Problem 37 - Physics

Question

The 1200 -turn coil in a dc motor has an area per turn of \(1.1 \times 10^{-2} \mathrm{~m}^{2}\). The design for the motor specifies that the magnitude of the maximum torque is \(5.8 \mathrm{~N} \cdot \mathrm{m}\) when the coil is placed in a 0.20 -T magnetic field. What is the current in the coil?

Step-by-Step Solution

Answer

The current in the coil is 2.20 A.

Step 1: Understanding the Problem

We are given a coil in a DC motor with 1200 turns and an area per turn of \(1.1 \times 10^{-2} \mathrm{~m}^{2}\). The maximum torque produced is \(5.8 \mathrm{~N} \cdot \mathrm{m}\) in a magnetic field of \(0.20\) T. We need to find the current in the coil.

Step 2: Formula for Torque

The formula for torque \(\tau\) on a current-carrying coil in a magnetic field is given by \(\tau = n \cdot A \cdot B \cdot I \cdot \sin(\theta)\), where \(n\) is the number of turns, \(A\) is the area per turn, \(B\) is the magnetic field, and \(I\) is the current.\(\theta\) is the angle between the magnetic moment and the field, which is \(90^\circ\) (for max torque), meaning \(\sin(\theta) = 1\).

Step 3: Reorganizing the Formula

To find the current \(I\), reorganize the torque formula: \(I = \frac{\tau}{n \cdot A \cdot B}\).

Step 4: Substituting the Values

Substitute the given values into the equation: \(I = \frac{5.8}{1200 \cdot 1.1 \times 10^{-2} \cdot 0.20}\).

Step 5: Calculating the Current

Perform the calculations: \(I = \frac{5.8}{2.64} = 2.20\) A. Therefore, the current in the coil is 2.20 amperes.

Key Concepts

Torque Calculation in DC Motors

DC motors transform electrical energy into mechanical energy, and torque is a key aspect of this process. Torque is essentially the twisting force that causes the motor's rotor to spin. In a coil inside a DC motor, torque is generated due to the interaction between the electric current flowing through the coil and the magnetic field it's placed within.
The mathematical formula to calculate torque (\( \tau \)) on a current-carrying coil in a magnetic field is:\[ \tau = n \cdot A \cdot B \cdot I \cdot \sin(\theta) \]where:

• \( n \)= number of turns in the coil,
• \( A \)= area of each turn (or loop),
• \( B \)= magnetic field strength,
• \( I \)= current flowing through the coil,
• \( \sin(\theta) \)= sine of the angle between the magnetic field and the coil.

In most cases with maximum torque, the angle \( \theta \) equals \( 90^\circ \), simplifying the equation since \( \sin(90^\circ) = 1 \). This results in:\[ \tau = n \cdot A \cdot B \cdot I \]Understanding this formula allows us to calculate any unknown variable when others are provided, such as determining the current using the known torque, as done in the problem above.

The Role of Magnetic Field in DC Motors

A magnetic field is fundamental to the operation of a DC motor because it interacts with the electric current to produce motion. In a DC motor, the magnetic field can be generated by permanent magnets or electromagnets.
When the coil in the motor, which carries an electric current, is placed within this magnetic field, a force is exerted on the current according to the right-hand rule. This force generates the torque needed to rotate the motor.The strength of the magnetic field greatly influences the motor's performance. A stronger magnetic field (\( B \)) results in greater torque, assuming all other variables remain constant. This is why designing an efficient motor involves careful consideration of the magnetic field's origin and intensity.By understanding the interplay between the current and magnetic field, we gain insight into the mechanics behind the motion generated within DC motors. This interaction exemplifies fundamental physics principles applied to create practical mechanical systems.

Current in the Coil and Its Importance

The current flowing through the coil in a DC motor is vital because it directly impacts the generated torque and, thus, the motor’s overall performance. The current (\( I \)) running through the coil interacts with the magnetic field, allowing the torque calculation as seen in the formula above.
Too much current can cause overheating, reducing efficiency or damaging the motor. Conversely, insufficient current means not enough torque is produced, limiting the motor's capabilities.When calculating the ideal current for a specific torque, as shown in the problem-solving steps, it's essential to understand how each variable—like the coil’s turns and area, and the magnetic field—plays a role. Properly controlling current is critical to optimizing the motor's function, efficiency, and longevity. The solution above illustrates how we can manipulate and determine current to achieve the motor’s desired performance, ensuring that the motor operates efficiently and effectively under various conditions.