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Chapter 21: Problem 48

You have a wire of length \(L=1.00 \mathrm{m}\) from which to make the square coil of a dc motor. The current in the coil is \(I=1.7 \mathrm{A},\) and the magnetic field of the motor has a magnitude of \(B=0.34 \mathrm{T}\) . Find the maximum torque exerted on the coil when the wire is used to make a single-turn square coil and a two-turn square coil.

Short Answer

Expert verified
Single-turn torque: 0.0361 Nm; Two-turn torque: 0.0180 Nm.

Step by step solution

01

Understanding Variables and Known Data

We have a wire of length \(L = 1.00 \, \text{m}\). We need to form a coil, which can either be a single-turn or a two-turn square coil. The current through the coil is given as \(I = 1.7 \, \text{A}\), and the magnetic field strength is \(B = 0.34 \, \text{T}\). We'll calculate the maximum torque for each situation.
02

Calculating Side Length for Single-Turn Coil

For a single-turn square coil, the coil will have four sides with each side having the length \( \frac{L}{4} \). This is because the total length of the wire is used up in forming the square.
03

Formula Application for Single-Turn Coil

The side length \(a\) of the square coil for the single-turn is \(a = \frac{L}{4} = \frac{1.00}{4} = 0.25 \, \text{m}\). The area \(A\) of the coil is \(a^2 = (0.25)^2 = 0.0625 \, \text{m}^2\). The torque is given by \( \tau = n imes I \times A \times B \sin(\theta) \), where \(n\) is the number of turns, \(\theta\) is the angle between the normal to the coil and the magnetic field. Maximum torque occurs at \(\theta = 90^\circ\), hence \(\sin(\theta) = 1\).
04

Torque Calculation for Single-Turn Coil

Substitute \(n = 1\), \(A = 0.0625 \, \text{m}^2\), \(I = 1.7 \, \text{A}\), \(B = 0.34 \, \text{T}\) into the torque formula. \[ \tau = 1 \times 1.7 \times 0.0625 \times 0.34 \approx 0.036125 \, \text{Nm}.\]
05

Calculating Side Length for Two-Turn Coil

For a two-turn square coil, each loop uses half the wire length, so \(2a \times 4 = L\). Therefore, each side length \(a\) is \(\frac{L}{8} = \frac{1.00}{8} = 0.125 \, \text{m}\).
06

Formula Application for Two-Turn Coil

For the two-turn coil, \(a = 0.125 \, \text{m}\). The area \(A = (0.125)^2 = 0.015625 \, \text{m}^2\). Substitute \(n = 2\), \(A = 0.015625 \, \text{m}^2\), into the torque formula, with maximum at \(\theta = 90^\circ\), \(\sin(\theta) = 1\).
07

Torque Calculation for Two-Turn Coil

Substitute into the formula for the two-turn coil: \[ \tau = 2 \times 1.7 \times 0.015625 \times 0.34 \approx 0.018025 \, \text{Nm}.\]
08

Comparison of Results

The torque for the single-turn coil is approximately 0.036125 Nm, while for the two-turn coil, it's about 0.018025 Nm.

