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

Choices:  
A: The power radiated from the directional antenna is increased by the gain of the antenna  
B: The power radiated from the directional antenna is stronger by its front-to-back ratio  
C: They are the same  
D: The power radiated from the isotropic radiator is 2.15 dB greater than that from the directional antenna    

Correct Answer is  C: They are the same  

Correct answer: C — They are the same

1) Why the correct answer is correct
- For a lossless antenna, all input power is radiated: $P_{\text{rad}} = P_{\text{in}}$.
- Antenna gain does not create power; it redistributes the same total radiated power into certain directions.
- Therefore, a lossless directional antenna and an isotropic radiator, driven with the same power, radiate the same total power. The directional antenna just concentrates it.

2) Why the other choices are incorrect
- A: “Increased by the gain” — Incorrect. Gain increases power density in some directions, not total radiated power. Total power remains $P_{\text{in}}$ for a lossless antenna.
- B: “Stronger by its front-to-back ratio” — Incorrect. Front-to-back ratio describes the pattern (forward vs backward gain), not total radiated power.
- D: “Isotropic is 2.15 dB greater” — Incorrect. The 2.15 dB figure is the difference between dBi and dBd references ($\text{dBi} = \text{dBd} + 2.15$ dB), not a statement about total radiated power.

3) Useful formulas and rules of thumb
- Total radiated power (lossless): $$P_{\text{rad}} = P_{\text{in}}$$
- Isotropic power density at distance $r$: $$S_{\text{iso}} = \frac{P_{\text{rad}}}{4\pi r^2}$$
- Directional antenna power density: $$S(\theta,\phi) = \frac{P_{\text{rad}}\, G(\theta,\phi)}{4\pi r^2}$$ with the average gain condition $$\frac{1}{4\pi}\int_{4\pi} G(\theta,\phi)\, d\Omega = 1$$
- EIRP (lossless): $$\text{EIRP} = P_{\text{in}}\times G_{\text{linear}}$$
- Reference conversion: $$G_{\text{dBi}} = G_{\text{dBd}} + 2.15\ \text{dB}$$

Key idea: Gain focuses, it doesn’t add power.