The electric field component of an electromagnetic wave in vacuum is \( \vec{E} = 3\cos(1.8y + 5.4 \times 10^{8}t)\,\hat{\imath}\,\mathrm{N/C} \), where y is in meter and t is in seconds. The frequency of the wave is (approximately)
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 86 MHz
Compare the wave with the standard form \( E = E_0\cos(ky + \omega t) \). The coefficient of \( t \) is the angular frequency \( \omega \), and the frequency is \( \nu = \omega/2\pi \).
\( \omega = 5.4\times10^{8}\,\mathrm{rad\,s^{-1}} \)
\( k = 1.8\,\mathrm{m^{-1}} \)
\( \displaystyle \nu = \frac{\omega}{2\pi} = \frac{5.4\times10^{8}}{2 \times 3.14} \approx 8.6\times10^{7}\,\mathrm{Hz} = 86\,\mathrm{MHz} \)
As a check, the wave speed is \( \omega/k = 5.4\times10^{8}/1.8 = 3\times10^{8}\,\mathrm{m\,s^{-1}} \), the speed of light in vacuum.