The energy contained in a small volume through which an electromagnetic wave is passing oscillates with

Options

A. zero frequency
B. the frequency of the wave
C. half the frequency of the wave
D. double the frequency of the waves.

Answer

To solve the problem, we need to analyze how the energy contained in a small volume through which an electromagnetic wave is passing oscillates. The energy density of an electromagnetic wave is given by the formula:

\[ U = \frac{1}{2} \epsilon_0 E^2 + \frac{1}{2 \mu_0} B^2 \]

where \( U \) is the energy density, \( E \) is the electric field, \( B \) is the magnetic field, \( \epsilon_0 \) is the permittivity of free space, and \( \mu_0 \) is the permeability of free space.

### Step-by-Step Solution:

1. **Write the expressions for electric and magnetic fields**:
- The electric field \( E \) can be expressed as:
\[ E = E_0 \sin(kz - \omega t) \]
- The magnetic field \( B \) can be expressed as:
\[ B = B_0 \sin(kz - \omega t) \]

2. **Substitute the fields into the energy density formula**:
- Substitute \( E \) and \( B \) into the energy density equation:
\[ U = \frac{1}{2} \epsilon_0 (E_0 \sin(kz - \omega t))^2 + \frac{1}{2 \mu_0} (B_0 \sin(kz - \omega t))^2 \]

3. **Simplify the expression**:
- This becomes:
\[ U = \frac{1}{2} \epsilon_0 E_0^2 \sin^2(kz - \omega t) + \frac{1}{2 \mu_0} B_0^2 \sin^2(kz - \omega t) \]

4. **Use the identity for sine squared**:
- Recall that \( \sin^2(x) = \frac{1 - \cos(2x)}{2} \):
\[ U = \frac{1}{2} \epsilon_0 E_0^2 \left(\frac{1 - \cos(2(kz - \omega t))}{2}\right) + \frac{1}{2 \mu_0} B_0^2 \left(\frac{1 - \cos(2(kz - \omega t))}{2}\right) \]

5. **Combine the terms**:
- This can be simplified to:
\[ U = \left(\frac{\epsilon_0 E_0^2}{4} + \frac{B_0^2}{4 \mu_0}\right) - \left(\frac{\epsilon_0 E_0^2 + B_0^2}{4} \cos(2(kz - \omega t))\right) \]

6. **Identify the oscillation frequency**:
- The term \( \cos(2(kz - \omega t)) \) indicates that the energy oscillates at a frequency of \( 2\omega \), which is double the frequency of the electromagnetic wave.

### Conclusion:
The energy contained in the small volume oscillates with a frequency that is double that of the electromagnetic wave.

### Final Answer:
The energy contained in a small volume through which an electromagnetic wave is passing oscillates with double the frequency of the wave.

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