Electric and Magnetic fields oscillate sinusoidally in an Electromagnetic Waves. The oscillating electric field cannot be represented in this way, where the symbols have their usual meaning:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( E = E_0\sin\left[\displaystyle\frac{2\pi}{T}(vz - t)\right] \)
A wave travelling along \( +z \) has the form \( E = E_0\sin(kz - \omega t) \) with the wave number \( k = 2\pi/\lambda \), the angular frequency \( \omega = 2\pi/T \) and the speed \( v = \lambda/T \). Every valid form must reduce to this, and the quantity inside the sine must be dimensionless.
Option (1) \( E = E_0\sin(kz - \omega t) \) — Correct form. This is the standard form of a wave moving along \( +z \).
Option (2) \( E = E_0\sin\left[2\pi\left(\displaystyle\frac{z}{\lambda} - \frac{t}{T}\right)\right] \) — Correct form. \( 2\pi\left(\frac{z}{\lambda} - \frac{t}{T}\right) = \frac{2\pi}{\lambda}z - \frac{2\pi}{T}t = kz - \omega t \).
Option (3) \( E = E_0\sin\left[\displaystyle\frac{2\pi}{\lambda}(z - vt)\right] \) — Correct form. \( \frac{2\pi}{\lambda}(z - vt) = kz - \frac{2\pi v}{\lambda}\,t = kz - \omega t \), since \( v/\lambda \) is the frequency.
Option (4) \( E = E_0\sin\left[\displaystyle\frac{2\pi}{T}(vz - t)\right] \) — Cannot represent the wave. Inside the bracket \( vz \) has the unit \( \mathrm{m^{2}\,s^{-1}} \) while \( t \) has the unit s, so the two cannot be subtracted; the form is dimensionally wrong. The valid form would be \( \frac{2\pi}{T}\left(\frac{z}{v} - t\right) \).
NCERT (2026-27):
Summary – “… 4. Electric and magnetic fields oscillate sinusoidally in space and time in an electromagnetic wave. … For a wave of frequency ν, wavelength λ, propagating along z-direction, we have
\( \displaystyle E = E_x(t) = E_0 \sin(kz - \omega t) = E_0 \sin 2\pi\left(\frac{z}{\lambda} - \nu t\right) = E_0 \sin 2\pi\left(\frac{z}{\lambda} - \frac{t}{T}\right) \)”