Magnification
The magnification of an object, m, is simply the ratio of the image height to the object height. A sign convention is introduced so that m>0 is an upright image while m<0 is an inverted image.
Mirror
Consider the central ray (reflected at equal angle):
One can see that:
tan θ = y1/u = y2/v
The magnification is by definition:
m = y2 / y1 = v/u
or
m = Vi / Vf.
That is the magnification is the ratio of the initial vergence to the final vergence. It is customary to set the magnification as positive if the image is upright and negative if the image is inverted. The above equation using the vergences satisfies this requirement.
Return to the problems in the section on mirrors and calculate the magnifications. Do they agree with your diagrams?
Interface
Consider the central ray (refracted according to Snell's Law):
One can see that:
tan θ1 = y1/u and tan θ2 = y2/v
Assuming small angles
tan θ ~ sin θ ~ θ.
Snell's Law becomes:
n1θ1 = n2θ2
Combining these gives
n1y1/u = n2y2/v
m = y2 / y1 = (n1/u) / (n2/v)
or
m = Vi / Vf.
Thus the magnification has exactly the same form as for the mirror (including sign convention).
Return to the problems in the section on interfaces and calculate the magnifications. Do they agree with your diagrams?