Figure44.15.Special rays for a convex mirror. Only a cross-section is shown here. (c) Incident ray 3 is incident at the vertex of the mirror. The reflected ray \(3^\prime\) is reflected symmetrically with respect to the axis.
Special Ray 1: Bounce Back Ray
Special ray in Figure 44.15(a) has incident ray 1 directed towards the center of the sphere, which is the same as the center of the circle of the cross-section view in the figure. Since normal is perpendicular to the tangent to the circle at the point of intersection of the ray with the circle, the angle of incidence is zero. That will mean angle of reflection will also be zero. Hence, this type of ray will just bounce back in the opposite direction of the incident direction as shown by \(1^\prime\) here.
Special Ray 2: Parallel to Optical Axis Ray
Special ray in Figure 44.15(b) hs incident ray 2 parallel to the symmetry axis, called the optical axis. The normal again is the line from center of the sphere to the point where the ray intersects the sphere/circle. The reflected ray is \(2^\prime\text{.}\) When you extend this ray backward behind the convex mirror, it crosses the optical axis at F, the focal point, which is half-way between the vertex and the center. Reflected ray from every parallel ray, e.g., ray 4 in Figure 44.16, when extended will meet the optical axis at F.
Figure44.16.
Reverse of Ray 2 is another special ray: if you take a ray that is pointed towards the focal point then, the reflected ray is parallel to the axis. In Figure 44.17, Ray 5 is coming towards the focal point, which upon reflection goes in the direction parallel to the optical axis.
Figure44.17.
Special Ray 3: Vertex Symmetric Ray
Finally, we have the third type of special rays in Figure 44.15(c). You can immediately see that normal for these rays is the optical axis. Therefore, these rays will just come out symmetrically on the other side of the optical axis. If the incident ray strikes the vertex from above the optical axis, the reflected ray will go below the axis and vice-versa.
Figure44.18.
Subsubsection44.2.1.1Locating Image of an Arbitrary Point Object By a Convex Mirror
Now, we will use special rays in a drawing to locate the image of a point object placed in front of a convex mirror. Figure 44.19 shows the image formation of a point P by using the special rays 1 and 2 described above.
When we extrapolate reflected rays \(1^{\prime}\) and \(2^{\prime}\) they meet at the image point Q. The image is a virtual image since only the extrapolated rays, not the real rays, meet there.
Figure44.19.Using special rays.
When you look into this convex mirror it will appear that a likeness of the object P is located at point Q behind the mirror. Here, two additional rays are shown. Since we know the location of image, these additional rays upon reflection appear to come from the image Q.
Figure44.20.
Unlike the image in a plane mirror, the image behind a convex mirror appears nearer in the mirror than the actual distance of the object from the mirror. That makes convex mirros ideal for situations where you would want to cover a large area such as in the mirrors of a car and the security systems of stores.
Image by a convex mirror is alway behind the mirror. That is, the image is a virtual image, similar to the image by a plane mirror. The orientation is upright in the vertical direction and left-right switched in the horizontal direction.
Subsection44.2.2Image by Concave Mirror
There are three special rays here as well as shown in Figure 44.15. I have also marked the three special points in the figure - center at C, vertex at V, and focal point F which is located half way between C and V. The distance VF is called focal length of the mirror.
Figure44.21.Special rays for a concave mirror. (a) Incident ray 1 passes through the center of the sphere whose surface is coated. The reflected ray \(1^\prime\) bounces back along the same line. (b) Incident ray 2 is parallel to the symmetry axis, called the optical axis. The reflected ray \(2^\prime\) passes through the focal point F. (c) Incident ray 3 is incident at the vertex of the mirror. The reflected ray \(3^\prime\) is reflected symmetrically with respect to the axis.
Image by a concave mirror will behind the mirror if the horizontal distance from V to the object is less than the VF distance. On the otherhand, if the horizontal distance from V to the object is greater than the VF distance, the image will form in front of the mirror. Finally, if if the horizontal distance from V to the object is equal to the VF distance, the image will be at infinity, i.e., you cannot locate the image. We will see these below for a point object.
Subsubsection44.2.2.1Object Within Focal Length of Concave Mirror
I will use special rays 1 and 3. Figure 44.22 shows the construction with these rays from the object point P. The reflected rays \(1^\prime\) and \(3^\prime\) diverge, so they will not cross ever. But, when we extend them back, their extension crosses behind the mirror, where image Q will be located.
Figure44.22.Virtual image of point object within a focal length of a concave mirror.
In Figure 44.22, we see that vertical distance of the image from the axis is more than that of the object. That will mean, the image of ordinary objects will be enlarged by this mirror if placed in this configuration.
Subsubsection44.2.2.2Object Farther than Focal Length of Concave Mirror
Again, I will use special rays 1 and 3. Figure 44.23 shows the construction with these rays from the object point P. The reflected rays \(1^\prime\) and \(3^\prime\) cross at point Q in space which is where image will be located. Since, this is just a space point, you will have to place a screen, such as cardboard, at Q to see the image. This will be a real image.
Figure44.23.Real image of point object within a focal length of a concave mirror.
In Figure 44.23, we see that vertical distance of the image from the axis is less than that of the object. That will mean, the image of ordinary objects will be diminished by this mirror if placed in this configuration. The image will also have an upside down orienatation since image point Q is below the axis while object point Q was above the axis.
Exercises44.2.3Exercises
1.Image of a Point on Axis of a Concave Mirror Within a Focal Point.
Use a ruler and a protractor to draw rays to find images of point object located on the axis of a concave mirror located at a point within the focal length from the vertex.
Hint.
Draw two rays from the point source.
Answer.
See the solution.
Solution.
Drawing two rays from the point source that strike the mirror gives diverging rays \(1'\) and \(2'\text{.}\) When we extend them back, they meet at Q, which is the virtual image of P.
2.Image of a Point on Axis of a Concave Mirror Farther than a Focal Point.
Use a ruler and a protractor to draw rays to find images of point object located on the axis of a concave mirror located at a point farther than the focal length from the vertex.
Hint.
Draw two rays from the point source.
Answer.
See the solution.
Solution.
Drawing two rays from the point source that strike the mirror givesrays \(1'\) and \(2'\text{,}\) which meet at Q. Therefore, Q is a real image of P.
3.Image of a Point on Axis of a Concave Mirror Within a Focal Point.
Use a ruler and a protractor to draw rays to find images of point object located on the axis of a convex mirror located at a point within the focal length from the vertex.
Hint.
Draw two rays from the point source.
Answer.
See the solution.
Solution.
Drawing two rays from the point source that strike the mirror gives diverging rays \(1'\) and \(2'\text{.}\) When we extend them back, they meet at Q, which is the virtual image of P.
4.Image of a Point on Axis of a Convex Mirror Farther than a Focal Point.
Use a ruler and a protractor to draw rays to find images of point object located on the axis of a convex mirror located at a point farther than the focal length from the vertex.
Hint.
Draw two rays from the point source.
Answer.
See the solution.
Solution.
Drawing two rays from the point source that strike the mirror gives diverging rays \(1'\) and \(2'\text{.}\) When we extend them back, they meet at Q, which is the virtual image of P.
