Question

Diffuse Reflection

The law of reflection is quite useful for mirrors and other flat, shiny surfaces. (This sort of reflection is called specular reflection). However, you've likely been told that when you look at something, you are seeing light reflected from the object that you are looking at. This is reflection of a different sort: diffuse reflection. In this problem, you will see how diffuse reflection actually arises from the same law of reflection that you are accustomed to for  reflections from mirrors.

here are the images

Consider a spotlight shining onto a horizontal mirror (PartA figure) . If the light from the spotlight strikes the mirror at an angle θa to the normal, what angle θr to the normal would you expect for the reflected rays?

Express your answer in terms of θa.
θr = ?

This simple rule of reflection no longer seems to hold for diffuse reflection. Consider the same spotlight but now reflecting from the surface of a table (Intro1 figure) . Unlike the light reflected from the mirror, the light reflected from the table seems to go in all directions. If it didn't, then you'd only be able to see tables when you were at a specific angle to the lights above you! To understand why the light reflects in all directions, you must first look at a slightly simpler problem.

Consider a flat surface, inclined downward from the horizontal by an angle α (Intro2 figure) . The red line represents the surface and the red dotted line indicates the normal to this surface (the normal line). The two blue dashed lines represent horizontal and vertical. The angle between the incoming ray and the vertical is θa. Throughout this problem, assume that θa is larger than α but smaller than 2α. (If you wish, you can determine the correct sign rules to generalize your results later.)

Find the angle θt between the reflected ray and the vertical.

Express the angle between the reflected ray and the vertical in terms of α and θa.

θr =  ?

Answer 1

Concepts and reason

The concept required to solve this problem is laws of reflection.

Use the law of reflection to solve for the angle of reflection.

Fundamentals

The law of reflection states that angle of incidence of the light ray with the normal to surface is equal to the angle of reflection.

From the law of reflection, the angle of incidence is equal to the angle of reflection. The spotlight is incident at angle ${\theta _{\rm{a}}}$ from the normal to the mirror. The angle of reflection ${\theta _{\rm{r}}}$ to the normal of the reflected rays is equal to the angle of incidence ${\theta _{\rm{a}}}$ .

${\theta _{\rm{r}}} = {\theta _{\rm{a}}}$

Draw the figure and use the geometry of the figure to solve for the angle of reflection. The angle of incidence from the normal is ${\theta _{\rm{i}}}$ . From law of reflection angle of reflection is equal to angle of reflection ${\theta _{\rm{b}}} = {\theta _{\rm{i}}}$ .

${\theta _{\rm{i}}}$ $90^\circ - \alpha$

Diagram 1: The ray diagram of the reflection.

The sum of angle of incidence and normal is equal to sum of angle of incident light ray from vertical and angle between vertical and surface.

$\begin{array}{c}\\{\theta _{\rm{i}}} + 90^\circ = {\theta _{\rm{a}}} + 90^\circ - \alpha \\\\{\theta _{\rm{i}}} = {\theta _{\rm{a}}} - \alpha \\\end{array}$

Substitute ${\theta _{\rm{b}}}$ for ${\theta _{\rm{i}}}$ in the above equation ${\theta _{\rm{i}}} = {\theta _{\rm{a}}} - \alpha$ as angle of reflection is equal to angle of incidence.

${\theta _{\rm{b}}} = {\theta _{\rm{a}}} - \alpha$

The angle between the normal and vertical is $\alpha$ as the angle of inclination of the plane is $\alpha$ . So, the angle between the vertical and reflected ray is,

${\theta _{\rm{r}}} = \alpha - {\theta _{\rm{b}}}$

Substitute ${\theta _{\rm{a}}} - \alpha$ for ${\theta _{\rm{b}}}$ in the above equation ${\theta _{\rm{r}}} = \alpha - {\theta _{\rm{b}}}$ .

$\begin{array}{c}\\{\theta _{\rm{r}}} = \alpha - \left( {{\theta _{\rm{a}}} - \alpha } \right)\\\\ = 2\alpha - {\theta _{\rm{a}}}\\\end{array}$

Ans:

The angle to the normal expected for the reflected rays is ${\theta _{\rm{r}}} = {\theta _{\rm{a}}}$ .