An electromagnetic wave has an electric field given by $\vec { E } = - \left( 9.0 \times 10 ^ { 5 } \mathrm {~V} / \mathrm { m } \right) \hat { j } \cos \left[ \left( 9.42 \times 10 ^ { 15 } \mathrm { rad } / \mathrm { s } \right) t - k z \right]$ . The wave number of the magnetic field associated with this wave is
A) $3.1 \times 10 ^ { 7 } \mathrm {~m} ^ { - 1 }$
B) $3.0 \times 10 ^ { - 3 } \mathrm {~m} ^ { - 1 }$
C) $3.0 \times 10 ^ { 5 } \mathrm {~m}$
D) $3.1 \times 10 ^ { 7 } \mathrm {~m}$

Question

Quizplus · Accepted Answer

The wave number $k$ is the same for both the electric and magnetic fields of an electromagnetic wave. Given the electric field equation, the term $kz$ indicates the wave number $k$ is part of the wave's spatial variation, but its value isn't directly provided. However, we can relate $k$ to the frequency $\omega$ using the relationship $\omega = ck$, where $c$ is the speed of light ($3.0 \times 10^8 \, \text{m/s}$). Given $\omega = 9.42 \times 10^{15} \, \text{rad/s}$, solving for $k$ gives $k = \omega / c = (9.42 \times 10^{15} \, \text{rad/s}) / (3.0 \times 10^8 \, \text{m/s}) = 3.14 \times 10^7 \, \text{m}^{-1}$, which is closest to option A.

An Electromagnetic Wave Has an Electric Field Given By the Wave Number of the Magnetic Field Associated with