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Chapter 22: Problem 31

The magnetic field of an EM wave is given by \(B_{y}=\) $B_{\mathrm{m}} \sin (k z+\omega t), B_{x}=0,\( and \)B_{z}=0 .$ (a) In what direction is this wave traveling? (b) Write expressions for the components of the electric field of this wave.

Short Answer

Expert verified
Answer: The wave is traveling in the x-z plane direction. The electric field components are \(E_x = cB_m\sin(kz+\omega t)\), \(E_y = 0\), and \(E_z = 0\).

Step by step solution

01

(a) Direction of Wave Propagation

To find in which direction the wave is traveling, we need to find the direction perpendicular to the magnetic field \(B_y\) which is given. The magnetic field has components \(B_x = 0\), \(B_y = B_m\sin(kz+\omega t)\), and \(B_z = 0\). The wave vector \(k\) is aligned with the propagation direction of the wave. Since \(B_x\) and \(B_z\) are both zero, the wave vector should be aligned along the x and z directions. Hence, the wave is traveling in the x-z plane direction.
02

(b) Electric Field Component Expressions

Following the relation between the electric field and magnetic field in an EM wave: \(E = cB\) We can write the equation for electric field components: 1. For \(E_x\): Since \(B_y \neq 0\) and the wave is propagating in the x-z plane, we have \(E_x = cB_y = cB_m\sin(kz+\omega t)\) 2. For \(E_y\): Since \(B_x = 0\), the electric field component along the y direction should have no dependence on the magnetic field, \(E_y = 0\) 3. For \(E_z\): Since \(B_z = 0\), the electric field component along the z direction should have no dependence on the magnetic field, \(E_z = 0\) Therefore, the components of the electric field of this wave are given by: \(E_x = cB_m\sin(kz+\omega t)\), \(E_y = 0\), and \(E_z = 0\).

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