The rotating loop in an AC generator is a square l on each side. It is rotated at frequency f in a uniform field of B. The generator is loaded with a resistor of resistance R. Give formulas in terms of l, B, f , and R for the following quantities:
a. the flux Φ (t) through the loop as a function of time
b. the emf Σ(t) induced in the lop
c. the current I(t) induced in the resistor
d. the power R(t) delivered to the resistor
e. the torque τ(t) that must be exerted to rotate the loop.

Explanation:

Area of square loop = L²

Flux Φ = area x magnetic field

= L²B

Frequency = f

angular velocity ω = 2πf

a )

Let at time t = 0 , the magnetic field is making 90 degree with the face of the loop

flux through loop = L²B

After time t , coil will turn by angle ω t = 2πft

Flux through the loop = L²B cosω t

Φ (t) = L²B cosω t

= L²B cos2πft

b )

emf induced e

= - d/dt [Φ (t)]

= - d/dt [ L²B cosω t]

= L²B ω sinω t

= L²B 2πf sin2πft

c )

current = e / R

(L²B ω/ R ) sinω t

Power delivered

P(t) = VI ,

VOLT X CURRENT

= AB ω sinω t X ( AB ω/ R ) sinω t

= L⁴B² 4π²f²/R sin²2πft

e )

torque = MB sinω t

τ(t) = i(L²B ) sinω t

= (L²B ω/ R ) sinω t x (L²B ) sinω t

= (L²B  )²ω/ R sin²ω t

= (L²B  )² 2πf/ R sin²2πft

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