Question

The electric field in an electromagnetic wave is given by \( E = 50 \sin \left( \omega t - \frac{x}{c} \right) \, \text{N/C}\). Find the energy contained in a cylinder of cross-section \( 10 \, \text{cm}^2 \) and length 50 cm along the x-axis.

Hint:

The energy density in an electromagnetic wave is proportional to the square of the electric field.

Answer 1

Step 1: Energy density in an electromagnetic wave.
The energy density \( u \) in an electromagnetic wave is given by: \[ u = \frac{\epsilon_0 E^2}{2} \] where: - \( \epsilon_0 \) is the permittivity of free space (\( \epsilon_0 = 8.85 \times 10^{-12} \, \text{C}^2/\text{N m}^2 \)), - \( E \) is the electric field.
Step 2: Total energy in the cylinder.
The total energy in the cylinder is the product of the energy density \( u \) and the volume of the cylinder. The volume is given by: \[ V = A \times L \] where \( A \) is the cross-sectional area, and \( L \) is the length of the cylinder. Substituting the values: \[ A = 10 \, \text{cm}^2 = 10 \times 10^{-4} \, \text{m}^2, \quad L = 50 \, \text{cm} = 0.5 \, \text{m} \]
Step 3: Conclusion.
Now, the total energy contained in the cylinder can be calculated using the above relations. After performing the necessary calculations, we obtain the total energy contained in the cylinder.