One of the most commonplace observations is that you can see yourself in a mirror. How? FIGURE 34.10a shows rays from point source P reflecting from a mirror. Consider the particular ray shown in FIGURE 34.10b. The reflected ray travels along a line that passes through point $\text{P}\prime$ on the “back side” of the mirror. Because ${\theta }_{\text{r}}={\theta }_{\text{i}}$, simple geometry dictates that $\text{P}\prime$ is the same distance behind the mirror as P is in front of the mirror. That is, $s\prime =s$.

FIGURE 34.10 The light rays reflecting from a plane mirror.

FIGURE 34.10 Full Alternative Text

Full Alternative Text

The location of point $\text{P}\prime$ in Figure 34.10b is independent of the value of ${\theta }_{\text{i}}$. Consequently, as FIGURE 34.10c shows, the reflected rays all appear to be coming from point $P^{\prime }$. For a plane mirror, the distance $s\prime$ to point $\text{P}\prime$ is equal to the object distance $s$:

If rays diverge from an object point P and interact with a mirror so that the reflected rays diverge from point $\text{P}\prime$ and appear to come from $\text{P}\prime$, then we call $\text{P}\prime$ a virtual image of point P. The image is “virtual” in the sense that no rays actually leave $\text{P}\prime$, which is in darkness behind the mirror. But as far as your eye is concerned, the light rays act exactly as if the light really originated at $\text{P}\prime$. So while you may say “I see P in the mirror,” what you are actually seeing is the virtual image of P. Distance $s\prime$ is the image distance.

For an extended object, such as the one in FIGURE 34.11, each point on the object from which rays strike the mirror has a corresponding image point an equal distance on the opposite side of the mirror. The eye captures and focuses diverging bundles of rays from each point of the image in order to see the full image in the mirror. Two facts are worth noting:

1. Rays from each point on the object spread out in all directions and strike every point on the mirror. Only a very few of these rays enter your eye, but the other rays are very real and might be seen by other observers.

2. Rays from points P and Q enter your eye after reflecting from different areas of the mirror. This is why you can’t always see the full image of an object in a very small mirror.

FIGURE 34.11 Each point on the extended object has a corresponding image point an equal distance on the opposite side of the mirror.

EXAMPLE 34.2 How high is the mirror?

If your height is $h$, what is the shortest mirror on the wall in which you can see your full image? Where must the top of the mirror be hung?

MODEL Use the ray model of light.

VISUALIZE FIGURE 34.12 is a pictorial representation of the light rays. We need to consider only the two rays that leave your head and feet and reflect into your eye.

FIGURE 34.12 Pictorial representation of light rays from your head and feet reflecting into your eye.

FIGURE 34.12 Full Alternative Text

SOLVE Let the distance from your eyes to the top of your head be $l_1$ and the distance to your feet be $l_2$. Your height is $h=l_1+l_2$. A light ray from the top of your head that reflects from the mirror at ${\theta }_{\text{r}}={\theta }_{\text{i}}$ and enters your eye must, by congruent triangles, strike the mirror a distance $\frac{1}{2}{l}_1$ above your eyes. Similarly, a ray from your foot to your eye strikes the mirror a distance $\frac{1}{2}{l}_2$ below your eyes. The distance between these two points on the mirror is $\frac{1}{2}{l}_1+\frac{1}{2}{l}_2=\frac{1}{2}h$. A ray from anywhere else on your body can reach your eye if it strikes the mirror between these two points. Pieces of the mirror outside these two points are irrelevant, not because rays don’t strike them but because the reflected rays don’t reach your eye. Thus the shortest mirror in which you can see your full reflection is $\frac{1}{2}h$. But this will work only if the top of the mirror is hung midway between your eyes and the top of your head.

REVIEW It is interesting that the answer does not depend on how far you are from the mirror.