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)

Short answer

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

Step-by-step solution

Given Information.

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

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.

Use Euler’s formula.

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

Use Euler’s formula.

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

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.

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

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.