antennaelectromagnetism
For this question in my view the the output power should be \$1 mW\$ since the antenna is lossless,and so no ohmic power should be dissipated and the radiated power should equal the power fed.
\$\eta_r=\frac{P_{rad}}{P_{in}}\$,here antenna is lossless so \$\eta_r=1\$ so \$P_{rad}=P_{in}=1mW\$
But the answer is \$4mW\$.Why is it like that can anyone explain please?
Putting the antenna behind the board affects its performance.
Your board seems to contain a ground plane directly below the antenna, this can also adversely affect its performance, the ground plane in this case stops part of the signal from reaching the antenna.
The board size and layout also affects the antenna performance. If I am not mistaken, you have a monopole antenna from Lynx technologies (Or Laird, or Antenna Factor, same company), This antenna requires large (>10cm) ground for optimal performance.
If you can rotate the jack 90 degrees counter clockwise, you might get a significant improvement in reception range.
Personally I'm also almost certain that this is a folded loop antenna. It's really similar to this one (found at http://ok1ike.c-a-v.com/soubory/200anten.htm):
The only differences between the two lie in the fact that yours doesn't use capacitive coupling for tuning (because it's obviously tuned for a fixed frequency) and the target frequencies are completely different too. In fact I'm almost certain that what you have is not an FM (nor VHF) antenna because it's way too small for that. My guess would be some early Wi-Fi, some GSM or (since you said that it comes from older equipment) even WiMAX design whose wavelengths could roughly match the dimensions of the antenna based on your picture. The ground plane could be there for extra shielding against interference, perhaps to improve reception (obviously this is not a TX antenna).
Best Answer
As you've correctly surmised, the amount of power radiated from the antenna, physically, is 1 milliwatt. This makes sense since the antenna is lossless (100% efficiency), you've fed in 1 milliwatt, so by conservation of energy the antenna is radiating at 1 milliwatt.
What I think the question is means to ask for is the EIRP, or Effective Isotropic Radiated Power. The EIRP is a measure of much power a hypothetical isotropic (perfectly omnidirectional) antenna must radiate so that the power measured in the far field of the main lobe of the directional antenna is equal to the power of the isotropic antenna measured at the same distance. This is a useful figure for doing a link budget.
If the question is asking for EIRP, then 4 milliwatts is correct:
$$10log_{10}(1 mW) = 0 dBm$$ $$0 dBm + 6 dB = 6 dBm$$ $$10^\frac{6 dBm}{10} = 4 mW$$
If this is a homework assignment, you should write a note to your instructor. The question is ambiguous and should clearly state that it's looking for the EIRP.