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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?

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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.

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    \$\begingroup\$ I don't even think it is ambiguous, I think it is just wrong! Total power radiated is not EIRP it is the power passing thru the surface of a sphere surrounding the antenna, and as such is clearly 1mW (And is by definition independent of the directivity). I would certainly be querying this. \$\endgroup\$ – Dan Mills Sep 19 '17 at 10:17
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The key is there is gain from the antenna. Every +3 dB is 2x power so the radiated power for 6dB (3dB+3dB) is 4x (2x • 2x)

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  • \$\begingroup\$ See my edit in the question and tell me whats wrong with that? \$\endgroup\$ – Rohit Sep 19 '17 at 2:05

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