Mirror and Magnification Equations
In addition to ray diagrams, we can use equations to predict the characteristics of an
image on a concave mirror.
variable
meaning
There are two equations that you can use:
do
distance of object
di
distance of image
Mirror Equation: allows you to calculate the
ho
height of object
location of the image
hi
height of image
f
focal length
The image distance, di, is negative if the image is behind the mirror
Magnification Equation (m): allows you to find the magnification from the
object and image distances
The image height, hi, is negative if the image is inverted relative to the object
when using this equation, signs are very important!
do
positive
Object distances are always positive
di
di
positive
negative
For real images (when the image is on the same side of the mirror)
For virtual images (when the image is on the opposite side of the mirror)
ho
positive
Object height is always positive
hi
hi
positive
negative
Image is upright
Image is inverted
f
f
positive
negative
For concave mirrors
For convex mirrors
Sample Problem:
A concave mirror has a focal length of 12cm. An object with a height of 2.5cm is placed
40.0cm in front of the mirror.
A. Calculate the image distance
Use the mirror equation to find the image distance. Use the GRASS method.
G: f = 12 cm
do = 40.0 cm
R: di = ?
A:
Rearrange the equation to isolate di.
1 =
di
1 f
1
do
S: 1 = _1__ di
12 cm
_1__
40 cm
Use the inverse function on your calculator, it looks like this: x-1
1 = 12 x-1
di
-
40 x-1
1 = 0.0583
di
Solve for di:
di = 0.0583 x-1
= 17.14 cm
Don’t forget about significant digits!!
S: Therefore the image distance is 17 cm and in front of the mirror. (because the
distance image is positive).
B. Calculate the image height
Use the magnification equation to solve for the image height. Use the GRASS method.
G: ho = 2.5 cm
di = 17.14 cm
do = 40.0 cm
R: hi = ?
A:
S:
hi
=
2.5 cm
-17.14 cm
40.0 cm
Cross-Mulitply
40.0hi = -42.85 cm
Isolate hi
40.0hi = -42.85 cm
40.0
40.0
40.0 on left side cancel each other out.
hi = - 1.07 cm
S: Therefore, the image height is -1.1 cm and is inverted. (because the image height is
negative).