A circular loop of radius "a" is rotated at constant angular velocity ω in a uniform magnetic field B.
Find the magnitude of the net induced emf on the loop at any given time
A similar example is below. In your case use pi a^2 instead of Nab
and omega = 2 pi f
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A rectangular coil of N turns and of length a and width b is rotated at frequency f in a
uniform magnetic field B as indicated. The coil is connected to co rotating cylinders against
which metal brushes slide to make contact. (a) Show that the emf matches the given expression.
(b) What value of Nab gives an emf such that its maximum value is 150V. Take the angular
speed to be 60 rev/s and the field to be 0.5 T.
This is a classic induced emf problem. You are being asked to calculate the expression that
describes how a generator works.
We begin by writing out the flux for the loop in the configuration below (shown edge on).
q
B
ΦB = BAcosθ
= BNabcosθ
θ =ω t = 2π f t
ΦB = BAcos2π f t
Note that since the loop is rotating, we have substituted in for the angle in terms of the angular
velocity and then frequency. We can now find the induced emf
ΦB = B Nabcos2π f t
ε = − dΦB/d t
= 2π f BNabsin(2π f t)
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in your case
= omega B pi a^2 sin (omega t)
To find the magnitude of the net induced emf on the loop, we can use Faraday's law of electromagnetic induction, which states that the induced emf in a loop is equal to the rate of change of magnetic flux through the loop.
The magnetic flux through a loop is given by the product of the magnetic field strength, the area of the loop, and the cosine of the angle between the magnetic field and the normal to the loop.
In this case, the magnetic field is uniform and the loop is rotating in the plane perpendicular to the field, so the angle between the magnetic field and the normal to the loop remains constant at 90 degrees. Therefore, the flux through the loop is given by:
Φ = B * A
where B is the magnetic field strength and A is the area of the loop.
The area of a circular loop is given by the formula:
A = π * (radius)^2
In this case, the radius of the loop is "a", so the area of the loop is:
A = π * a^2
Therefore, the flux through the loop is:
Φ = B * π * a^2
As the loop rotates, the magnetic flux changes over time, resulting in an induced emf in the loop. The rate of change of magnetic flux is given by:
dΦ/dt
So, the magnitude of the induced emf is:
|ε| = |dΦ/dt| = |dB/dt * π * a^2|
where the negative sign indicates that the induced emf opposes the change in magnetic flux.
Please note that this equation assumes that the angle between the magnetic field and the normal to the loop remains constant during rotation, and that the loop is a perfect conductor.