(a) Write down a 1D wave equation for an electromagnetic wave in vacuum for an electric field polarized along the y-axis and propagating along the z-axis.(b) Write down a solution for the electric field that satisfies this equation.(c) What is the phase velocity of this wave?(d) Assuming that the E-field solution in (b) is a sine function, find the B-field.

Answered: (a) Write down a 1D wave equation for…

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

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(a) Write down a 1D wave equation for an electromagnetic wave in vacuum for an electric field polarized along the y-axis and propagating along the z-axis.

(b) Write down a solution for the electric field that satisfies this equation.

(c) What is the phase velocity of this wave?

(d) Assuming that the E-field solution in (b) is a sine function, find the B-field.

Expert Solution

Step 1

Hello. Since your question has multiple sub-parts, we will solve the first three sub-parts for you. If you want the remaining sub-part to be solved, then please resubmit the whole question and specify the sub-part you want us to solve.

(a)

For the wave polarized along the y-axis, its electric field vector (E) varies along the y-axis.

Also, if the wave propagates along the z-axis, this means that the k-vector is along the z-axis. Also, the mutually perpendicular nature of the E-field, the B-field, and the k-vector implies that the B-field must be along the x-axis.

The one-dimensional wave equation of such a wave can be deduced from its three-dimensional wave equation obtained by taking the curl of E twice and by using Maxwell’s equations in the vacuum as follows:

$\begin{array}{rcl} \nabla \times (\nabla \times E) & = & \nabla \times (- \frac{\partial B}{\partial t}) (∵ \nabla \times E = - \frac{\partial B}{\partial t}) \\ \nabla (\nabla \cdot E) - \nabla^{2} E & = & - \frac{\partial}{\partial t} (\nabla \times B) (∵ \nabla \times (\nabla \times E) = \nabla (\nabla \cdot E) - \nabla^{2} E) \\ \nabla^{2} E & = & \frac{\partial}{\partial t} (μ_{0} ε_{0} \frac{\partial E}{\partial t}) (∵ \nabla \cdot E = 0, \nabla \times B = μ_{0} ε_{0} \frac{\partial E}{\partial t}) \\ \frac{\partial^{2} E}{\partial z^{2}} = μ_{0} ε_{0} \frac{\partial^{2} E}{\partial t^{2}} \end{array}$

Here, μ₀ and ε₀ are the standard vacuum permeability and permittivity, respectively.

Since the wave propagates along the z-axis, only the z-term of the Laplacian is non-zero.

Step 2

(b)

The solution of the above wave equation is harmonic.

The simplest of its forms may be expressed as follows:

$E = E_{0} \cos (k x - ω t) \hat{y}$

Here, E₀ and ω are the E-field’s amplitude and the wave’s angular frequency, respectively.

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