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Obtain the relation for the current I=I(...

Obtain the relation for the current `I=I_(o)sin(omegat-pi//2)` for a pure inductor across which an alternating `emfe=e_(o)sinomegat` is applied.

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Let an alternating emf is applied acorss across a pure inductor.
`E=E_(0)sin omega t " " ….(i)`
As the current i in the coil varies continuously, an opposing emf is induced which is `(Ldi)/(dt)`.

`:.E_(0)sin omega t-"L"(di)/(dt)=0`
`(Ldi)/(dt)=E_(0)sin omegat`
Integrating both sides of equation
`int di= int(E_(0))/(L) sin omega t dt `
`i=(E_(0))/(L)((-cos omega t)/(omega))= (-E)/(omega L)cos omega t`
`=(E_(0))/(omegaL) sin (omega t -pi//2)`
A maximum valueof `sin(omega t-pi//2)` is 1, so `E_(0)//omegaL` is the maximum current in the circuit.
So, `i=i_(0)sin (omegat-pi//2)" " ....(ii)`
On comparing equation (i) and (ii), we find that in case of a pure inductor the current lags behind the applied emf by a phase angle of `pi//2` i.e., `90^(@)`.
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