The coil of the ac generator rotates at a frequency of f =60.0 Hz and
develops an emf of 120 V (rms). The coil has an area of A = 3.0 x 10 -3 m 2 and consists of N = 500 turns.
a. Find the magnitude of the magnetic field in which the coil rotates.
b. If the total resistance of the wire is 12Ω, what is the maximum induced current when the output terminals
are connected to a low resistance conductor?
To find the magnitude of the magnetic field in which the coil rotates, we can use Faraday's law of electromagnetic induction. According to this law, the induced emf (ε) is equal to the rate of change of magnetic flux (Φ) through the coil.
The equation for Faraday's law is:
ε = - N * dΦ/dt
Where:
ε = induced emf
N = number of turns
dΦ/dt = rate of change of magnetic flux
In this case, the emf is given as 120 V, the frequency as 60.0 Hz, and the number of turns as 500. The area of the coil is also given as 3.0 x 10^(-3) m^2.
First, let's find the rate of change of magnetic flux, dΦ/dt.
Since the area and magnetic field (B) are related by the equation:
Φ = B * A
where:
Φ = magnetic flux
A = area
B = magnetic field
We can rearrange this equation to solve for B:
B = Φ / A
The rate of change of magnetic flux is given by:
dΦ/dt = ΔΦ / Δt
For rotational motion, ΔΦ is simply the change in angle (Δθ) multiplied by the magnetic flux per unit angle (dΦ/dθ). In this case, since the coil rotates at a frequency of 60.0 Hz, we have:
ΔΦ = dΦ/dθ * Δθ = (2π/N) * Δθ
Substituting this into the equation for dΦ/dt, we get:
dΦ/dt = (2π/N) * Δθ / Δt
Since f = 1 / T, where T is the period, we have:
Δt = 1 / f
Substituting this into the equation for dΦ/dt, we get:
dΦ/dt = (2π/N) * Δθ * f
Finally, substituting this into Faraday's law, we have:
ε = - N * (2π/N) * Δθ * f
Simplifying, we get:
ε = - 2π * Δθ * f
Solving for the magnetic field B, we get:
B = ε / (2π * Δθ * f * A)
Now we can substitute the given values into this equation.
For part (a), the magnitude of the magnetic field in which the coil rotates:
B = (120 V) / (2π * Δθ * (60.0 Hz) * (3.0 x 10^(-3) m^2))
To find the maximum induced current when the output terminals are connected to a low resistance conductor (part b), we can use Ohm's law. The maximum current (I) is given by:
I = ε / R
Where:
I = maximum current
ε = induced emf
R = total resistance
Substituting the given values, we can find the maximum induced current.
a. To find the magnitude of the magnetic field in which the coil rotates, we can use the equation for the induced emf in a generator:
ε = NABω
Where ε is the induced emf, N is the number of turns in the coil, A is the area of the coil, B is the magnetic field strength, and ω is the angular frequency (2πf).
Plugging in the given values:
ε = 120 V
N = 500 turns
A = 3.0 x 10^(-3) m^2
f = 60.0 Hz
We can rearrange the equation to solve for B:
B = ε / (NAω)
Substituting the values:
B = (120 V) / (500 turns * (3.0 x 10^(-3) m^2) * (2π * 60.0 Hz))
Calculating this expression will give us the magnitude of the magnetic field in which the coil rotates.
b. To find the maximum induced current, we need to consider the total resistance of the wire. The maximum current occurs when the resistance in the circuit is minimized.
Using Ohm's Law, we can find the maximum current:
I = V / R
Where I is the current, V is the voltage, and R is the resistance.
Plugging in the values:
V = 120 V
R = 12 Ω
Calculating this expression will give us the maximum induced current when the output terminals are connected to a low resistance conductor.