PhysicsLAB: Mirror Equation

If we wanted to calculate the numerical position of an image or the focal length of a spherical mirror, we would use the mirror equation. It may be stated in one of these two alternative forms:

In this equation each variable has a special meaning.

d_o represents the distance from the vertex of the mirror to the position of the object as measured along the axis
d_i represents the distance from the vertex of the mirror to the position of the image as measured along the axis
f represents the focal length of the mirror. Remember that 2f = CV.

These sign conventions are summarized in the following table.

d_o	positive	when the object is "in front of the mirror"
d_i	positive	real images (inverted - "in front of the mirror")
d_i	negative	virtual images (upright - "behind the mirror")
f	positive	converging (concave) mirrors
f	negative	diverging (convex) mirrors

In this equation, d_o, d_i, and f must be measured in the same unit - usually all three are either expressed in centimeters or in meters.

The formula used to calculate the magnification of an image is:

where I and O represent the sizes of the image and object respectively.

As opposed to merely calculating the magnification of a mirror system, this equation is often used to compare the sizes of objects and images with their locations. For example, when a problem states that a real image is twice as large as an object this requires that you use the relationship d_i = 2d_o in the mirror equation. A virtual image twice as large as the object would need you to use the value d_i = -2d_o.

Let's work a few examples to show how these equations complement the ray diagrams that we have already learned how to construct.

Refer to the following information for the next seven questions.

A candle is placed 30 cm in front of a concave mirror that has a radius of curvature of 15 cm.

	What type of image will be formed, real or virtual?

	Which of these properties will the image manifest: upright/inverted? enlarged/equal/reduced in size?

	Where will the image be formed?

	If the candle is 35 cm tall, then how tall is its image?

	If the mirror were changed to a convex mirror with the same radius of curvature, what properties would the new image manifest: real/virtual? upright/inverted? enlarged/equal/reduced in size?

	Where would the new image be formed?

	How tall is this second image?