Question

Figure 34-30 shows four thin lenses, all of the same material, with sides that either are flat or have a radius of curvature of magnitude $\mathbf{10}\mathbf{}\mathbf{}\mathbf{\text{cm}}$. Without written calculation, rank the lenses according to the magnitude of the focal length, greatest first.

Short answer

The rank of the lenses according to the magnitude of the focal length, greatest first is $f_d>f_a=f_b>f_c$.

Step-by-step solution

Given data

• Magnitude of the radius of curvature $R=10\text{cm}$,
• Figure 34-30 showing four lenses are given.

Understanding the concept of lens

The focal length of each lens by using the lens maker formula is calculated. Then rank them according to their magnitudes.

Formula:

The lens maker equation,$\frac{1}{f}=\left(\mu -1\right)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)······\left(1\right)$

Here, $\mu =\text{refractiveindexoflens}$

$R_1=\text{radiusofcurvatureofincidentlight}$

$R_2=\text{radiusofcurvaturethroughwhichlightgoesout}$

Calculation of the rank of the lenses according to the magnitude of their focal length

Refractive index of glass,${\mu }_{glass}=1.5$

Case 1:

$R_1=10\text{cm};R_2=\infty$

The focal length for the above data can be given using equation (1) as follows:

$\begin{array}{rcl}\frac{1}{f_a}& =& \left(1.5-1\right)\left(\frac{1}{10}-\frac{1}{\infty }\right)\\ f_a& =& 20\text{cm}\end{array}$

Case 2:

$R_1=\infty ;R_2=-10\text{cm}$

The focal length for the above data can be given using equation (1) as follows:

$\begin{array}{rcl}\frac{1}{f_b}& =& \left(1.5-1\right)\left(\frac{1}{\infty }-\frac{1}{-10}\right)\\ f_b& =& -20\text{cm}\\ & & \end{array}$

Case 3:

$R_1=10\text{cm};R_2=-10\text{cm}$

The focal length for the above data can be given using equation (1) as follows:

$\begin{array}{rcl}\frac{1}{f_d}& =& \left(1.5-1\right)\left(\frac{1}{10}-\frac{1}{-10}\right)\\ f_d& =& 10\text{cm}\\ & & \end{array}$

Here,$R_2<R_1$

Hence, the focal length$f_d$will be greater than the focal lengths$f_a,f_b\text{and}{f}_c$.

Therefore, the rank of the lenses according to the magnitude of the focal length is $f_d>f_a=f_b>f_c$.