Chapter 37, Problem 12P

(a)

To determine

The angle at which the first maximum of sound intensity located.

Answer to Problem 12P

The angle at which the first maximum of sound intensity located is $\underset{\_}{34.9°}$.

Explanation of Solution

Write the expression to find the wavelength.

    $\lambda =\frac{v}{f}$                                                                                                                        (I)

Here, $\lambda$ is the wavelength, $v$ is the velocity if the sound and $f$ is the frequency.

Write the expression for location of maxima.

    $d\sin \theta =m\lambda$                                                                                                            (II)

Here, $d$ is the distance between the slits, $\theta$ is the angle and $m$ is the order.

Conclusion:

Substitute $343\text{m}/\text{s}$ for $v$ and $2000{\text{s}}^{-1}$ for $f$ in equation (I).

    $\begin{array}{c}\lambda =\frac{343\text{m}/\text{s}}{2000{\text{s}}^{-1}}\\ =0.1715\text{m}\end{array}$

Write the expression to find the angle of maxima from equation (II).

    $\theta ={\sin }^{-1}\left(\frac{m\lambda }{d}\right)$                                                                                                      (III)

Substitute $1$ for $m$, $0.1715\text{m}$ for $\lambda$ and $0.300\text{m}$ for $d$ in equation (III).

    $\begin{array}{c}\theta ={\sin }^{-1}\left(\frac{\left(1\right)\left(0.1715\text{m}\right)}{0.300\text{m}}\right)\\ =34.9°\end{array}$

Therefore, the angle at which the first maximum of sound intensity located is $\underset{\_}{34.9°}$.

(b)

To determine

The slit separation if the sound wave is replaced by microwave.

Answer to Problem 12P

The slit separation if the sound wave is replaced by microwave is $\underset{\_}{5.25\text{cm}}$.

Explanation of Solution

Write the expression for location of maxima.

    $d\sin \theta =m\lambda$                                                                                                            (II)

Here, $d$ is the distance between the slits, $\theta$ is the angle and $m$ is the order.

Conclusion:

Substitute $1$ for $m$, $0.0300\text{m}$ for $\lambda$ and $34.9°$ for $\theta$ in equation (II).

    $\begin{array}{c}d\sin 34.9°=\left(1\right)\left(0.0300\text{m}\right)\\ d=5.25\text{cm}\end{array}$

Therefore, the slit separation if the sound wave is replaced by microwave is $\underset{\_}{5.25\text{cm}}$.

(c)

To determine

The frequency of light that gives the same angle to the first maximum intensity.

Answer to Problem 12P

The frequency of light that gives the same angle to the first maximum intensity is $\underset{\_}{5.24\times {10}^{14}\text{Hz}}$.

Explanation of Solution

Write the expression for location of maxima.

    $d\sin \theta =m\lambda$                                                                                                             (II)

Here, $d$ is the distance between the slits, $\theta$ is the angle and $m$ is the order.

Conclusion:

Substitute $1.00\times {10}^{-6}\text{m}$ for $d$, $1$ for $m$ and $34.9°$ for $\theta$ in equation (II).

    $\begin{array}{c}\left(1.00\times {10}^{-6}\text{m}\right)\sin 34.9°=\left(1\right)\lambda \\ \lambda =5.24\times {10}^{14}\text{Hz}\end{array}$

Therefore, the frequency of light that gives the same angle to the first maximum intensity is $\underset{\_}{5.24\times {10}^{14}\text{Hz}}$.