Question

Compare the light-gathering power of the lens described in Problem $7.16$ with one that has a diameter of $36.6cm$ and a focal length of $15.8cm$.

Short answer

The light-gathering power of the lens$F_1$is greater.

Step-by-step solution

Given information

$$\begin{gathered} f_1=15.6cm,d_1=5.2cm \\ {f}_2=15.8cm,d_2=36.6cm \end{gathered}$$

Explanation

As we know the formula

$f_1=15.6cm,d_1=5.2cm$

role="math" localid="1646558603620" $$\begin{gathered} F_1=\frac{f_1}{d_1} \\ {F}_1=\frac{15.6\mathrm{cm}}{5.2\mathrm{cm}} \\ {F}_1=3 \end{gathered}$$

When$f_2=15.8cm,d_2=36.6cm$

$$\begin{gathered} F_2=\frac{f_2}{d_2} \\ {F}_2=\frac{15.8\mathrm{cm}}{36.6\mathrm{cm}} \\ {F}_2=0.43 \end{gathered}$$

Comparison

As the $F_1=3$and $F_2=0.43$.

So the light-gathering power of the lens $F_1$is greater when$f_1=15.6cm,d_1=5.2cm$.