Question

Find the focal length of the meniscus polystyrene plastic lens in FIGURE $\mathrm{EX}34.28$.

Short answer

The focal length of meniscus polystyrene plastic lens is$203\mathrm{cm}$.

Step-by-step solution

Formula for focal length

The lens-makers' equation gives the inverse of the focal length of a narrow lens surrounded by air.

$\frac{1}{\mathrm{f}}=(\mathrm{n}-1)\left(\frac{1}{{\mathrm{R}}_1}-\frac{1}{{\mathrm{R}}_2}\right)$

For surfaces convex toward the object, the radius of curvaturelocalid="1649148884725" $\mathrm{R}$is positive.

For surfaces that are concave toward the object, the radius of curvature localid="1649148894449" $\mathrm{R}$is negative.

Calculation of focal length of lens

Assuming the object is on the left, the lens's initial surface is planar, and we can think of it as a part of a sphere with$\mathrm{R}=-40\mathrm{cm}$.

With $\mathrm{R}=-30\mathrm{cm}$, the second surface serves as a concave surface.

The focal length of the lens is,

$\frac{1}{\mathrm{f}}=(\mathrm{n}-1)\left(\frac{1}{{\mathrm{R}}_1}-\frac{1}{{\mathrm{R}}_2}\right)$

Substitute the following numerical values:

$\frac{1}{\mathrm{f}}=(1.59-1)\left(\frac{1}{-40\mathrm{cm}}-\frac{1}{30\mathrm{cm}}\right)$

$\frac{1}{\mathrm{f}}=\frac{59}{12000\mathrm{cm}}$

$\mathrm{f}=\frac{12000}{59}\mathrm{cm}=203\mathrm{cm}$