A Concave Reflector

We are given that the concave mirror projects a real image of the coils at an image distance $d_{\text{i}}=\mathrm{3.00\; m}$. The coils are the object, and we are asked to find their location—that is, to find the object distance $d_{\text{o}}$. We are also given the radius of curvature of the mirror, so that its focal length is $f=R/2=\mathrm{25.0\; cm}$ (positive since the mirror is concave or converging). Assuming the mirror is small compared with its radius of curvature, we can use the thin lens equations, to solve this problem.

Since $d_{\text{i}}$ and $f$ are known, thin lens equation can be used to find $d_{\text{o}}$:

Rearranging to isolate $d_{\text{o}}$ gives

Entering known quantities gives a value for $\mathrm{1/}{d}_{\text{o}}$:

This must be inverted to find $d_{\text{o}}$:

Note that the object (the filament) is farther from the mirror than the mirror’s focal length. This is a case 1 image ($d_{\text{o}}>f$ and $f$ positive), consistent with the fact that a real image is formed. You will get the most concentrated thermal energy directly in front of the mirror and 3.00 m away from it. Generally, this is not desirable, since it could cause burns. Usually, you want the rays to emerge parallel, and this is accomplished by having the filament at the focal point of the mirror.

Note that the filament here is not much farther from the mirror than its focal length and that the image produced is considerably farther away. This is exactly analogous to a slide projector. Placing a slide only slightly farther away from the projector lens than its focal length produces an image significantly farther away. As the object gets closer to the focal distance, the image gets farther away. In fact, as the object distance approaches the focal length, the image distance approaches infinity and the rays are sent out parallel to one another.