An analytic expression to describe spherical aberration

[Nabin kalauni]

Homework Statement

Derive an analytic expression for the distance from the vertex to the focus for a particular ray in terms of (i) the radius of curvature R of the concave mirror (ii) the angle of incidence θ between incident ray and radius of the mirror. Hence show that the focus moves closer to the mirror as rays move towards the outer edge of a concave mirror.

Homework Equations

A. The law of reflection
B. Some basic trigonoteric identities

The Attempt at a Solution

I considered a ray parallel to the optic axis and a height h above the axis. The reflected ray meets the axis at a distance f from the vertex. The sketch is linked to the thread.
I could see pretty easily that θ increases as we take a ray further from the axis. As a result, the point D moves closer to the vertex. But I could not derive a mathematical expression. So I looked at the solution and they used quite a few approximations that baffled me like cos θ = ½R/(R-f). How can we get this identity from the given sketch and what approximations do we use?

 

[kuruman]

So I looked at the solution and they used quite a few approximations that baffled me like cos θ = ½R/(R-f). How can we get this identity from the given sketch and what approximations do we use?

The equation you mentioned is not an approximation and can be derived from simple geometrical considerations. Perhaps if you posted the diagram you used, we will be able to diagnose where your problem lies.

 

[Nabin kalauni]

The equation you mentioned is not an approximation and can be derived from simple geometrical considerations. Perhaps if you posted the diagram you used, we will be able to diagnose where your problem lies.

I have diagnosed my problem. Thank you.