Resource Lesson
Mirror Equation
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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?
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