GRADES 9–12
NGSS: Waves and Their Applications in Technologies for Information Transfer:
Use mathematical representations to support a claim regarding relationships among the frequency, wavelength, and speed of waves traveling in various media.
CCSS: Modeling with Geometry:
Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).
Deriving the magnification of a telescope is a fairly simple exercise and can show the students a use for their analytic geometry. We begin by drawing an (x, y) coordinate system with the larger lens on the left at an xcoordinate
of “–F_{L}” (the focal length of the lefthand lens) and the smaller lens at an xcoordinate of “F_{R}” (the focal length of the righthand lens).
Consider a light ray coming in from the left. It will follow a straightline path with a slope “m_{1}” which we can describe with the equation
We specify that the ray hits the lefthand lens at a ycoordinate of “y_{L}”. (For the purposes of this calculation, the lenses are infinitely thin. This is an approximation.) This allows us to specify the constant “b”:
The lefthand lens focuses light coming in from the left with zero slope at its focal point. If the light coming in from the left is horizontal, with a zero slope, then the point at which it focuses is the origin, with a ycoordinate of zero and a distance “F_{L}”
from the lens. Thus the change in slope “Dm_{L}” of a ray of light hitting the lefthand lens at a ycoordinate of “y_{L}” is
Our ray of light that follows the line of “y = m_{L} ( x + F_{L} ) + y_{L}” will have a new slope “m_{2}” when it has passed the lefthand lens:
The ray passes through the point ( – F_{L}, y_{L} ) and so follows the line with the equation
The ray hits the righthand lens at the point x = F_{R}. Its ycoordinate at that point is
Similar to the lefthand lens, the righthand lens bends a horizontal ray of light that hits it at “y_{R}” so that it passes through the focal point. The change in slope is therefore
To the right of the righthand lens, the ray of light (which hit the lens at ( F_{R}, y_{R}) ) will follow a line with the equation
The slope of the line, m_{3}, is given by the formula
Now let us express m_{3} in terms of the known quantities. We substitute for y_{R}:
Now we substitute for m_{2}:
For small angles, the slope of a line is proportional to its angle from the horizontal. What this means is that whatever angle the ray of light is coming into the objective lens at is multiplied by the ratio of the lenses’ focal lengths to give the angle at which the ray of light comes out
of the eyepiece. The apparent size of an object is given by the angle between a ray of light coming from one side of it and a ray of light coming from its other side. A telescope increases this angle, and thus magnifies the object, by a factor equal to the ratio of the focal lengths of the two lenses. The minus sign in the ratio means that the
object appears to be upside down.
In this derivation, it is critical that the distance between the lenses be equal to the sum of their focal lengths. If the distance between the lenses is some other value, then the “y_{L}” does not cancel out of the equation and the final angle depends on where on the
objective lens the ray of light falls. In this case, the object is seen in the telescope as being outoffocus and blurry.
Sixty Years Ago in the Space Race:
January 31:
The American Explorer 1 became the first American spacecraft
successfully launched into orbit.
