Question

I need 3 & 4... free free to answer 1 & 2 if you want!

Lenses may be simple ones with two spherical curved surfaces on a piece of transparent material like glass or plastic, or much more complex and compounded of different elements each with sometimes a different material. The surfaces do not have to be spherical, and manufacturing techniques today allow combining these "aspheric" lenses in designs that produce exquisite detail in an image. Your cell phone camera lens is an example, as are the lenses of a larger digital or photographic camera. This week's problem is chosen to get to the basics of lenses and how they work because they are the most common essential component of optical instruments. Starting with Snell's law, you can show that a lens has a property called a "focal length" such that 1/f = 1/p + 1/q where p is the distance to the object in front of the lens, and \(q\) is the distance from the lens to the image it forms. This applies to a lens so thin that the thickness of the glass is small compared to these distances. Light from infinity must form an image at q = f Written this way, there is a convention to measure the distance to the object as positive to the left of the lens, and the distance to the image as positive to the right of the lens.

1. Where does light coming from a distance f in front of the lens form an image? Explain.

2. If I want a lens to be halfway between an object and a screen where the image forms, what is the focal length of the lens? You may answer generally, or if you prefer a specific case let the object and the screen be 10 cm apart.

3. The focal length of a thin lens is given by the "lens maker's equation" 1 divided by f space equals space left parenthesis n minus 1 right parenthesis space left parenthesis 1 divided by R subscript 1 space minus space 1 divided by R subscript 2 right parenthesis This works when you can neglect the spacing between the surfaces, that is, when the radii are much bigger than the thickness of the lens. It is simple enough, but perilous for problems because of how the signs have to be interpreted. A lens surface that curves outward so that it is thicker at the center on that surface is "convex". One that curves inward, making it thinner at the center on that side, is concave. By convention, the sign of \(R\) is positive if the lens is convex to the incoming light, and negative if it is concave. Here n is the index of refraction of the glass relative to the medium it is in (say air), and the \(R\)'s are the radii of the surfaces of the lens. Thinking of light as coming from the left, the radius is positive if it is convex to the left, concave to the right. For example, a lens that has convex surfaces on both sides with radii 10 cm, an index of 1.5, would have a focal length of 1/f = (1.5 - 1) (1/10 - (-1/10)) = 0.1 f = 10 cm The second radius is negative because it is concave to the left and convex to the right. The shape of the surface and the index on both sides determine whether the lens converges the light, or diverges it.

--> 3. What would be the radius of curvature of the surfaces of a double convex lens with the same shape on both sides and a focal length of 1 meter? Assume an index of 1.5.

--> 4. Suppose you made a lens in which the first surface was convex to the left with a radius of 50 cm. Immediately after it the back surface is exactly the same, also convex to the left, with the same radius of curvature. Now take this lens outside and let sunlight fall on it. What happens to the light that goes through the lens? Explain it with these equations for a thin lens, and also with the wave theory of light.

Answer 1