Problem 42 - Physics for Scientists and Engineers with Modern Physics

Question

(1) A motor has an armature resistance of 3.05\(\Omega .\) If it draws 7.20 A when running at full speed and connected to a \(120-\mathrm{V}\) line, how large is the back emf?

Step-by-Step Solution

Answer

The back emf is 98.04 V.

Step 1: Apply Ohm's Law

To find the back emf (Electromotive Force) of the motor, we start by applying Ohm's Law, which relates voltage (V), current (I), and resistance (R): \[ V = I \times R \]Here, the armature resistance \( R \) is given as \( 3.05\, \Omega \), and the current \( I \) is \( 7.20\, A \). Substituting these values gives:\[ V_r = 7.20 \times 3.05 \]This calculates the voltage drop (\( V_r \)) across the armature.

Step 2: Calculate Voltage Drop Across Armature

Perform the multiplication to find the voltage drop across the armature:\[ V_r = 7.20 \times 3.05 = 21.96\, V \]

Step 3: Determine Back EMF

The motor is connected to a \(120\, V\) line. The back emf \( (E_b) \) is the difference between the line voltage and the voltage drop across the armature:\[ E_b = 120\, V - 21.96\, V \]

Step 4: Calculate Back EMF

Subtract the voltage drop from the line voltage to find the back emf:\[ E_b = 120 - 21.96 = 98.04\, V \]

Step 5: Conclusion: Back EMF Value

The back emf of the motor, when running at full speed, is \(98.04\, V\).

Key Concepts

Understanding Ohm's Law

Ohm's Law is a fundamental concept in electrical engineering. It establishes the relationship between voltage, current, and resistance in an electrical circuit. The formula given by Ohm's Law is \[ V = I \times R \]where:

• \(V\) stands for voltage, measured in volts.
• \(I\) is the current, which is measured in amperes (A).
• \(R\) refers to resistance, measured in ohms (\(\Omega\)).

According to the law, the voltage across a conductor is directly proportional to the current flowing through it and the resistance of the conductor. In practical terms, this means if you know any two of these values, you can compute the third. A higher resistance or current will affect the voltage across an element in a circuit. This principle is crucial when determining other valuable measures like the voltage drop across a motor's armature.

Armature Resistance Explained

Armature resistance is a specific type of electrical resistance in a motor's armature winding. The armature is the rotating part of an electric motor, and it plays a critical role in converting electrical energy into mechanical energy.The resistance of the armature winding affects the current flowing through and the motor's efficiency. In the given problem, the armature resistance is 3.05\(\Omega\). This value is essential because it influences the overall voltage drop and helps in calculating the back electromotive force (emf) of a motor.An understanding of armature resistance is essential in designing efficient electrical motors. Lower resistance generally means less energy loss and more efficient performance. In our example, this resistance value is factored into Ohm's Law to find how much voltage is used up in the armature winding.

What is Voltage Drop?

A voltage drop is the reduction in voltage in an electrical circuit when current flows through it. It's important in circuits as it indicates how much potential energy is being used across different components.In our problem, the voltage drop across the armature is calculated using Ohm's Law: \[ V_r = I \times R \]where \( V_r \) is the voltage drop, and substituting the values from the problem, \[ V_r = 7.20 \times 3.05 = 21.96\, V \]This means that 21.96 volts of the total supplied voltage drops over the armature resistance. Understanding voltage drop simplifies down the calculation of other parameters, such as the back emf, by showing how much potential is "consumed" before reaching other parts of the circuit.

Deciphering Electric Current

Electric current is a flow of electrical charge carriers, usually electrons or electron-deficient atoms. In simple terms, it's the flow of electricity along a wire or through a component, like the armature of an electric motor. It is measured in amperes (A). In the context of the problem, a current of 7.20 A is flowing through the motor armature. This current influences how much voltage is utilized across the armature, and plays a part in defining the power being consumed. Electric current can be adjusted by changing either the resistance or the voltage within a circuit, governed by the principles of Ohm's Law. Understanding current, and how it affects circuits, is crucial when designing and troubleshooting electrical systems. In relation to motors, it impacts the efficiency and performance of these devices when drawing power from an external supply.