The coil of a generator has a radius of 0.12 m. When this coil is unwound, the wire from which it is made has a length of 5.5 m. The magnetic field of the generator is 0.10 T, and the coil rotates at an angular speed of 25 rad/s. What is the peak emf of this generator?
Number of turns in the coil :
N = 5.5/(2 pi R) = 7.3
Coil area = A = pi R^2
B = 0.1 T
R = 0.12 m
Phi = magnetic flux = B A N cos theta
Induced EMF =d(Phi)/dt = - B A N sin theta * w
where w = d(theta) dt = 25 radians per second
Peak EMF = B A N w
To determine the peak emf of the generator, we can use Faraday's law of electromagnetic induction, which states that the induced emf in a wire is equal to the rate of change of magnetic flux through the wire.
The magnetic flux through a coil can be calculated using the formula:
Φ = B * A * cosθ
Where:
Φ = magnetic flux
B = magnetic field strength
A = area of the coil
θ = angle between the magnetic field and the perpendicular to the coil
In this case, the angle between the magnetic field and the perpendicular to the coil is 0° (cos 0° = 1).
The area of the coil can be calculated using the formula:
A = π * r^2
Where:
A = area of the coil
r = radius of the coil
Let's calculate the area of the coil first:
A = π * (0.12 m)^2
A = 0.0451 m^2
Now, we can calculate the magnetic flux through the coil:
Φ = (0.10 T) * (0.0451 m^2)
Φ = 0.00451 Wb
The next step is to calculate the rate of change of the magnetic flux with respect to time, which is the angular speed of the coil (ω) multiplied by the number of turns in the coil (N):
dΦ/dt = N * ω
The number of turns in the coil is equal to the length of the wire divided by the circumference of the coil:
N = wire length / coil circumference
Circumference of the coil = 2 * π * r
N = 5.5 m / (2 * π * 0.12 m)
N = 7.279 turns
Now, we can calculate the rate of change of the magnetic flux:
dΦ/dt = (7.279 turns) * (25 rad/s)
dΦ/dt = 181.975 Wb/s
Finally, the peak emf of the generator is equal to the rate of change of the magnetic flux:
Emf = dΦ/dt = 181.975 V
Therefore, the peak emf of this generator is 181.975 volts.