Q12P - Mathematical Methods in Physical Sciences

Question

In optics, the following expression needs to be evaluated in calculating the intensity of light transmitted through a film after multiple reflections at the surfaces of the film:

${\left(\sum _{n=0}^{\infty }{r}^{2n}cosn\theta \right)}^{\mathbf{2}}\mathbf{+}{\left(\sum _{n=0}^{\infty }{r}^{2n}sin\theta \right)}^{\mathbf{2}}$

Show that this is equal to ${\left|\sum _{n=0}^{\infty }{r}^{2n}{e}^{in\theta }\right|}^{\mathbf{2}}$ and so evaluate it assuming |r| < 1 (r is the fraction of light reflected each time)

Step-by-Step Solution

Answer

The value of the complex number is $z=-1$.

Step 1: Given Information.

The given expression is ${\left(\sum _{\mathrm{n}=0}^{\infty }{\mathrm{r}}^{2\mathrm{n}}\mathrm{cosn\theta }\right)}^2+{\left(\sum _{\mathrm{n}=0}^{\infty }{\mathrm{r}}^{2\mathrm{n}}\mathrm{sin\theta }\right)}^2$.

Step 2: Definition of complex series.

The numbers that are presented in the form of $\mathit{a}\mathbf{+}\mathit{i}\mathit{b}$ where, a,b are real numbers and $\mathbf{\text{'}}\mathit{i}\mathbf{\text{'}}$ is an imaginary number called complex numbers.

Example: 3+2i.

Step 3: Use Euler&rsquo;s formula.

Consider $S=\sum _{n=0}^{\infty }{r}^{2n}{e}^{\left(in\theta \right)}$

 

Use Euler’s formula.

$$\begin{gathered} \begin{array}{ll}S=\sum _{n=0}^{\infty }{r}^{2n}\left[\left(\cos \right(n\theta )+i \sin \left(n\theta \right)\right]& \end{array} \\ \begin{array}{ll}S=\sum _{n=0}^{\infty }{r}^{2n}\cos \left(n\theta \right)+i\sum _{n=0}^{\infty }{r}^{2n} \sin \left(n\theta \right)& \end{array} \end{gathered}$$

Step 4: Find the magnitude.

Find the magnitude of the above equation.

$$\begin{gathered} \begin{array}{ll}\left|S\right|=\left|\sum _{n=0}^{\infty }{r}^{2n}\cos \left(n\theta \right)+i\sum _{n=0}^{\infty }{r}^{2n} \sin \left(n\theta \right)\right|& \end{array} \\ \left|S\right|=\sqrt{{\left(\sum _{n=0}^{\infty }{r}^{2n}\cos \left(n\theta \right)\right)}^2+{\left(i\sum _{n=0}^{\infty }{r}^{2n} \sin \left(n\theta \right)\right)}^2} \\ {\left|S\right|}^2={\left(\sum _{n=0}^{\infty }{r}^{2n}\cos \left(n\theta \right)\right)}^2+{\left(i\sum _{n=0}^{\infty }{r}^{2n} \sin \left(n\theta \right)\right)}^2 \end{gathered}$$

 

Rewrite S.

$\begin{array}{l}S=\sum _{n=0}^{\infty }{\left[r^2{e}^{\left(i\theta \right)}\right]}^n\\ S=\sum _{n=0}^{\infty }{\left[M\right]}^n\\ S=1+{\left[r^2{e}^{\left(i\theta \right)}\right]}^1+{\left[r^2{e}^{\left(i\theta \right)}\right]}^2+{\left[r^2{e}^{\left(i\theta \right)}\right]}^3+···\end{array}$

Step 5: Use the formula for Geometric series.

Use the formula for geometric series

$$\begin{gathered} S=\frac{1}{1-M} \\ S=\frac{1}{1-\left(r^2 exp\left(i\theta \right)\right)} \\ S=\frac{1}{\left[1-r^2 \cos \left(\theta \right)\right]-i{r}^2 \sin \left(\theta \right)} \end{gathered}$$

 

Find its magnitude.

$$\begin{gathered} \left|S\right|=\frac{1}{\left[1-r^2 \cos \left(\theta \right)\right]-i{r}^2 \sin \left(\theta \right)} \\ \left|S\right|=\frac{1}{\sqrt{{\left(1-r^2 \cos \theta \right)}^2-{\left(r^2 \sin \theta \right)}^2}} \\ \left|S\right|=\frac{1}{\sqrt{1-2r^2 \cos {\theta }^2+r^4 {\cos }^2\theta +r^4 {\sin }^{2 }\theta }} \\ \left|S\right|=\frac{1}{\sqrt{1-2r^2 \cos \theta +r^4}} \\ \left|S\right|=\frac{1}{1-2r^2 \cos \theta +r^4} \end{gathered}$$

 

Hence, the given criterion has been proved.