4.1 The Concepts of Force and Mass

Chapter 25: THE REFLECTION OF LIGHT: MIRRORSThe Reflection of Light2. The Formation of Images by a Plane Mirror3. Reflection by Spherical Mirrors4. The Formation of Images by Spherical Mirrors5. The Mirror Equation and the Magnification EquationCOLLEGE PHYSICS, Part IIReflection and Refraction

When an incident light beam hits a surface that separates two transparent materials, such as air and glass, the light is being partially reflected and partially transmitted (refracted). (There are other effects, such as absorption, scattering, diffraction, but for now we consider only reflection and transmission.) Usually consideration of just one ray is enough. The directions of the incident, reflected, and refracted rays are usually presented by the angles they make with the normal to the surface, as shown in the lower right part of the Figure.

LAW OF REFLECTION

The incident ray, the reflected ray, and the normal to the surface all lie in the same plane, and the angle of incidence equals the angle of reflection.

This means the incident and reflected rays are reversible.

At large distances from the source, the wavefronts become less and less curved. Waves whose wavefronts are flat surfaces are called plane waves. Naturally, the rays of a plane wave are parallel to each other.Wavefront is a surface that is perpendicular to the direction of propagation of the wave, and all rays have the same phase at the wavefront.

In specular reflection, the reflected rays are parallel to each other.Reflection at a defined angle from a smooth planar surface is called specular reflection. Random reflection from a rough surface is called diffuse reflection.

The image obtained by reflection is not identical to the object:In the mirror, the sense of right and left is reversed.The persons right hand becomes the images left hand.A right handed helix becomes a left handed helix, etc.

Right-handed helixLeft-handed helix

The reflection in a plane (flat) mirror:is upright.is the same size as the object.the image is as far behind the mirror as the object in front of it.

A ray of light from the top of the chess piece reflects from the mirror.To the eye, the ray seems to come from behind the mirror.

Because none of the rays directly comes from the image, it is called a virtual image.Formation of Images by a Plane Mirror

The image distance is equal to the object distance.

andbut

hence

The triangles ABC and CBD are identical (congruent) .

hSo, the object and the image are at equal distances from the surface of the mirror. Using similar simple geometric arguments, we can show that the object and the image have the same height and that the image has the same orientation (is upright).

Conceptual Example 1 Full-Length Versus Half-Length Mirrors

What is the minimum mirror height necessary for her to see her full image?The mirror should be at least half the persons height

Conceptual Example 2 Multiple Reflections

A person is sitting in front of two mirrors that intersect at a right angle.The person sees three images of herself. Why are there three, rather than two, images?Images in mirrors 1 and 2 are created because of simple reflections from the corresponding mirror. Image 3 is created because of two reflections, as shown in part (b), from mirror 1 and then from mirror 2 (right hand side of image 3), and from mirror 2 and then from mirror 1 (left hand side of image 3),

A spherical mirror is a piece of spherical surface that can reflect light. If the inside surface of the spherical surface is polished and works as a mirror, it is a concave mirror. If the outside surface is polished, is it a convex mirror. C is the center of curvature, and R is the radius.

The law of reflection applies, just as it does for a plane mirror.

The principal axis (aka optic axis) of the mirror is a straight line drawn through the center and the midpoint of the mirror, perpendicular to the surface.Spherical Mirrors

A point on the tree lies on the principal axis of the concave mirror.

Rays from that point that are near the principal axis cross the axisat the image point.

After passing the image point, light rays continue traveling and for the viewer they really come from the image. Therefore the image is a real image.

When the object is far away from the mirror, the incident rays are (nearly) parallel to the principal axis. The rays near and parallel to the principal axis are reflected from the concave mirror and converge at the focal point F. Thus, objects far from the concave mirror create images at the focal point. The distance from F to the middle of the mirror is called the focal length f.The focal point of a concave mirror is halfway between the center of curvature of the mirror C and the mirror:

Consider rays coming from a far away object. Suppose the rays are parallel to and near the principal axis. According to the reflection law, the angle of incidence, q, equals the angle of reflection. The angle ACB is also q because the line AC is the transversal of two parallel lines. So, the triangle ACF is an isosceles triangle:

For incident rays close to the principal axis, the angle q is small and

The point B at the center of the mirror is called the vertex.

aFor small a, FB = AFcosa AF

Rays that lie close to the principal axis are called paraxial rays.

