Rays, Reflection and the Plane Mirror

Light goes straight

The waves of the unit on Waves spread round corners, because their wavelengths are comparable to doors and walls. Light is also a wave, but its wavelength is tiny. Visible light has wavelengths from about \(400\,\text{nm}\) (violet) to \(700\,\text{nm}\) (red). A gap of \(1\,\text{mm}\) is \(2000\) times as wide as a wavelength of \(500\,\text{nm}\). Through such openings light bends round the edges very little, and it travels in straight lines.

This straight-line travel is the whole basis of geometrical optics. We draw the path of light as a ray, a straight line with an arrow along the direction in which the light energy travels. A bundle of rays is a beam. The spreading of light round an edge, called diffraction, is left to the chapter on diffraction later in this unit.

A beam can have three shapes. In a parallel beam the rays never meet; sunlight reaching the Earth and the light of a laser pointer are parallel beams. In a diverging beam the rays spread out from a point, like the light of a bulb or a diya. In a converging beam the rays close in towards a point, like sunlight after a magnifying glass.

A tiny hole in a card makes an image with no lens at all. Each point of the object sends light in all directions. Only one narrow ray from each point gets through the hole, and it goes straight on. The ray from the top of the object therefore lands low on the screen, and the ray from the bottom lands high. In Fig 1.1 the image is \(4 \times 20/30 = 2.67\,\text{cm}\) tall, and it is inverted. This is a pinhole camera.

How the eye locates a point

The eye does not reach out to the object. Each point of an object sends out a diverging cone of rays, and the eye catches a thin part of that cone. The eye and brain trace the caught rays backwards, in straight lines, to the point from which they seem to come. We see the point there. The eye has no way to know what happened to the rays before they arrived.

This leads to two kinds of image. A real image is formed where rays actually meet after reflection or refraction. Light energy reaches that point, so a screen placed there shows the image. A virtual image is formed where rays only seem to meet. The rays diverge from the mirror, and their backward extensions, drawn dashed, meet behind it. No light reaches a virtual image, so a screen there stays dark.

The same idea applies to objects. A real object is a point from which rays actually diverge, like a lamp or a lit face. Sometimes a mirror or a lens receives a converging beam before its rays have met. The point where they would have met is then a virtual object. Virtual objects matter when two lenses or mirrors act one after the other, in the chapter on lenses.

The two laws of reflection

When a ray meets a polished surface, draw the normal, the line perpendicular to the surface at the point of incidence. The angle of incidence \(i\) is the angle between the incident ray and the normal. The angle of reflection \(r\) is the angle between the reflected ray and the normal. Both angles are measured from the normal, never from the surface.

The first law says that reflection never throws a ray sideways out of the plane of incidence. The second law fixes the direction inside that plane. Together they give the reflected ray completely. The laws hold at every point of every surface, flat or curved, because each tiny patch of a surface is flat.

A mirror gives regular reflection: its surface is smooth, all the normals are parallel, and a parallel beam stays parallel. A sheet of paper gives diffuse reflection. Each tiny patch still obeys \(r = i\), but the patches tilt every way, so the reflected rays scatter in all directions. That is why you can read a page from any seat in a room.

The image in a plane mirror

Place a point object O at a distance \(d\) in front of a plane mirror. Take two rays from O that reflect and reach the eye. The reflected rays diverge, so they never meet in front of the mirror. Extend them backwards, dashed, behind the mirror. They meet at a point I, and the eye sees O there.

Where exactly is I? Each reflected ray makes the same angle with the mirror as its incident ray. So each ray, with its extension, forms two triangles on either side of the mirror that are congruent. Every such pair gives the same point I: on the normal through O, at the same distance \(d\) behind the mirror. Every ray from O, extended backwards, passes through I, whatever its angle.

That fixed meeting point is what makes I an image. An eye anywhere in front of the mirror traces the rays back to the same place. The image of a body is the set of images of all its points. So the image is virtual, erect and the same size as the object. It lies as far behind the mirror as the object is in front, on the same normal.

One more property surprises people. Raise your right hand in front of a mirror. The image raises the hand on its right side as you look at it, which is the image's left hand. A plane mirror reverses front and back along its normal, and this makes right and left swap. The effect is called lateral inversion. It is why the word AMBULANCE is painted reversed on the front of an ambulance: in a car's mirror it reads correctly.

A clock seen in a mirror is laterally inverted about its 12 to 6 line. The hour hand of the image points to the mirror position of the real hand. To read the real time, subtract the image reading from 11:60. An image showing 4:30 means the real time is \(11{:}60 - 4{:}30 = 7{:}30\).

Now back to the neem tree. Each gap between the leaves acts as a pinhole, and each spot on the ground is an image of the Sun, not of the gap. The Sun is round, so the spots are round. The Sun's diameter is about \(1.39 \times 10^9\,\text{m}\), at a distance of about \(1.5 \times 10^{11}\,\text{m}\). A gap \(5\,\text{m}\) above the ground therefore gives a spot about \(5 \times 1.39/150 \approx 0.046\,\text{m}\) across, about \(4.6\,\text{cm}\). During an eclipse the Sun itself is a crescent, and so are its images.