Question

The electric field in an electromagnetic wave is given as \[ \vec{E} = 20 \sin \left( \omega t - \frac{x}{c} \right) \, \hat{j} \, \text{N/C} \] where \( \omega \) and \( c \) are angular frequency and velocity of electromagnetic wave respectively. The energy contained in a volume of \( 5 \times 10^4 \, \text{m}^3 \) will be (Given \( \epsilon_0 = 8.85 \times 10^{-12} \, \text{C}^2 / \text{Nm}^2 \)):

• A. \( 8.85 \times 10^{-13} \, \text{J} \)
• B. \( 17.7 \times 10^{-13} \, \text{J} \)
• C. \( 8.85 \times 10^{-10} \, \text{J} \)
• D. \( 28.5 \times 10^{-13} \, \text{J} \)

Hint:

For electromagnetic waves, use the formula for energy density \( u = \frac{1}{2} \epsilon_0 E_0^2 \) and multiply by volume to find the total energy.

Answer 1

The Correct Option is A

The energy density in an electromagnetic wave is given by the formula: \[ u = \frac{1}{2} \epsilon_0 E_0^2 \] Where \( E_0 \) is the peak electric field. Substituting the given values: \[ u = \frac{1}{2} \times 8.85 \times 10^{-12} \times (20)^2 = 8.85 \times 10^{-13} \, \text{J/m}^3 \] The total energy contained in a volume \( V = 5 \times 10^4 \, \text{m}^3 \) is: \[ \text{Energy} = u \times V = 8.85 \times 10^{-13} \times 5 \times 10^4 = 8.85 \times 10^{-13} \, \text{J} \] Thus, the energy is \( 8.85 \times 10^{-13} \, \text{J} \).