For a given lens, we could try to create a particular focal length. For that reason, the equation
that we have here which tells us how to make that focal length is often called the lens maker’s equation.
If I know what focal length I would like my lens to have, I can try to achieve that focal length
using the geometric properties of my lens by making the variables in this equation match
my focal length. The variables are the index of refraction of the lens in a particular medium,
so we have both of those indices of refraction minus 1 times the inverse of each radius subtracted
from each other. So, what are these radii, this r1 and this r2? We talked about our lenses
as being sort of part of a circle. So, if we imagine that the curvature of the lens on one side is part
of an entire circle if we drew the entire thing, we can imagine each side of our lens as in principle
being able to be part of two different circles. Maybe one side of the lens is more curved
and the other one is more shallow. These two radii, we just simply put into our lens maker’s equation.
In fact, I actually do the opposite in terms of procedure. We find a particular focal length. We pick
our materials. Maybe we know that our lens will be always in air. We know we would like
to use glass. Given everything else, we can figure out exactly what we would like these two radii
on either side of our lens to be. In that case, we simply have to grind the lens down,
which it turns out is a very tedious process. We can always figure out exactly what these radii have to be.
This perhaps is an equation that you would need to memorize and have committed completely
into mind but there are few important things to know. One is to be aware that there is a lens maker’s
equation, that there’s a way to figure out what size your lens would need to be, the shape of it
on either side. But there’s another thing you should be aware of when you’re looking at an equation
like this one which is how we talk about the radii. Notice in this equation, I have 1 over r1
minus 1 over r2. That might be confusing. What if we had a symmetric lens where r1 and r2
were the same, it might seem like we simply get 1 over f equals zero after we subtracted
these two equal terms. But in fact, we define the radii differently. Since one of the radii
is pointing in one direction and the other radius is pointing in the other direction, we define one
of them to be negative. Typically by convention, we would say that r1 is a positive number
because if we drew the radius from this left side, that radius would go towards the right, towards
the center of the circle. All the radius from the second lens would instead be going towards the left.
For this reason, for any convex lens like this one, these two radii could be the same on either side.
They would be negatives of each other. They would add normally. So, we don’t have to
worry about this term being zero as long as we’re careful to keep our conventions for r1
and r2 straight for the lens maker’s equation. Again, the important thing to take away
from this equation is simply that if we know the focal length that we have desired,
we can find the geometrical properties of the material in order to pick that focal length.