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प्रश्न
When an AC source is connected to an ideal inductor show that the average power supplied by the source over a complete cycle is zero.
उत्तर
In an AC circuit containing only an ideal inductor, the current i lags behind the emf e by a phase angle of π/2 rad. Here, for e = e0 sin ωt, we have,
i = i0 sin (ωt - π/2)
Instantaneous power, P = ei
= (e0 sin ωt)[i0(sin ωt cos π/2 - cos ωt sin π/2)]
= - e0i0 sin ωt cos ωt as cos π/2 = 0 and sin π/2 = 1
Average power over one cycle,
`"P"_"av" = "work done in one cycle"/"time for one cycle"`
`= (int_0^"T" "P dt")/("T") = (- int_0^"T" "e"_0"i"_0 sin omega"t" cos omega"t" "dt")/("T")`
`= (- "e"_0"i"_0)/"T" int_0^"T"` sin ωt cos ωt dt
Now, `int_0^"T"` sin ωt cos ωt dt = 0
∴ Pav = 0
[Note: For reference, see the answer to Q. 6. The proof should be written as part of the answer.]
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