Problem 6.93: A lens is made of glass having an index of refraction of $1.5$. Sides of the lens are convex with a radius of curvature of $20 \:cm$. $(a)$ Find the focal length of the lens.

For a thin lens we have the relation $\frac{1}{f}\:=\:(n-1)\left(\frac{1}{R_1}\:-\frac{1}{R_2}\right)$ where $f$ is the lens’s focal length, $n$ is the index of refraction of the lens material, and $R_1$ and $R_2$ are the radii of curvature of the two sides of the lens, which are spherical surfaces.

This equation is used with the following sign convention.

1. A convex lens surface that faces the object has a positive radius of curvature.

2. concave lens surface that faces the object has a negative radius of curvature.

Real images form on the side of a lens that is opposite the object, and virtual images form on the same side as the object.

**Now calculation:**

$\frac{1}{f}\:=\:(1.5-1)\left(\frac{1}{+20}\:-\frac{1}{-20}\right)$

So, $\frac{1}{f}\:=\:(0.5)\left(\frac{2}{+20}\right)$

Or focal length of the lens, $f\:=\:20\:cm$.