Chapter 7, Problem 72E

At night the pupils of a certain woman’s eyes are 8 mm in diameter. (a) How many kilometers away from a car facing her will the woman be able to distinguish its headlights from each other? (b) What would the distance be if her pupils were 4 mm in diameter (say at twilight)? Assume that the headlights are 1.5 m apart, that the average wavelength of their light is 600 nm, and that her eyes attain half their theoretical resolving power.

To determine

The range at which car’s head light can be distinguishable.

Answer to Problem 72E

The range at which car’s head light can be distinguishable is $8.2\text{km}$.

Explanation of Solution

Given info:

The separation between the headlights is $1.5\text{m}$, the diameter of pupil is $8\text{mm}$ and the average wavelength is $600\text{nm}$.

Since the eyes attain only half of the resolving power, the distance will be half of the range obtain.

Write an expression to calculate the range at which car’s head light can be distinguishable.

$L=\frac{1}{2}\left(\frac{d_{0}D}{1.22\lambda }\right)$

Here,

$D$ is the diameter of pupil

$\lambda$ is the wavelength of the light

$L$ is the range at which car’s head light can be distinguishable

$d_0$ is the separation between the headlights

Substitute $600\text{nm}$ for $\lambda$, $8\text{mm}$ for $D$ and $1.5\text{m}$ for $d_0$ to find $L$.

$\begin{array}{c}L=\frac{1}{2}\left(\frac{\left(1.5\text{m}\right)\left(\left(8\text{mm}\right)\left(\frac{1\text{m}}{{10}^3\text{mm}}\right)\right)}{1.22\left(\left(600\text{nm}\right)\left(\frac{1\text{m}}{{10}^9\text{nm}}\right)\right)}\right)\\ =8.2\times {10}^3\text{m}\\ \text{=}\left(8.2\times {10}^3\text{m}\right)\left(\frac{1\text{km}}{{10}^3\text{m}}\right)\\ =8.2\text{km}\end{array}$

Thus, the range at which car’s head light can be distinguishable is $8.2\text{km}$.

Conclusion:

The range at which car’s head light can be distinguishable is $8.2\text{km}$.

To determine

The range at which car’s head light can be distinguishable.

Answer to Problem 72E

The range at which car’s head light can be distinguishable is $4.1\text{km}$.

Explanation of Solution

Given info:

The separation between the headlights is $1.5\text{m}$, the diameter of pupil is $4\text{mm}$ and the average wavelength is $600\text{nm}$.

Since the eyes attain only half of the resolving power, the distance will be half of the range obtained.

Write an expression to calculate the range at which car’s head light can be distinguishable.

$L=\frac{1}{2}\left(\frac{d_{0}D}{1.22\lambda }\right)$

Here,

$D$ is the diameter of pupil

$\lambda$ is the wavelength

$L$ is the range at which car’s head light can be distinguishable

$d_0$ is the separation between the headlights

Substitute $600\text{nm}$ for $\lambda$, $4\text{mm}$ for $D$ and $1.5\text{m}$ for $d_0$ to find $L$.

$\begin{array}{c}L=\frac{1}{2}\left(\frac{\left(1.5\text{m}\right)\left(\left(4\text{mm}\right)\left(\frac{1\text{m}}{{10}^3\text{mm}}\right)\right)}{1.22\left(\left(600\text{nm}\right)\left(\frac{1\text{m}}{{10}^9\text{nm}}\right)\right)}\right)\\ =4.1\times {10}^3\text{m}\\ \text{=}\left(4.1\times {10}^3\text{m}\right)\left(\frac{1\text{km}}{{10}^3\text{m}}\right)\\ =4.1\text{km}\end{array}$

Thus, the range at which car’s head light can be distinguishable is $4.1\text{km}$.

Conclusion:

The range at which car’s head light can be distinguishable is $4.1\text{km}$.