For an electromagnetic wave, which one of the following statements is…

The correct answer is Option (3) → Average energy density of an electromagnetic wave is \( \frac{1}{4}\varepsilon_{0}E_{\mathrm{rms}}^{2} \)

For an electromagnetic wave the direction of propagation is along \( \vec{E}\times\vec{B} \), and the electric and magnetic fields carry equal average energy, each \( \frac{1}{4}\varepsilon_{0}E_{0}^{2} = \frac{1}{2}\varepsilon_{0}E_{\mathrm{rms}}^{2} \).

Option (1) Direction of propagation of electromagnetic wave is same as the direction of \( \vec{E}\times\vec{B} \) vector. — Correct. This is the defining property: \( \vec{E}\times\vec{B} \) points along the direction of travel.

Option (2) Direction of propagation of electromagnetic wave is opposite to the direction of \( \vec{B}\times\vec{E} \) vector. — Correct. \( \vec{B}\times\vec{E} = -\vec{E}\times\vec{B} \), so the propagation direction is opposite to \( \vec{B}\times\vec{E} \).

Option (3) Average energy density of an electromagnetic wave is \( \frac{1}{4}\varepsilon_{0}E_{\mathrm{rms}}^{2} \) — Incorrect. The total average energy density is twice this value (see Option 4); \( \frac{1}{4}\varepsilon_{0}E^{2} \) would hold for the peak value \( E_{0} \) and the electric part only. This is the incorrect statement asked for.

Option (4) Average energy density of electromagnetic wave is \( \varepsilon_{0}E_{\mathrm{rms}}^{2} \) — Correct. Electric and magnetic parts together:

    \( \displaystyle u_{\mathrm{av}} = \frac{1}{2}\varepsilon_{0}E_{\mathrm{rms}}^{2} + \frac{B_{\mathrm{rms}}^{2}}{2\mu_{0}} = \varepsilon_{0}E_{\mathrm{rms}}^{2} = \frac{1}{2}\varepsilon_{0}E_{0}^{2} \)