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 oscillates at double the frequency of a passing electromagnetic wave.

### Step-by-Step Solution:

1. **Understanding the Electromagnetic Wave**:
An electromagnetic wave consists of an oscillating electric field (E) and an oscillating magnetic field (B). The electric field oscillates in one direction (say y-axis) and the magnetic field oscillates in a direction perpendicular to it (say z-axis). The wave travels in the x-direction.

2. **Equations of the Fields**:
The electric field can be represented as:
\[ E_y = E_0 \sin(\omega t - kx) \]
The magnetic field can be represented as:
\[ B_z = B_0 \sin(\omega t - kx) \]
Here, \( \omega \) is the angular frequency and \( k \) is the wave number.

3. **Energy in the Magnetic Field**:
The energy density (energy per unit volume) in the magnetic field is given by:
\[ u_B = \frac{B^2}{2\mu_0} \]
where \( \mu_0 \) is the permeability of free space.

4. **Substituting the Magnetic Field**:
Substituting \( B_z \) into the energy density equation:
\[ u_B = \frac{(B_0 \sin(\omega t - kx))^2}{2\mu_0} \]
This simplifies to:
\[ u_B = \frac{B_0^2 \sin^2(\omega t - kx)}{2\mu_0} \]

5. **Using the Trigonometric Identity**:
We can use the identity \( \sin^2(x) = \frac{1 - \cos(2x)}{2} \):
\[ u_B = \frac{B_0^2}{4\mu_0} (1 - \cos(2(\omega t - kx))) \]

6. **Frequency Analysis**:
The term \( \cos(2(\omega t - kx)) \) indicates that the oscillation occurs at a frequency of \( 2\omega \), which is double the frequency of the original electromagnetic wave.

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

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