The magnetic field in a plane electromagnetic wave is given by \( B_y=2\times10^{-7}\sin\left(0.5\times10^{3}\,x+1.5\times10^{11}\,t\right)\,\mathrm{T} \) (x in metres, t in seconds). The electromagnetic wave is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → a microwave
For a wave written as \( \sin(kx+\omega t) \), the wave number is \( k=2\pi/\lambda \) and the angular frequency is \( \omega=2\pi\nu \). The wavelength decides the band of the spectrum.
From the equation:
\( k=0.5\times10^{3}\,\mathrm{rad\,m^{-1}} \)
\( \omega=1.5\times10^{11}\,\mathrm{rad\,s^{-1}} \)
The wavelength is:
\( \begin{aligned} \lambda &= \frac{2\pi}{k} \\ &= \frac{2\times3.14}{0.5\times10^{3}\,\mathrm{m^{-1}}} \\ &\approx 1.26\times10^{-2}\,\mathrm{m} \end{aligned} \)
The frequency is:
\( \begin{aligned} \nu &= \frac{\omega}{2\pi} \\ &= \frac{1.5\times10^{11}}{2\times3.14}\,\mathrm{Hz} \\ &\approx 2.4\times10^{10}\,\mathrm{Hz} \end{aligned} \)
From the above, \( \lambda\approx1.26\,\mathrm{cm} \). This lies inside the microwave band, 0.1 m to 1 mm. Hence, the wave is a microwave.
NCERT (2026-27):
Electromagnetic Spectrum
