Discussion
In an electromagnetic wave, the electric field of amplitude 6.2 V/m oscillates with a frequency of 2.4 × 1010 Hz. Estimate energy density of the wave :
(ε = 8.85 × 10-12 C2 / Nm2)
Answer: C
The key gives \(3.4 \times 10^{-10}\) J/m\(^{3}\), which is the peak total energy density, \(\varepsilon_{0}E_{0}^{2}\).
In a plane wave the electric and magnetic fields carry equal energy, so the total energy density is \(u = \varepsilon_{0}E^{2}\). At the instant the field is at its amplitude:
\[ \begin{aligned} u &= \varepsilon_{0} E_{0}^{2} \\ &= 8.85 \times 10^{-12} \times (6.2)^{2} \\ &= 8.85 \times 10^{-12} \times 38.44 \\ &\approx 3.4 \times 10^{-10}\ \text{J/m}^{3} \end{aligned} \]
The frequency does not enter the calculation. The time-averaged value would be half of this, about \(1.7 \times 10^{-10}\) J/m\(^{3}\), which is not among the options. 1.4, 2.4 and 4.4 do not come from either formula.
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