A 600-loop circular armature coil with a diameter of 9.0 cm rotates at 190 rev/s in a uniform magnetic field of strength 0.55 T .
What is the rms voltage output of the generator?
Express your answer to two significant figures and include the appropriate units.
To calculate the rms voltage output of the generator, we can use Faraday's law of electromagnetic induction, which states that the induced voltage (V) in a coil is equal to the rate of change of magnetic flux (Φ) through the coil.
The magnetic flux through a coil is given by the equation:
Φ = B * A * cos(θ)
where B is the magnetic field strength, A is the area of the coil, and θ is the angle between the magnetic field and the normal to the coil's surface.
In this case, the coil is circular, so its area (A) can be calculated using the formula:
A = π * r^2
where r is the radius of the coil.
Given: Number of loops (N) = 600 Diameter of the coil = 9.0 cm Radius (r) = diameter / 2 = 9.0 cm / 2 = 4.5 cm = 0.045 m Rotation frequency (f) = 190 rev/s Magnetic field strength (B) = 0.55 T
First, let's calculate the area of the coil:
A = π * r^2 = π * (0.045 m)^2
Next, let's calculate the rate of change of magnetic flux (dΦ/dt) by multiplying the magnetic field strength (B) by the change in the magnetic flux, which is the product of the area (A) and the rotation frequency (f):
dΦ/dt = B * A * f
Finally, the rms voltage output (V) can be calculated by multiplying the rate of change of magnetic flux (dΦ/dt) by the number of loops (N):
V = N * dΦ/dt
Let's plug in the values and calculate the rms voltage output:
A = π * (0.045 m)^2 dΦ/dt = B * A * f V = N * dΦ/dt
A = π * (0.045 m)^2 dΦ/dt = (0.55 T) * A * (190 rev/s) V = (600) * dΦ/dt
Calculating A:
A = π * (0.045 m)^2 ≈ 0.0063617 m^2
Calculating dΦ/dt:
dΦ/dt = (0.55 T) * (0.0063617 m^2) * (190 rev/s) ≈ 0.038599 V·s
Calculating V:
V = (600) * (0.038599 V·s) ≈ 23.16 V
Therefore, the rms voltage output of the generator is approximately 23.16 V.
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