Consider the following two statements regarding a linearly polarized, plane electromagnetic wave:
The electric field and the magnetic field have equal average values.
The electric energy and the magnetic energy have equal average values.

Options

A. Both A and B are true.
B. A is false but B is true.
C. B is false but A is true.
D. Both A and B are false.

Answer

To solve the question regarding the two statements about a linearly polarized plane electromagnetic wave, we will analyze each statement step by step.

### Step 1: Analyze Statement A
**Statement A:** The electric field and the magnetic field have equal average values.

1. **Understanding the Fields:** In a linearly polarized plane electromagnetic wave, both the electric field (E) and magnetic field (B) oscillate sinusoidally.
2. **Average Values:** The average value of a sinusoidal function over one complete cycle is zero. Therefore, the average value of the electric field (E) over one cycle is zero, and the average value of the magnetic field (B) over one cycle is also zero.
3. **Conclusion for Statement A:** Since both average values are zero, we can conclude that the statement is correct in the context of average values being equal.

### Step 2: Analyze Statement B
**Statement B:** The electric energy and the magnetic energy have equal average values.

1. **Understanding Energy in Electromagnetic Waves:** The energy density of the electric field (U_E) is given by \( U_E = \frac{1}{2} \epsilon_0 E^2 \) and the energy density of the magnetic field (U_B) is given by \( U_B = \frac{1}{2} \frac{B^2}{\mu_0} \).
2. **Relationship Between E and B:** In an electromagnetic wave, the magnitudes of the electric field and magnetic field are related by the equation \( E = cB \), where \( c \) is the speed of light.
3. **Average Energy Calculation:** When we calculate the average energy densities over one cycle, we find that the average electric energy density is equal to the average magnetic energy density. This means that the total electromagnetic energy is equally divided between the electric and magnetic fields.
4. **Conclusion for Statement B:** Therefore, the statement is also correct.

### Final Conclusion
Both statements A and B are true.

### Summary of the Solution
- Statement A: True (average values of E and B are both zero).
- Statement B: True (average electric energy equals average magnetic energy).