Which of the following physical quantities contained in a small volume oscillates at double the frequency of passing electromagnetic wave?

Options

A. Electric field
B. Magnetic field
C. Magnetic energy
D. All of these

Answer

To solve the question, we need to identify which physical quantity in a small volume oscillates at double the frequency of the passing electromagnetic wave.

### Step-by-Step Solution:

1. **Understanding Electromagnetic Waves**:
- An electromagnetic wave consists of oscillating electric (E) and magnetic (B) fields. These fields are perpendicular to each other and to the direction of wave propagation.
- For our analysis, let's assume the electromagnetic wave is traveling along the x-axis, with the electric field oscillating along the y-axis and the magnetic field oscillating along the z-axis.

2. **Frequency of the Electromagnetic Wave**:
- Let the angular frequency of the electromagnetic wave be denoted as \( \omega \). The frequency \( f \) is related to angular frequency by the equation \( f = \frac{\omega}{2\pi} \).

3. **Expressions for Electric and Magnetic Fields**:
- The electric field can be expressed as:
\[ E_y = E_0 \sin(\omega t - kx) \]
- The magnetic field can be expressed as:
\[ B_z = B_0 \sin(\omega t - kx) \]
- Here, \( E_0 \) and \( B_0 \) are the amplitudes of the electric and magnetic fields, respectively.

4. **Energy Densities**:
- The energy density associated with the electric field is given by:
\[ u_E = \frac{1}{2} \epsilon_0 E^2 \]
- The energy density associated with the magnetic field is given by:
\[ u_B = \frac{1}{2} \frac{B^2}{\mu_0} \]

5. **Magnetic Energy Density**:
- For a small volume \( dV \), the magnetic energy density can be expressed as:
\[ u_B = \frac{B_0^2}{2\mu_0} \sin^2(\omega t - kx) \]
- Using the identity \( \sin^2(x) = \frac{1 - \cos(2x)}{2} \), we can rewrite the magnetic energy density:
\[ u_B = \frac{B_0^2}{4\mu_0} (1 - \cos(2(\omega t - kx))) \]

6. **Frequency of Magnetic Energy**:
- The term \( \cos(2(\omega t - kx)) \) indicates that the magnetic energy density oscillates at a frequency of \( 2\omega \), which is double the frequency of the electromagnetic wave.

7. **Conclusion**:
- Therefore, the physical quantity that oscillates at double the frequency of the passing electromagnetic wave is the **magnetic energy density**.

### Final Answer:
The physical quantity that oscillates at double the frequency of the passing electromagnetic wave is **magnetic energy density**.

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