A closely wound rectangular coil of 80 turns has dimensions of 25.0 cm by 40.0 cm. The plane of the coil is rotated from a position where it makes an angle of 43.0 ∘ with a magnetic field of 1.40 T to a position perpendicular to the field. The rotation takes 0.0700 s . What is the average emf induced in the coil?

Answer :

Answer:

The average induced emf is 50.88 V

Explanation:

Given;

number of turns, N = 80 turns

angle between the plane of the coil and magnetic field, θ =  43°

strength of magnetic field, B = 1.40 T

time to make 80 turns = 0.0700 s

[tex]Emf_{avg} = \frac{N \delta \phi}{dt} = N(\frac{BACos \theta {_f}-BACos \theta {_i}}{dt} )\\\\Emf_{avg} = \frac{NBA(Cos \theta {_f}-Cos \theta {_i})}{dt}[/tex]

where;

A is the area of the coil = 0.25 x 0.4 = 0.1 m²

If the plane makes an angle of  43° with a magnetic field to a position perpendicular to the field, then the initial angle θi = 90 - 43 = 47° and the final angle θf  with the field = 0

[tex]Emf_{avg} = \frac{NBA(Cos \theta {_f}-Cos \theta {_i})}{dt}\\\\Emf_{avg} = \frac{80*1.4*0.1(Cos(0)-Cos(47))}{0.07} = 50.88 \ V[/tex]

Therefore, the average induced emf is 50.88 V

The average emf induced in the coil is; Emf = 50.88 V

We are given;

Number of turns; N = 80 turns

Length of rectangle; L = 40 cm = 0.4 m

Width of rectangle; W = 25 cm = 0.25 m

Magnetic field; B = 1.4 T

Change in rotation time; Δt = 0.07 s

Angle made by the plane of the coil with the magnetic field; θ =  43°

Magnetic field; B = 1.4 T

Now, formula for the average EMF Induced in the coil is given as;

Emf = NΔΦ/Δt

Where; ΔΦ = BAcos θ

where A is area = length x width = 0.25 * 0.4

A = 0.1 m²

Now, since the Angle made by the plane of the coil with the magnetic field was θ =  43°, then we can say that;

Initial angle; θ_i = 90° - 43° = 47°  

final angle; θ_f  = 0°

Thus; ΔΦ = BA(cos θ_f - cos θ_i)

∴Emf = (NBA(cos θ_f - cos θ_i))/Δt

∴Emf = [80 × 1.4 × 0.1(cos 0° - cos 47°)]/0.07

Emf = (80 × 1.4 × 0.1(1 - 0.6820))/0.07

Emf = 50.88 V

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