Chapter 19.1, Problem 13SSC

(a)

To determine

The second thinnest width of the soap film.

Answer to Problem 13SSC

The second thinnest width of the soap film is $324.0\text{ nm}$ .

Explanation of Solution

Given:

The index of refraction of the soap solution is $n_{\text{film}}=1.33$ .

The wavelength of light is $\lambda =575\text{ nm}$ .

Formula used:

The expression for the extra distance that the second wave must travel through the thin film to create the needed interference is,

  $2d=\left(m+\frac{1}{2}\right)\frac{\lambda }{n_{\text{film}}}$

Here, $m$ is an integer, $\lambda$ is the wavelength, and $n_{\text{film}}$ is the index of refraction of a thin film.

Calculation:

Consider the integer number as $m=1$ for second thinnest thickness. The second thinnest width of the soap film is,

  $\begin{array}{l}2d=\left(m+\frac{1}{2}\right)\frac{\lambda }{n_{\text{film}}}\\ 2d=\left(1+\frac{1}{2}\right)\frac{575\text{nm}}{1.33}\\ 2d=648\\ d=324.0\text{ nm}\end{array}$

Conclusion:

Thus, the second thinnest width of the soap film is $324.0\text{ nm}$ .

(b)

To determine

The other widths of the soap film.

Answer to Problem 13SSC

The other widths of the soap film is $108\left(2m+1\right)\text{ nm}$ .

Explanation of Solution

Given:

The index of refraction of the soap solution is $n_{\text{film}}=1.33$ .

The wavelength of light is $\lambda =575\text{ nm}$ .

Formula used:

The expression for the extra distance that the second wave must travel through the thin film to create the needed interference is,

  $2d=\left(m+\frac{1}{2}\right)\frac{\lambda }{n_{\text{film}}}$

Here, $m$ is an integer, $\lambda$ is the wavelength, and $n_{\text{film}}$ is the index of refraction of a thin film.

Calculation:

Consider the integer number as $m$ for other widths. The other widths of the soap film is,

  $\begin{array}{l}2d=\left(m+\frac{1}{2}\right)\frac{\lambda }{n_{\text{film}}}\\ 2d=\left(\frac{2m+1}{2}\right)\frac{575\text{nm}}{1.33}\\ 2d=216\left(2m+1\right)\\ d=108\left(2m+1\right)\text{ nm}\end{array}$

Conclusion:

Thus, the other widths of the soap film is $108\left(2m+1\right)\text{ nm}$ .