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# Obtain an expression for average power dissipated in a purely resistive A.C. circult. Define power factor of the circuit and state its value for purely resistive A.C. circult. - HSC Science (General) 12th Board Exam - Physics

ConceptPower in Ac Circuit: the Power Factor

#### Question

Obtain an expression for average power dissipated in a purely resistive A.C. circult. Define power factor of the circuit and state its value for purely resistive A.C. circult.

#### Solution

Expression for average power in purely resistive circuit:-

i. Let, e = e0 sinωt be applied e.m.f across a resistor of resistance ‘R’ as shown in figure. At certain instant, current I = I0 sin ωt is flowing through the resistor. In this case both ‘e’ and ‘I’ are in phase.

ii. Instantaneous power in circuit is given by,

P = eI = (e0 sinωt) (I0 sin ωt)

∴ P = e0I0 sin2 ωt                                                            ..........................(1)

iii. Average power for a complete cycle can be obtained by integrating equation (1).

thereforeP_"av"="work done in one cycle"/"time for one cycle"

=(int_0^TPdt)/T

=(int_0^Te_0I_0sin^2omegatdt)/T

therefore P_"av"=(e_0I_0)/T(int_0^Tsin^2omegatdt)                                               ...............(2)

iv. "But"int_0^Tsin^2omegatdt=int_0^T((1-cos2omegat)/2)dt

=1/2[int_0^Tdt-int_0^Tcos2omegatdt]

=1/2[t-(sine2omegat)/(2omega)]_0^T

=1/2[T-(sin2omegaxx(2pi)/omega)/(2omega)]

=1/2[T-(sin4pi)/(2omega)]

=T/2" ........................................."[becausesin4pi=0]

From equation (2),

thereforeP_"av"=(e_0I_0)/T(T/2)

=(e_0I_0)/2

=(1/2)e_0I_0

thereforeP_"av"=e_0/sqrt2xxI_0/sqrt2

= erms x Irms

∴ Pav = erms x Irms

Power factor:-

Power factor of an a.c circuit is defined as the cosine of the phase difference between the applied voltage and the circuit current.

For purely resistive circuit, power factor = 1

Is there an error in this question or solution?

#### APPEARS IN

2014-2015 (October) (with solutions)
Question 5.1 | 7.00 marks

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Solution Obtain an expression for average power dissipated in a purely resistive A.C. circult. Define power factor of the circuit and state its value for purely resistive A.C. circult. Concept: Power in Ac Circuit: the Power Factor.
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