Chapter 24, Problem 75GP

To determine

Thickness of the film.

Answer to Problem 75GP

Thickness of the film is $t=810\text{ nm}$

Explanation of Solution

Given:

Wavelength, ${\lambda }_1=512\text{ nm}$

Wavelength, ${\lambda }_2=640\text{ nm}$

Refractive index, $n=1.58$

Formula used:

Thickness can be given by: $t=\frac{m\lambda }{2n}$

Calculation:

For wavelength: ${\lambda }_1=512\text{ nm}$

Thickness is given as: $t=\frac{m_1{\lambda }_1}{2n}\mathrm{........................}(1)$

For wavelength: ${\lambda }_2=640\text{ nm}$

Thickness is given as: $t=\frac{m_2{\lambda }_2}{2n}\mathrm{........................}(2)$

From equations (1) and (2),

  $\begin{array}{l}\frac{m_1{\lambda }_1}{2n}=\frac{m_2{\lambda }_2}{2n}\\ m_1{\lambda }_1=m_2{\lambda }_2\\ \frac{m_1}{m_2}=\frac{{\lambda }_2}{{\lambda }_1}=\frac{640\text{ nm}}{\text{512 nm}}\\ \frac{m_1}{m_2}=\frac{5}{4}\end{array}$

Now, using the equation (1),

  $\begin{array}{l}t=\frac{m_1{\lambda }_1}{2n}=\frac{\left(5\right)\left(\text{512 nm}\right)}{2\left(1.58\right)}\\ t=810\text{ nm}\end{array}$

Conclusion:

Thus, the thickness of the film is $t=810\text{ nm}$