91Ó°ÊÓ

Each of these problems consists of Concept Questions followed by a related quantitative Problem. The Concept Questions involve little or no mathematics. They focus on the concepts with which the problems deal. Recognizing the concepts is the essential initial step in any problem-solving technique. Concept Question You have a wire of length \(L\) from which to make the square coil of a dc motor. In a given magnetic field a coil of \(N\) turns, each with area \(A\), produces more torque when its total effective area of \(N A\) is greater rather than smaller. This follows directly from Equation \(21.4 .\) Is more torque obtained by using the length of wire to make a single-turn coil or a two-turn coil, or is the torque the same in each case? Explain. Problem The length of the wire is \(L=1.00 \mathrm{~m} .\) The current in the coil is \(I=1.7 \mathrm{~A},\) and the magnetic field of the motor is \(0.34 \mathrm{~T}\). Find the maximum torque when the wire is used to make a single- turn square coil and a two-turn square coil. Verify that your answers are consistent with your answer to the Concept Question.

Step by step solution

01

Concept Understanding

The torque on a coil in a magnetic field is given by \( \tau = NIA B \sin \theta \), where \( N \) is the number of turns, \( I \) is the current, \( A \) is the area of the coil, \( B \) is the magnetic field, and \( \theta \) is the angle between the magnetic field and the normal to the coil. To maximize torque, \( \sin \theta \) is 1 (or \( \theta = 90^\circ \)). The effective area is given by \( NA \). More torque is obtained when the effective area \( NA \) is larger.

02

Calculate Single-Turn Coil Area

For a single-turn square coil, the wire forms a perimeter of \( 4s = L \), where \( s \) is the side length. Solving gives \( s = \frac{L}{4} = \frac{1.00}{4} = 0.25 \) m. The area \( A \) is \( s^2 = (0.25)^2 = 0.0625 \) m\(^2\).

03

Calculate Single-Turn Coil Torque

Using \( \tau = NIA B \), and substituting \( N = 1 \), \( A = 0.0625 \), \( I = 1.7 \), and \( B = 0.34 \), we get \( \tau = 1 \times 1.7 \times 0.0625 \times 0.34 = 0.036125 \) Nm.

04

Calculate Two-Turn Coil Side Length

For a two-turn coil, the total wire length per coil turn is \( \frac{L}{2} \), so \( 4s = \frac{L}{2} \). Solving gives \( s = \frac{L}{8} = \frac{1.00}{8} = 0.125 \) m. The area of each turn \( A \) is \( (0.125)^2 = 0.015625 \) m\(^2\).

05

Calculate Two-Turn Coil Torque

Substituting \( N = 2 \), \( A = 0.015625 \), \( I = 1.7 \), and \( B = 0.34 \) into \( \tau = NIA B \), we get \( \tau = 2 \times 1.7 \times 0.015625 \times 0.34 = 0.0180625 \) Nm.

06

Conclusion

The torque for a single-turn coil is greater (0.036125 Nm) than for a two-turn coil (0.0180625 Nm). This confirms that more torque is obtained when the effective coil area \( NA \) is larger, as with a single-turn coil, since the area \( A \) for single-turn is larger than that of two smaller turns.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Single-Turn Coil

A single-turn coil refers to a loop of wire where the wire only wraps around once to complete a single loop. This configuration is straightforward and makes the calculation of its properties simple. The length of the wire determines the size of the coil. In this exercise, the wire length is given as 1 meter. Thus, if made into a square coil, each side of the square would be the total length divided by four, resulting in a single-turn coil with side lengths of 0.25 meters each.

• Perimeter of single-turn coil: 4s
• Side length of coil: s = L/4
• Total wire length: 1.00 m

This single-turn coil presents a relatively large coil area due to the larger side length, resulting in a larger effective area. A key aspect of single-turn coils is that they have fewer constraints, allowing the area per turn to be maximized, thus enhancing the torque generated in a magnetic field.

Two-Turn Coil

A two-turn coil involves the wire wrapping around twice to form two loops. This configuration doubles the number of turns but results in each individual loop having a smaller area compared to the single-turn coil, as the total wire length gets divided among both turns. With a total wire length of 1 meter, each turn now uses 0.5 meters of wire because:

• Total wire length per turn: L/2
• Perimeter of two-turn coil: 2 * 4s
• Side length per turn: s = L/8

For a two-turn square coil, each side of the square would be approximately 0.125 meters. This results in a smaller area for each turn, ultimately affecting the torque produced. While increasing the number of turns can enhance the coil’s magnetic effect, the resulting decrease in area limits the maximum torque achievable.

Effective Coil Area

The concept of effective coil area is vital in understanding the torque a coil can generate. It is the product of the number of turns, N, and the area of the coil, A, for each turn. Thus, effective coil area is mathematically expressed as \( NA \). This measurement is crucial because:

• It directly influences the torque in the magnetic field.
• Larger effective areas can produce greater torque.

When comparing coil setups, like a single- vs. a two-turn coil, the effective coil area helps in determining which configuration might produce more torque. In our scenario, the single-turn coil produced a greater effective area (0.0625 m²) than the two smaller turns (0.015625 m² each), meaning a larger total effective area, benefiting the generation of more torque.

Magnetic Field

A magnetic field is the environment around a magnetic material or electrical current in which magnetic forces act. In this problem, the coil is subjected to a magnetic field strength of 0.34 Tesla (T). The role of the magnetic field is pivotal as it interacts with the coil to produce torque. The force experienced by the coil in a magnetic field can be described using the equation: \[ \tau = NIA B \sin \theta \]where:

• \( B \) = magnetic field strength
• \( I \) = current through coil
• \( A \) = area of the coil
• \( \theta \) = angle between magnetic field and coil normal

A strong magnetic field, like 0.34 T here, amplifies the torque potential of the system, making maximization of the effective coil area even more critical for optimizing performance.

Coil Torque

Torque generated by a coil in a magnetic field demonstrates the coil’s potential to produce a rotating effect. Torque is a result of the interaction between the electric current flowing through the coil and the external magnetic field. The coil torque can be computed with the formula: \[ \tau = NIA B \sin \theta \]where \( \theta \) is the angle between the plane of the coil and the magnetic field. Maximum torque is achieved when the angle \( \theta \) is 90 degrees, making \( \sin \theta = 1 \).
To analyze for maximum torque in both single and two-turn coils, we see:

• Single-turn coil's torque: 0.036125 Nm
• Two-turn coil's torque: 0.0180625 Nm

The results indicate that a larger effective coil area, as seen in the single-turn coil, produces significantly greater torque. Understanding this relationship is fundamental in applications like electric motors, where efficient torque generation is crucial.