Introduction to reflection and refraction

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1.1 Introduction

This "book" is not intended to be a vast, definitive treatment of everything that is known

about geometric optics. It covers, rather, the geometric optics of first-year students,

whom it will either help or confuse yet further, though I hope the former. The part of

geometric optics that often causes the most difficulty, particularly in getting the right

answer for homework or examination problems, is the vexing matter of sign conventions

in lens and mirror calculations. It seems that no matter how hard we try, we always get

the sign wrong! This aspect will be dealt with in Chapter 2. The present chapter deals

with simpler matters, namely reflection and refraction at a plane surface, except for a

brief foray into the geometry of the rainbow. The rainbow, of course, involves refraction

by a spherical drop. For the calculation of the radius of the bow, only Snell's law is

needed, but some knowledge of physical optics will be needed for a fuller understanding

of some of the material in section 1.7, which is a little more demanding than the rest of

the chapter.

1.2 Reflection at a Plane Surface

The law of reflection of light is merely that the angle of reflection r is equal to the angle

of incidence r. There is really very little that can be said about this, but I'll try and say

what little need be said.

i. It is customary to measure the angles of incidence and reflection from the normal to

the reflecting surface rather than from the surface itself.

i r

FIGURE I.1

2

ii. Some curmudgeonly professors may ask for the lawS of reflection, and will give you

only half marks if you neglect to add that the incident ray, the reflected ray and the

normal are coplanar.

iii. A plane mirror forms a virtual image of a real object:

or a real image of a virtual object:

FIGURE I.2

• O I °

FIGURE I.3

• I O °

3

iv. It is usually said that the image is as far behind the mirror as the object is in front of

it. In the case of a virtual object (i.e. light converging on the mirror, presumably from

some large lens somewhere to the left) you'd have to say that the image is as far in front

of the mirror as the object is behind it!

v. If the mirror were to move at speed v away from a real object, the virtual image

would move at speed 2v. I'll leave you to think about what happens in the case of a

virtual object.

vi. If the mirror were to rotate through an angle θ (or were to rotate at an angular speed

ω), the reflected ray would rotate through an angle 2θ (or at an angular speed 2ω).

vii. Only smooth, shiny surfaces reflect light as described above. Most surfaces, such

as paper, have minute irregularities on them, which results in light being scattered in

many directions. Various equations have been proposed to describe this sort of

scattering. If the reflecting surface looks equally bright when viewed from all directions,

the surface is said to be a perfectly diffusing Lambert's law surface. Reflection

according to the r = i law of reflection, with the incident ray, the reflected ray and the

normal being coplanar, is called specular reflection (Latin: speculum, a mirror). Most

surfaces are intermediate between specular reflectors and perfectly diffusing surfaces.

This chapter deals exclusively with specular reflection.

viii. The image in a mirror is reversed from left to right, and from back to front, but is

not reversed up and down. Discuss.

ix. If you haven't read Through the Looking-glass and What Alice Found There, you are

missing something.

1.3 Refraction at a Plane Surface

I was taught Snell's Law of Refraction thus:

When a ray of light enters a denser medium it is refracted towards the normal in such

a manner than the ratio of the sine of the angle of incidence to the sine of the angle of

refraction is constant, this constant being called the refractive index...
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