Formulae for spherical mirrors
For an object distance u, image distance v and
focal length f we have:
1/u + 1/v = 1/f (Cartesian sign convention)
This
formula applies to both convex and concave minors although you must remember to enter real
distances as positive numbers and virtual distances as negative
numbers.
Newton's formula
Another useful formula is known as
Newton's formula.
Let the distance of the object and image from the principal focus be x and
y respectively.
Then
u= f+x and v= f+y
Using the mirror formula:
1/u + 1/v = 1/f gives
1/(f+x) + 1/(f+y) = 1/f and so:
Newton’s formula: f2 = xy
Transverse magnification
Just as
important as where the image is will be the size of the image. There are two ways of looking at
the size of an image: -
(a) its actual size and (b) its angular size
The linear size is fixed
but the angular size will vary depending on the distance of the observer from the image. To
understand this think about a telescope forming an image of the Moon. When we say that the
telescope magnifies it 30x this would mean a linear size of 1738x30 km = 52140 km. What we
actually mean is that its angular size has been increased by a factor of thirty - from 0.5 to 15 m.
It therefore looks thirty times closer.
M = image distance (v) / object distance
(u)
The transverse or linear magnification is defined as:-
Magnification (M) = image height/object height = Image distance/Object distance
Example problems
1. A concave mirror of focal length 9 cm forms an image of a real object 3 cm high placed 15 cm in front of it.
Find (a) the position (b) nature and (c) the size of the image.
(a) using the equation: 1/u + 1/v = 1/f we have 1/v 1/f-1/u = 1/f - 1/15 = 0.044 V = 22.5 cm
(b) since the image distance is positive the image is real and it is inverted
(c) magnification of the image = v/u = 22.5/15 = 1.5 therefore image size = 3 x 1.5 = 4.5 cm
2. A girl uses a convex car wing mirror of focal length 2 m to look at a lorry which is actually 30 m from the mirror. Calculate:-
(a) the position of the image
(b) the magnification of the image
(a) Using Newton’s formula F = xy with x = 32 m (remember that for a convex mirror the focal length is negative.
Therefore 4 = 32y giving y = 0.125 m so v = - 1.875 m (virtual)
Using 1/u+ 1/v= 1/f we have 1/v -1/2-1/30 =0.53 so v = - 1.875 m (virtual)
(b) magnification = v/u = -1.875/30 = 0.0625.
3. A concave mirror forms a sharply focused image of the Sun 1.5 cm in diameter on a screen. If the Sun subtends an angle of 0.5o at the Earth calculate the radius of curvature of the mirror.
tan 0.5 = 1.5/f and therefore
Focal length of mirror (f) = 172 cm
Radius of curvature = 344 cm = 3.44 m
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