Example: Flat mirrors

Understanding the situation

The webpage Flat mirrors says:

When an object is viewed in a mirror, the viewer's brain interprets it as being behind the mirror by the same distance (as measured along the perpendicular to the mirror) that the object is in front of it.

This seems a weird result. Let's work out a problem and see how it works.

Presenting a sample problem

A. Mel and Taylor are both looking directly at the same object through a large window. The direction that each of them is looking is indicated with an arrow in the diagram below. Where’s the object? How do you know? Show by using a ray diagram.

B. Now suppose that they are looking at a mirror, not a window. Now where's the object? Show by using a ray diagram.

Solving this problem

A. Our basic principle of the ray model of light is that the brain assumes that light that enters its eye has come to it along a straight line. Since the window just permits the transmission of light and doesn't bend it, we can infer where the object is by following the directions in which Mel and Angie see the objects. Mel's view indicates that the object must be somewhere along the red path shown. Angie's view indicates that the object must be somewhere along the blue path shown. Therefore, the object must be at their intersection where the small black square is shown. This is shown in the figure at the right.

B. Now what if it's a mirror, not a window? We can first see where Mel and Angie think the object appears to be. Again, since the brain assumes the rays to have come in straight lines, the object will still appear to be in the same place it was in the window. But now, the rays we are tracing back don't go through the mirror but reflect back. Therefore, in the figure at the left we've drawn the extensions to the (virtual) image with dotted lines. The web page suggest that the image in the mirror will be as far behind the mirror as the object is in front of in, alone a perpendicular to the mirror. 

Using this idea, we've put the object in a position consistent with this heuristic. If we look at the path of the actual rays (shown in red and blue) we seen that they follow the fundamental ray principle of mirrors, angle of incidence equals angle of reflection. (We can prove this by noting that triangle OMP and IMP are similar as are triangles OAP and IAP. See if you can complete this proof for yourself!)

Joe Redish 7/2/19