To find the back emf generated by the electric fan motor at its normal operating speed, we need to use Ohm's Law and the concept of power.
First, let's understand the relationship between back emf, current, and voltage:
1. Ohm's Law states that voltage (V) is equal to the product of current (I) and resistance (R): V = I * R.
2. In an electric motor, the back emf (E) is induced due to the rotation of the fan blade. The back emf opposes the flow of current, which is why it is often referred to as the counter-emf. Therefore, the back emf can be represented as E = I * R_back, where R_back is the effective resistance caused by the back emf.
Now, let's use the provided information to solve the problem:
1. We are given that the motor draws only 10.9% of the current it draws when starting. Let's assume the starting current is I_start. Therefore, the current at normal operating speed can be represented as I_normal = 0.109 * I_start.
2. The fan is plugged into a 120.0-V socket, which gives us the value of voltage (V) in Ohm's Law.
3. We need to determine the back emf (E) generated at normal operating speed. Since back emf opposes the flow of current, we can consider the back emf as a voltage drop, assuming the same effective resistance (R_back) for simplicity.
Plugging all the information into Ohm's Law (V = I * R), we get:
V = I_normal * R_back
120.0 V = (0.109 * I_start) * R_back
Now, note that we don't know the starting current (I_start) or the effective resistance (R_back). However, we have enough information to find the ratio between the normal operating speed current and the starting current:
I_normal = 0.109 * I_start
Dividing both sides of the equation by I_start, we have:
I_normal / I_start = 0.109
Now, consider the ratio of the back emf to the voltage:
E / V = R_back / R
Since the effective resistance caused by the back emf should remain constant, we can equate the ratio to the one we found above:
E / V = 0.109
Now, rearrange the equation to find the back emf (E):
E = V * 0.109
E = 120.0 V * 0.109
E = 13.08 V
Therefore, the back emf generated by the electric fan motor at its normal operating speed is approximately 13.08 V.