A plane electromagnetic wave travelling along the x-direction has a wavelength of 3 mm. The variation in the electric field occurs in the y-direction with an amplitude of \( 66\,\mathrm{V\,m^{-1}} \). The equations for the electric and magnetic fields as functions of x and t are, respectively,
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( E_y=66\cos2\pi\times10^{11}\left(t-\frac{x}{c}\right) \), \( B_z=2.2\times10^{-7}\cos2\pi\times10^{11}\left(t-\frac{x}{c}\right) \)
In an electromagnetic wave, the fields \( \vec{E} \) and \( \vec{B} \) are perpendicular to each other and to the direction of propagation. Their product \( \vec{E}\times\vec{B} \) points along the direction of travel. For a wave along x with \( \vec{E} \) along y, \( \vec{B} \) is along z. The amplitudes are related by \( B_0=E_0/c \), and each field varies as \( \cos\omega(t-x/c) \) with \( \omega=2\pi\nu \).
Taking \( c=3\times10^{8}\,\mathrm{m\,s^{-1}} \) and \( \lambda=3\,\mathrm{mm}=3\times10^{-3}\,\mathrm{m} \):
\( \begin{aligned} \nu &= \frac{c}{\lambda} \\ &= \frac{3\times10^{8}\,\mathrm{m\,s^{-1}}}{3\times10^{-3}\,\mathrm{m}} \\ &= 10^{11}\,\mathrm{Hz} \end{aligned} \)
\( \begin{aligned} \omega &= 2\pi\nu \\ &= 2\pi\times10^{11}\,\mathrm{rad\,s^{-1}} \end{aligned} \)
\( \begin{aligned} B_0 &= \frac{E_0}{c} \\ &= \frac{66\,\mathrm{V\,m^{-1}}}{3\times10^{8}\,\mathrm{m\,s^{-1}}} \\ &= 2.2\times10^{-7}\,\mathrm{T} \end{aligned} \)
From the above, \( \vec{E} \) is along y with amplitude 66 V/m, and \( \vec{B} \) is along z with amplitude \( 2.2\times10^{-7}\,\mathrm{T} \). Both vary with \( \omega=2\pi\times10^{11}\,\mathrm{rad\,s^{-1}} \). Hence, the fields are \( E_y=66\cos2\pi\times10^{11}\left(t-\frac{x}{c}\right) \) and \( B_z=2.2\times10^{-7}\cos2\pi\times10^{11}\left(t-\frac{x}{c}\right) \).
NCERT (2026-27):
Nature of Electromagnetic Waves – “… It is also seen from Maxwell’s equations that the magnitude of the electric and the magnetic fields in an electromagnetic wave are related as
\( B_0=(E_0/c) \)\( \qquad (8.10) \)”
Nature of Electromagnetic Waves – “To find the direction, we note that E is along y-direction and the wave propagates along x-axis. Therefore, B should be in a direction perpendicular to both x- and y-axes. Using vector algebra, E × B should be along x-direction.”