*Question E9B07:
What is the difference in radiated power between a lossless antenna with gain and an isotropic radiator driven by the same power?*  

*Correct Answer is  C: They are the same*  

### AI Response: (may contain errors)  

To understand the difference in radiated power between a lossless antenna with gain and an isotropic radiator driven by the same power, let's break down the concepts:

1. **Isotropic Radiator**: This is a theoretical antenna that radiates power equally in all directions. It is used as a reference point for measuring antenna gain.

2. **Directional Antenna with Gain**: This antenna focuses energy in a particular direction, which increases the power density in that direction compared to an isotropic radiator.

### Explanation:

- **Correct Answer (C: They are the same)**:  
  The total power radiated by both antennas is the same because they are driven by the same input power. The gain of a directional antenna does not increase the total power output; it merely redistributes the power to focus more in certain directions.

### Why Other Choices Are Incorrect:

- **Choice A**: "The power radiated from the directional antenna is increased by the gain of the antenna"  
  This is incorrect because the gain does not increase the total power radiated. It only increases the power density in the direction of the gain.

- **Choice B**: "The power radiated from the directional antenna is stronger by its front-to-back ratio"  
  This is incorrect because the front-to-back ratio describes the difference in power between the front and back of the antenna, not the total power radiated.

- **Choice D**: "The power radiated from the isotropic radiator is 2.15 dB greater than that from the directional antenna"  
  This is incorrect because 2.15 dB is the difference between dBi (gain relative to an isotropic radiator) and dBd (gain relative to a dipole), not a difference in total radiated power.

### Useful Formula:

- **Antenna Gain**:  
  Gain is often expressed in decibels (dB) and can be calculated as:  
  $$ G = 10 \log_{10} \left( \frac{\text{Power density in the direction of gain}}{\text{Power density of isotropic radiator}} \right) $$

This formula shows that gain is a measure of how much more power is radiated in a specific direction compared to an isotropic radiator, not an increase in total power.