Simple lens law

(metal clanking) (ball thudding) (bell dings) - Great work! Now that you know what's going on to make a sharp image, let's calculate exactly where the point of sharp focus is. We know the point where the medial and parallel rays intersect. Again, let's put the origin of our coordinate system at the center of the lens. And let the point to be focused have coordinates X zero and Y zero. The medial ray passes through the origin and the point X zero Y zero. So the line equation for it is just Y equals Y zero divided by X zero times X. The parallel ray passes through two points we know. The point on the lens it hits, which is at zero Y zero, and the focal point of the lens, which is at minus F zero. It has a slope Y zero over F, and a Y-intercept at Y zero. So the equation of this line is Y equals Y zero over F times X plus Y zero. To intersect these lines we set these equations equal to one another. We have Y zero over X zero times X equals Y zero over F times X plus Y zero. Then we divide through by Y zero to give one over X zero times X equals one over F times X plus one. And now let's divide through by X, which give us one over X zero equals one over F, plus one over X. Notice the X in this equation is the X coordinate where the lines intersect. According to our diagram this happens when X equals minus I. Our diagram also tells us that X zero is equal to the object distance, O. So we have one over O equals one over F minus one over I, which we usually rewrite as one over O plus one over I equals one over F. Ta-da! This is called the simple lens law. It says that an object at distance O from the lens, with focal length F, comes into focus at distance I. In the next exercise we'll test your understanding of this idea.