The armature of an ac generator has 100 turns. Each turn is a rectangular loop measuring 8.0 cm by 12 cm. The generator has a sinusoidal voltage output with an amplitude of 24 V. If the magnetic field of the generator is 250 mT,
a) With what frequency does the armature turn?
b) If you double the B and cut in half, how does this affect the amplitude?
To answer these questions, we need to use the formula for the frequency of an AC generator and understand the relationship between magnetic field strength and voltage amplitude.
a) To calculate the frequency (f) with which the armature turns, we can use the formula:
f = N * f_m
Where N represents the number of turns on the armature and f_m represents the frequency of the magnetic field. In this case, N = 100 and f_m is what we are trying to find.
To find f_m, we need to know the time it takes for one complete revolution of the armature. Let's calculate the length of the rectangular loop:
Perimeter of the rectangular loop = 2 * (length + width)
= 2 * (8.0 cm + 12.0 cm)
= 2 * 20.0 cm
= 40.0 cm
To find the duration of one complete revolution, we divide the perimeter of the rectangular loop by the linear speed (v) at which it rotates:
v = v_t = 2πr / T
Where r is the radius and T is the time period of one complete rotation. Since we have a rectangular loop, the radius is the average of the length and width:
r = (8.0 cm + 12.0 cm) / 2
= 10.0 cm
Let's convert the radius to meters:
r = 10.0 cm * (1 m / 100 cm)
= 0.1 m
Now, we can solve for T:
T = 2πr / v_t
= 2π(0.1 m) / v_t
The linear speed (v_t) is the distance traveled divided by the time taken:
v_t = distance / time
The distance traveled in one revolution is the perimeter of the rectangular loop:
distance = 40.0 cm
= 0.4 m
Let's assume the time taken for one revolution is T seconds.
v_t = 0.4 m / T
Now, we can substitute v_t into the equation for T:
T = 2π(0.1 m) / (0.4 m / T)
= 2π(0.1 m) / (0.4 m) * T
= 5πT/2
Solving for T:
2πT = 5πT/2
4πT = 5πT
T = (4πT) / (5π)
T = 4/5
Therefore, the time taken for one complete revolution is T = 4/5 seconds.
Now, we can find the frequency of the armature turning using the formula mentioned earlier:
f = N * f_m
f_m = f / N
f_m = 1 / T
f_m = 1 / (4/5)
f_m = 5/4
Therefore, the frequency with which the armature turns is f_m = 5/4 Hz.
b) If we double the magnetic field strength (B) and cut the number of turns in half, we can determine the effect on the amplitude (A) of the generator's output voltage.
We know that the voltage amplitude is directly proportional to the magnetic field strength (B) and the number of turns (N), and inversely proportional to the frequency (f).
Mathematically, we can express this relationship as:
A ∝ B * N / f
If we double the magnetic field strength (B) and halve the number of turns (N):
New B = 2 * 250 mT = 500 mT
New N = 100 turns / 2 = 50 turns
Plugging in the new values, the equation becomes:
New A ∝ (2 * 500 mT * 50 turns) / f
Since the frequency (f) remains the same, the new amplitude (New A) will be:
New A = 2 * A
Doubling B and halving N will result in doubling the amplitude (A). Therefore, the amplitude will increase by a factor of 2.
To summarize:
a) The armature turns with a frequency of 5/4 Hz.
b) If the magnetic field strength (B) is doubled and the number of turns (N) is halved, the amplitude (A) will be doubled.