Chapter 37, Problem 20P

(a)

To determine

The position of the $n=50$ fringe.

Answer to Problem 20P

The position of the $n=50$ fringe is $\underset{\_}{22.6\text{cm}}$.

Explanation of Solution

Write the expression to find the position of the $n=50$ fringe.

    $y=50y_{\text{bright}}$

Here, $y_{\text{bright}}$ is the position of the bright fringe.

Conclusion:

Substitute $4.52\times {10}^{-3}\text{m}$ for $y_{\text{bright}}$ in the above equation.

    $\begin{array}{c}y=50\left(4.52\times {10}^{-3}\text{m}\right)\\ =22.6\text{cm}\end{array}$

Therefore, the position of the $n=50$ fringe is $\underset{\_}{22.6\text{cm}}$.

(b)

To determine

The tangent of the angle the first order bright fringe makes with respect to the line extending from the point midway between the slits to the center of the central maximum.

Answer to Problem 20P

The tangent of the angle the first order bright fringe makes with respect to the line extending from the point midway between the slits to the center of the central maximum is $\underset{\_}{2.51\times {10}^{-3}}$.

Explanation of Solution

The following figure gives the system description.

Write the expression for the tangent of the angle the first order bright fringe makes with respect to the line extending from the point midway between the slits to the center of the central maximum.

    $\tan {\theta }_1=\frac{y_{\text{bright}}}{L}$

Conclusion:

Substitute $4.52\times {10}^{-3}\text{m}$ for $y_{\text{bright}}$ and $1.80\text{m}$ for $L$ in the above equation.

    $\begin{array}{c}\tan {\theta }_1=\frac{4.52\times {10}^{-3}\text{m}}{1.80\text{m}}\\ =2.51\times {10}^{-3}\end{array}$

Therefore, the tangent of the angle the first order bright fringe makes with respect to the line extending from the point midway between the slits to the center of the central maximum is $\underset{\_}{2.51\times {10}^{-3}}$.

(c)

To determine

The wavelength of the light.

Answer to Problem 20P

The wavelength of the light is $\underset{\_}{6.03\times {10}^{-7}\text{m}}$.

Explanation of Solution

Write the expression for wavelength of the light.

    $\lambda =\frac{d\sin \theta }{m}$

Here, $d$ is the separation between the slits.

Conclusion:

Substitute $2.40\times {10}^{-4}\text{m}$ for $d$, ${\tan }^{-1}\left(2.51\times {10}^{-3}\right)$ for ${\theta }_1$ and $1$ for $m$ in the above equation.

    $\begin{array}{c}\lambda =\frac{\left(2.40\times {10}^{-4}\text{m}\right)\sin \left({\tan }^{-1}\left(2.51\times {10}^{-3}\right)\right)}{1}\\ =6.03\times {10}^{-7}\text{m}\end{array}$

Therefore, the wavelength of the light is $\underset{\_}{6.03\times {10}^{-7}\text{m}}$.

(d)

To determine

The angle for the 50^th-order bright fringe.

Answer to Problem 20P

The angle for the 50^th-order bright fringe is $\underset{\_}{7.21°}$.

Explanation of Solution

Write the expression for angle for the 50^th-order bright fringe.

    ${\theta }_{50}={\sin }^{-1}\left(\frac{50\lambda }{d}\right)$

Conclusion:

Substitute $6.03\times {10}^{-7}\text{m}$ for $\lambda$ and $2.40\times {10}^{-4}\text{m}$ for $d$ in the above equation.

    $\begin{array}{c}{\theta }_{50}={\sin }^{-1}\left(\frac{50\left(6.03\times {10}^{-7}\text{m}\right)}{2.40\times {10}^{-4}\text{m}}\right)\\ =7.21°\end{array}$

Therefore, the angle for the 50^th-order bright fringe is $\underset{\_}{7.21°}$.

(e)

To determine

The position of 50^th-order bright fringe.

Answer to Problem 20P

The position of 50^th-order bright fringe is $\underset{\_}{2.28\text{cm}}$.

Explanation of Solution

Write the expression to find the position of 50^th-order bright fringe

    $y_{50}=L\tan {\theta }_{50}$

Here, $L$ is the distance between the screen and the slits.

Conclusion:

Substitute $7.21°$ for ${\theta }_{50}$ and $1.80\text{m}$ for $L$ in the above equation.

    $\begin{array}{c}{y}_{50}=\left(1.80\text{m}\right)\tan \left(7.21°\right)\\ =2.28\text{cm}\end{array}$

Therefore, the position of 50^th-order bright fringe is $\underset{\_}{2.28\text{cm}}$.

(f)

To determine

The comparison of the answers in part (a) and (e).

Answer to Problem 20P

The answers in part (a) and (e) are close but are not equal.

Explanation of Solution

The non linearity of the fringes in the screen are the reasons that the two answers are close but do not agree exactly with each other.

Therefore, the answers in part (a) and (e) are close but are not equal.