# 2.2 Spherical mirrors  (Page 5/20)

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${d}_{\text{o}}=\text{−}{d}_{\text{i}}$

which is the same as [link] obtained earlier.

Notice that we have been very careful with the signs in deriving the mirror equation. For a plane mirror, the image distance has the opposite sign of the object distance. Also, the real image formed by the concave mirror in [link] is on the opposite side of the optical axis with respect to the object. In this case, the image height should have the opposite sign of the object height. To keep track of the signs of the various quantities in the mirror equation, we now introduce a sign convention.

## Sign convention for spherical mirrors

Using a consistent sign convention is very important in geometric optics. It assigns positive or negative values for the quantities that characterize an optical system. Understanding the sign convention allows you to describe an image without constructing a ray diagram. This text uses the following sign convention:

1. The focal length f is positive for concave mirrors and negative for convex mirrors.
2. The image distance ${d}_{\text{i}}$ is positive for real images and negative for virtual images.

Notice that rule 1 means that the radius of curvature of a spherical mirror can be positive or negative. What does it mean to have a negative radius of curvature? This means simply that the radius of curvature for a convex mirror is defined to be negative.

## Image magnification

Let’s use the sign convention to further interpret the derivation of the mirror equation. In deriving this equation, we found that the object and image heights are related by

$-\frac{{h}_{\text{o}}}{{h}_{\text{i}}}=\frac{{d}_{\text{o}}}{{d}_{\text{i}}}.$

See [link] . Both the object and the image formed by the mirror in [link] are real, so the object and image distances are both positive. The highest point of the object is above the optical axis, so the object height is positive. The image, however, is below the optical axis, so the image height is negative. Thus, this sign convention is consistent with our derivation of the mirror equation.

[link] in fact describes the linear magnification    (often simply called “magnification”) of the image in terms of the object and image distances. We thus define the dimensionless magnification m as follows:

$m=\frac{{h}_{\text{i}}}{{h}_{\text{o}}}.$

If m is positive, the image is upright, and if m is negative, the image is inverted. If $|m|>1$ , the image is larger than the object, and if $|m|<1$ , the image is smaller than the object. With this definition of magnification, we get the following relation between the vertical and horizontal object and image distances:

$m=\frac{{h}_{\text{i}}}{{h}_{\text{o}}}=\text{−}\frac{{d}_{\text{o}}}{{d}_{\text{i}}}.$

This is a very useful relation because it lets you obtain the magnification of the image from the object and image distances, which you can obtain from the mirror equation.

## Solar electric generating system

One of the solar technologies used today for generating electricity involves a device (called a parabolic trough or concentrating collector) that concentrates sunlight onto a blackened pipe that contains a fluid. This heated fluid is pumped to a heat exchanger, where the thermal energy is transferred to another system that is used to generate steam and eventually generates electricity through a conventional steam cycle. [link] shows such a working system in southern California. The real mirror is a parabolic cylinder with its focus located at the pipe; however, we can approximate the mirror as exactly one-quarter of a circular cylinder.

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