Question

The amplitude of a light wave which is travelling in a medium can be represented by $A \exp \{i(n \omega x / c-\omega t)\}$, where $n$ is the refractive index and $c$ is the velocity of light in vacuo. Show that if the light is attenuated as it passes through the medium then this can be represented by taking $n$ as complex, i.e. in the form $n+i k$. Express $\varepsilon^{\prime}$ and $\varepsilon^{\prime \prime}$ in terms of $n$ and $k$.

Name: Problem 7, Chapter 13: The Solid State: An Introduction to the Physics of Crystals for Students of Physics, Materials Science, and Engineering
Uploaded: 2021-06-13T14:51:19-08:00
Duration: 1 min 18 s
Channel: Ajay Singhal
Description: The amplitude of a light wave which is travelling in a medium can be represented by A {i(n ωx / c-ωt)}, where n is the refractive index and c is the velocity of light in vacuo. Show that if the light is attenuated as it passes through the medium then this can be represented by taking n as complex, i.e. in the form n+i k. Express ε^' and ε^'' in terms of n and k.

   The amplitude of a light wave which is travelling in a medium can be represented by $A \exp \{i(n \omega x / c-\omega t)\}$, where $n$ is the refractive index and $c$ is the velocity of light in vacuo. Show that if the light is attenuated as it passes through the medium then this can be represented by taking $n$ as complex, i.e. in the form $n+i k$. Express $\varepsilon^{\prime}$ and $\varepsilon^{\prime \prime}$ in terms of $n$ and $k$.

The Solid State: An Introduction to the Physics of Crystals for Students of Physics, Materials Science, and Engineering

Harold Max Rosenberg 2nd Edition

Chapter 13, Problem 7

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Solved by Expert Ajay Singhal

06/13/2021

Step 1

Step 1: Start by considering the given expression for the amplitude of a light wave traveling in a medium: \[ A \exp \left\{i\left(\frac{n \omega x}{c} - \omega t\right)\right\} \] where $ A $ is the amplitude, $ \omega $ is the angular frequency, $ x $ is Show more…

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The amplitude of a light wave which is travelling in a medium can be represented by $A \exp \{i(n \omega x / c-\omega t)\}$, where $n$ is the refractive index and $c$ is the velocity of light in vacuo. Show that if the light is attenuated as it passes through the medium then this can be represented by taking $n$ as complex, i.e. in the form $n+i k$. Express $\varepsilon^{\prime}$ and $\varepsilon^{\prime \prime}$ in terms of $n$ and $k$.

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Key Concepts

Complex Permittivity

Complex permittivity characterizes how an electromagnetic field interacts with a material. It is expressed as ? = ?' + i?'', where the real part (?') relates to the stored energy in the medium, and the imaginary part (?'') accounts for the energy loss due to absorption. The relationship between the complex refractive index and complex permittivity is given by ? = (n + ik)², which leads to the expressions ?' = n² - k² and ?'' = 2nk.

Complex Refractive Index

In the description of light propagation through a medium, the refractive index is extended to be a complex number to account for both the phase change and the attenuation of the wave. The real part (n) determines the phase velocity of the wave, while the imaginary part (k) represents the absorption or damping of the wave's amplitude as it travels through the medium.

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