Rays that are far from the principal axis do not converge to a single point. The fact that a spherical mirror does not bring all parallel rays to a single point is known as spherical aberration.

Concave parabolic (not spherical) mirrors are able to focus all parallel rays at the focal point irrespective to their proximity to the principal axis. With the aid of such parabolic mirrors the energy of sunlight can be accumulated and used for heating, creating high temperature steam that is used to run the turbines of electric generators. Parabolic mirrors are also used to create parallel beam of light, e.g. in car headlights.

For convex mirrors, the focal length is again one-half of the radius of curvature, and a negative sight is assigned to it. When paraxial light rays that are parallel to the principal axis strike a convex mirror, they diverge from the mirror and appear to come from a point F behind the mirror, which is the focal point of the mirror. The distance from F to the vertex is the focal length f. Convex Mirrors

This ray is initially parallel to the principal axisand passes through the focal point.This ray initially passes through the focal point,then emerges parallel to the principal axis.This ray travels along a line that passes through the center.The Formation of Images by Spherical MirrorsThe image can be constructed using the ray tracing technique. Consider three special rays:Concave Mirrors

Image Formation and the Principle of Reversibility(a) Object between F and the center of curvature C: image is larger than object and inverted.(b) Object beyond the center of curvature: image is smaller than object and inverted.In both cases the image is real.The image can be constructed using two of the three special types of rays, as shown in the diagrams below. If the direction of light is reversed, the light retracts its original path.

When an object is located between the focal point and a concave mirror,an enlarged, upright, and virtual image is produced.

Ray 2 heads towards the focal point, emerging parallel to the principal axis.

Ray 3 travels toward the center of curvature and reflects back on itself.Convex MirrorsThe image is virtual, diminished in size, and upright. Convex mirrors always create virtual images. They provide a wider field of view than plane or concave mirrors. Ray 1 is initially parallel to the principal axis and appears to originate from the focal point.The Mirror Equation and the Magnification Equation

The mirror equation and the magnification equation provide relationships between: Concave Mirrors

The two colored triangles are similar because they have same angles. Therefore:

For paraxial rays, these two colored triangles are also similar. Therefore:

m = magnification

and yield:

Dividing both sides by do we get the mirror equation:The magnification, m, of a mirror is the ratio of the image height to the object height:

Using , we get the magnification equation:

Upright image:di and do have opposite signs, i.e. image is at the opposite side of mirror (virtual image).Inverted image:di and do have same signs, i.e. image is at the same side of mirror (real image).Example: A 2.0 cm high object is placed at do = 7.1 cm from a concave mirror with a radius of curvature of 10.2 cm. Find (a) the location of the image, di, and (b) its size, hi.

The focal distance is

The object is located between the focal point F and the center of curvature C. The image is real and inverted. (a) Use the mirror equation to find di :

di > 0 means the image is real.(b) Use the magnification equation to find hi :

hi < 0 means the image is inverted.Given: ho, do, RFind: di, hi

Example: A 1.2 cm high object is placed at do = 6.0 cm from a concave mirror with a 10.0 cm focal length. Find (a) the location of the image, di, and (b) its height, hi.

The object is located between the focal point F and the mirror. The image is virtual and upright. (a) Use the mirror equation to find di :

di < 0 means the image is virtual (behind the mirror).(b) Use the magnification equation to find hi :

hi > 0 means the image is upright.Given: ho, do, fFind: di, hi

Example: A convex mirror is used to reflect light from an object placed at do = 66 cm in front of the mirror. The radius of the mirror is R = 92 cm. Find (a) the location of the image and (b) the magnification.

The mirror and magnification equations are equally applicable to convex mirrors. However, in this case the focal distance should be taken with a negative sign.Convex Mirrors

Negative di means the image is virtual (behind the mirror).a)b)Positive m means an upright image.

Given: do, RFind: di, m

Example: A convex mirror reflects light from a toy at do = 30.0 cm from it. The toy's height is 6.0 cm, and the height of the toy's image is 1.2 cm. What is the focal length of the mirror?Given: do, ho, hiFind: f

We use the mirror eqn.: (1)

and the magnification eqn.: (2)From (2) we get:Now we can find f using eqn. (1):

(virtual image)

Equation (2) indicates the image is upright.Summary of Sign Conventions for Spherical Mirrors

m is + for an upright image.m is - for an inverted image.