Suppose that a 0.275 m radius, 500 turn coil produces an average emf of 11800 V when rotated one-fourth of a revolution in 4.42 ms, starting from its plane being perpendicular to the magnetic field. Required:
Find the magnetic field strength needed to induce an average emf of 10,000 V.

The magnetic field strength : 0.372 T

Explanation:

The equation of the induced emf is given by the following equation,

( Equation 1 ) emf = - N ( ΔФ / Δt ) - where N = number of turns of the coil, ΔФ = change in the magnetic flux, and Δt = change in time

The equation for the magnetic flux is given by,

( Equation 2 ) Ф = BA( cos( θ ) ) - where B = magnetic field, A = area, and θ = the angle between the normal and the magnetic field

The area of the circular coil is a constant, as well as the magnetic field. Therefore the change in the magnetic flux is due to the angle between the normal and the magnetic field. Therefore you can expect the equation for the change in magnetic flux to be the same as the magnetic flux, but only that there must be a change in θ.

( Equation 3 ) ΔФ = BA( Δcos( θ ) )

Now as the coil rotates one-fourth of a revolution, θ changes from 0 degrees to 90 degrees. The " change in cos θ " should thus be the following,

Δcos( θ ) = cos( 90 ) - cos( 0 )

= 0 - 1 = - 1

Let's substitute that value in the third equation,

( Substitution of Δcos( θ ) previously, into Equation 3 )

ΔФ = BA( - 1 ) = - BA

Remember the first equation? Well if the change in the magnetic flux = - BA, then through further substitution, the emf should = - N( - BA ) / Δt. In other words,

emf =  - N( - BA ) / Δt,

emf = NBA / Δt,

B = ( emf )Δt / NA

Now that we have B, the magnetic field strength, isolated, let's solve for the area of the circular coil and substitute all known values into this equation.

Area ( A ) = πr²,

= π( 0.275 )² = 0.2376 m²,

B = ( 10,000 V )( 4.42 10⁻³ s ) / ( 500 )( 0.2376 m² ) = ( About ) 0.372 T

The magnetic field strength : 0.372 T

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