*Question E9D01:
How much does the gain of an ideal parabolic reflector antenna increase when the operating frequency is doubled?*  

*Correct Answer is  D: 6 dB*  

### AI Response: (may contain errors)  

To understand how the gain of an ideal parabolic reflector antenna changes when the operating frequency is doubled, we need to consider the relationship between antenna gain and frequency.

### Explanation:

1. **Gain and Frequency Relationship:**
   - The gain of a parabolic reflector antenna is proportional to the square of the frequency. This is because the gain $G$ is related to the effective aperture $A_e$ and the wavelength $\lambda$ by the formula:
     $$ G = \frac{4\pi A_e}{\lambda^2} $$
   - When the frequency is doubled, the wavelength $\lambda$ is halved (since $\lambda = \frac{c}{f}$, where $c$ is the speed of light and $f$ is the frequency).

2. **Effect of Doubling Frequency:**
   - If the frequency is doubled, the wavelength becomes half, and the gain increases by a factor of 4 (since $(\frac{1}{\frac{1}{2}})^2 = 4$).

3. **Gain in Decibels:**
   - Gain increase in decibels (dB) is calculated using:
     $$ \Delta G_{\text{dB}} = 10 \log_{10} \left(\frac{G_{\text{new}}}{G_{\text{old}}}\right) $$
   - Substituting the gain increase factor of 4:
     $$ \Delta G_{\text{dB}} = 10 \log_{10}(4) \approx 6 \text{ dB} $$

### Why the Correct Answer is 6 dB:

- **Correct Answer (D: 6 dB):**  
  Doubling the frequency results in a gain increase of approximately 6 dB due to the relationship between gain and the square of the frequency.

### Why Other Choices are Incorrect:

- **Choice A (2 dB):**  
  A 2 dB increase would correspond to a much smaller change in gain, not consistent with doubling the frequency.

- **Choice B (3 dB):**  
  A 3 dB increase represents a doubling of power, not gain, and does not apply to the frequency doubling scenario.

- **Choice C (4 dB):**  
  A 4 dB increase is less than the actual increase when frequency is doubled, as shown by the calculation.

By understanding these relationships, you can see why the gain increase is 6 dB when the operating frequency of a parabolic reflector antenna is doubled.