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Key Concepts

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

DC Motor
A DC motor is a type of electric motor that converts direct current electrical energy into mechanical energy. It works based on the principle of electromagnetism. In the context of our exercise, the DC motor uses one or two square coils made from a wire of length 1 meter. When current flows through these coils, they experience a force in the presence of a magnetic field. This force produces torque, which is the rotational force that allows the motor to spin. Understanding how to calculate this torque is essential to evaluating the motor's performance.
Magnetic Field
A magnetic field is an invisible field that exerts magnetic force on certain materials and charged particles in its vicinity. It is represented by the symbol \(B\) and measured in teslas (T). In our exercise, the magnetic field strength is given as 0.34 T.
  • The coil of a DC motor, when exposed to a magnetic field, will experience magnetic torque. This is due to the interaction between the current in the coil and the magnetic field.
  • The strength and direction of the magnetic field affect how much torque is applied to the coil.
  • Maximum torque is experienced when the coil is perpendicular to the magnetic field.
The evaluation of torque is crucial, as it directly relates to the motor's efficiency and performance.
Torque Calculation
Torque is the measure of rotational force and is calculated using the formula \(\tau = n \times I \times A \times B \times \sin(\theta)\). Each component of this formula plays a significant role:
  • \(n\) is the number of turns in the coil.
  • \(I\) represents the current flowing through the coil.
  • \(A\) is the area of the coil.
  • \(B\) is the magnetic field strength.
  • \(\theta\) is the angle between the coil's plane and the magnetic field.
For maximum torque, \(\theta\) is set to 90掳, making \(\sin(\theta) = 1\). This means the coil's plane is perfectly perpendicular to the magnetic field. Knowing how to apply this formula helps in designing and analyzing motor efficiency.
Single-Turn Coil
A single-turn coil means that the wire forms one full loop in the shape of a square. In our exercise, this configuration uses the entire 1 meter of available wire to make four equal sides. Each side is 0.25 meters long, resulting in a square coil.
  • The area \(A\) of this coil is calculated as \(0.25^2 = 0.0625 \, \text{m}^2\).
  • The coil experiences torque (calculated previously as \(0.036125 \, \text{Nm}\)) when a current flows and interacts with the magnetic field.
A single-turn coil provides a certain amount of torque but is limited in its ability to generate higher torque compared to multiple turns.
Two-Turn Coil
A two-turn coil means that the wire is used to make two loops, effectively doubling the turns but halving the side length due to the same total wire length. In this exercise, each side of the square is 0.125 meters, determined by dividing the wire evenly across two turns.
  • The area \(A\) of one loop is \(0.125^2 = 0.015625 \, \text{m}^2\).
  • The two turns increase the factor \(n\) in the torque calculation, compensating for the smaller coil area.
  • The torque for a two-turn coil in our exercise is calculated as \(0.018025 \, \text{Nm}\).
Despite the increased turns, the smaller coil area results in lower torque than the single-turn coil, showing the importance of geometry in torque optimization.

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Most popular questions from this chapter

The drawing shows two long, straight wires that are suspended from a ceiling. The mass per unit length of each wire is 0.050 \(\mathrm{kg} / \mathrm{m}\) . Each of the four strings suspending the wires has a length of 1.2 \(\mathrm{m}\) . When the wires carry identical currents in opposite directions, the angle between the strings holding the two wires is \(15^{\circ} .\) What is the current in each wire?

A copper rod of length 0.85 \(\mathrm{m}\) is lying on a frictionless table (see the drawing). Each end of the rod is attached to a fixed wire by an unstretched spring that has a spring constant of \(k=75 \mathrm{N} / \mathrm{m}\) . A magnetic field with a strength of 0.16 \(\mathrm{T}\) is oriented perpendicular to the surface of the table. (a) What must be the direction of the current in the copper rod that causes the springs to stretch? (b) If the current is \(12 \mathrm{A},\) by how much does each spring stretch?

mmh Two circular coils are concentric and lie in the same plane. The inner coil contains 140 turns of wire, has a radius of \(0.015 \mathrm{m},\) and carries a current of 7.2 \(\mathrm{A}\) . The outer coil contains 180 turns and has a radius of 0.023 \(\mathrm{m}\) . What must be the magnitude and direction (relative to the current in the inner coil) of the current in the outer coil, so that the net magnetic field at the common center of the two coils is zero?

Particle 1 and particle 2 have masses of \(m_{1}=2.3 \times 10^{-8} \mathrm{kg}\) and \(m_{2}=5.9 \times 10^{-8} \mathrm{kg},\) but they carry the same charge \(q .\) The two particles accelerate from rest through the same electric potential difference \(V\) and enter the same magnetic field, which has a magnitude \(B\). The particles travel perpendicular to the magnetic field on circular paths. The radius of the circular path for particle 1 is \(r_{1}=12 \mathrm{cm} .\) What is the radius (in cm) of the circular path for particle 2\(?\)

A horizontal wire is hung from the ceiling of a room by two massless strings. The wire has a length of 0.20 \(\mathrm{m}\) and a mass of 0.080 kg. A uniform magnetic field of magnitude 0.070 \(\mathrm{T}\) is directed from the ceiling to the floor.When a current of \(I=42 \mathrm{A}\) exists in the wire, the wire swings upward and, at equilibrium, makes an angle \(\phi\) with respect to the vertical, as the drawing shows. Find (a) the angle \(\phi\) and \((b)\) the tension in each of the two strings.
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