WHY does curvature of a convex lens affect focal length?

[Anf31ld]

Homework Statement

For my high school physics coursework I must investigate factors affecting the focal length of a lens. I have focused on radii of curvature and completed my data collection and verified the accuracy using the lens makers equation. However, in the conclusion I am really struggling to explain in common language both why and how the curvature affects the focal length. Clearly as the lens becomes more curved the light is bent more and the focal length is shorter but I cannot confidently explain why this happens. I understand that the refraction is caused at the boundary between the glass and air due to the different densities of the materials but I cannot explain how the curvature impacts this. Any help anyone could give me would be hugely appreciated.

The Attempt at a Solution

As the light waves pass from air into the glass they encounter a more dense material and as a result the average speed of the light decreases and this causes the light to refract towards the normal. A more curved surface bends the light more and as a consequence decreases the focal length. This is because ... this is where I get very confused

 

[Simon Bridge]

The curve means that the same direction light hits different parts of the lens at different angles wrt the normal and the normal is pointing in different directions too.

The way to investigate this is to draw a curved surface carefully - use a compass.
Decide on a refractive index.
Draw a bunch of parallel rays incedent - 4-5 will do.
Use a protractor to draw the refracted rays in each case.
See where they cross.

Repeat with a different curvature.