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Chapter 28: Problem 50

A toroidal solenoid has an inner radius of 12.0 \(\mathrm{cm}\) and an outer radius of 15.0 \(\mathrm{cm} .\) It carries a current of 1.50 A. How many equally spaced turns must it have so that it will produce a magnetic field of 3.75 mT at points within the coils 14.0 \(\mathrm{cm}\) from its center?

Short Answer

Expert verified
The toroid must have approximately 87500 turns.

Step by step solution

01

Understand the problem

The problem involves understanding the given toroidal solenoid's parameters and calculating the required number of turns to produce a specified magnetic field at a certain radius.
02

Identify relevant formulas

For a toroidal solenoid, the magnetic field at a radial distance r from the center is given by:\[B = \frac{\mu_0 N I}{2 \pi r}\]where \(B\) is the magnetic field, \(\mu_0\) is the permeability of free space \((4\pi \times 10^{-7} \, \mathrm{T\cdot m/A})\), \(N\) is the number of turns, \(I\) is the current, and \(r\) is the radial distance from the center.
03

Substitute known values

We are given that \(B = 3.75 \, \mathrm{mT} = 3.75 \times 10^{-3} \, \mathrm{T}\), \(I = 1.50 \, \mathrm{A}\), and \(r = 14.0 \, \mathrm{cm} = 0.14 \, \mathrm{m}\). Substitute these into the formula:\[3.75 \times 10^{-3} = \frac{4\pi \times 10^{-7} \times N \times 1.50}{2\pi \times 0.14}\]
04

Simplify the equation

First, cancel out \(2\pi\) from both sides. Then, solve for \(N\):\[3.75 \times 10^{-3} = \frac{4 \times 1.50 \times N \times 10^{-7}}{0.14}\]
05

Solve for N

Rearrange the equation to solve for \(N\):\[N = \frac{3.75 \times 10^{-3} \times 0.14}{4 \times 1.50 \times 10^{-7}}\]Calculate \(N\):\[N = \frac{3.75 \times 0.14}{6 \times 10^{-7}} = \frac{0.525}{6 \times 10^{-7}}\]\[N \approx 8.75 \times 10^4\]Therefore, the toroid must have approximately 87500 turns.

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Key Concepts

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

Magnetic Field
The magnetic field is a fascinating aspect of electromagnetic theory. It describes the invisible force that arises from moving charges or magnetic materials. In the case of a toroidal solenoid, the magnetic field is crucial because it is the desired output for various applications like transformers or inductors.

The strength of the magnetic field inside a solenoid is not only dependent on the current flowing through its wire but also on the number of turns of the wire and the solenoid's shape.
  • Uniformity: Inside a toroidal coil, the magnetic field is uniform along the path equidistant from the center.
  • Equations: In a toroidal solenoid, the magnetic field strength at a point inside the coil can be calculated using the formula: \[ B = \frac{\mu_0 N I}{2 \pi r} \] where \( \mu_0 \) is the permeability of free space, \( N \) is the number of turns, \( I \) is the current, and \( r \) is the radius from the center.
  • Factors Affecting: The magnetic field decreases as the radial distance from the center increases, following an inverse relationship with \( r \) as per the formula.

Understanding these principles helps in designing solenoids for specific magnetic field requirements.
Toroidal Coil
A toroidal coil, or toroidal solenoid, is constructed by winding wire into a doughnut-shaped form. This unique structure contrasts with a typical cylindrical solenoid and offers specific advantages.

The main feature of a toroidal coil is that it confines the magnetic field lines within the core material, minimizing external magnetic fields. This makes toroidal coils incredibly efficient in applications that require a compact magnetic field.
  • Design: The coil's doughnut shape leads the magnetic flux to circulate without escaping, except for minimal leakage.
  • Efficiency: Toroidal coils are popular for their ability to produce strong magnetic fields with minimal energy loss, thanks to reduced energy dissipation into the surrounding environment.
  • Applications: These coils are often used in electronic devices, power transformers, and inductors due to their compact size and high efficiency.
The design and characteristics of toroidal coils make them a preferred choice when designing intricate magnetic field systems.
Number of Turns
The number of turns in a coil refers to how many times the wire is wound around the solenoid's core. It plays a significant role in determining the properties of the produced magnetic field.

In relation to a toroidal solenoid, the turns determine not just the strength, but also the uniformity of the magnetic field within the solenoid's path.
  • Contribution to Field Strength: Increasing the number of turns increases the length of wire interacting with the magnetic field, thus increasing the field's strength proportionally.
  • Calculation: As given in the formula \( B = \frac{\mu_0 N I}{2 \pi r} \), the number of turns \( N \) is directly proportional to the magnetic field \( B \). This relationship means that doubling the turns will also double the magnetic field strength for a fixed current and radius.

  • Design Consideration: When designing a solenoid, engineers must balance the number of turns to achieve the required magnetic field strength while ensuring that the coil remains efficient and feasible.
Understanding the influence of the number of turns is essential for tailoring the solenoid to desired specifications.

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Most popular questions from this chapter

A \(-4.80-\mu \mathrm{C}\) charge is moving at a constant speed of \(6.80 \times 10^{5} \mathrm{m} / \mathrm{s}\) in the \(+x\) -direction relative to a reference frame. At the instant when the point charge is at the origin, what is the magnetic-field vector it produces at the following points: (a) \(x=0.500 \mathrm{m}, y=0, z=0 ;\) (b) \(x=0\) \(y=0.500 \mathrm{m}, \quad z=0 ; \quad\) (c) \(x=0.500 \mathrm{m}, \quad y=0.500 \mathrm{m}, \quad z=0\) (d) \(x=0, y=0, z=0.500 \mathrm{m} ?\)

A \(+6.00-\mu \mathrm{C}\) point charge is moving at a constant \(8.00 \times 10^{6} \mathrm{m} / \mathrm{s}\) in the \(+y\) -direction, relative to a reference frame. At the instant when the point charge is at the origin of this reference frame, what is the magnetic-field vector \(\vec{B}\) it produces at the following points: (a) \(x=0.500 \mathrm{m}, y=0, \quad z=0 ;\) (b) \(x=0\) \(y=-0.500 \mathrm{m}, z=0 ; \quad(\mathrm{c}) x=0, \quad y=0, z=+0.500 \mathrm{m} ;\) (d) \(x=0, y=-0.500 \mathrm{m}, z=+0.500 \mathrm{m} ?\)

BIO Currents in the Brain. The magnetic field around the head has been measured to be approximately \(3.0 \times 10^{-8}\) G. Although the currents that cause this field are quite complicated, we can get a rough estimate of their size by modeling them as a single circular current loop 16 \(\mathrm{cm}\) (the width of a typical head) in diameter. What is the current needed to produce such a field at the center of the loop?

A long, straight wire lies along the \(y\) -axis and carries a current \(I=8.00 \mathrm{A}\) in the \(-y\) -direction (Fig. \(\mathrm{E} 28.23\) ). In addition to the magnetic field due to the current in the wire, a uniform magnetic field \(\vec{\boldsymbol{B}}_{0}\) with magnitude \(1.50 \times 10^{-6} \mathrm{T}\) is in the \(+x\) -direction What is the total field (magnitude and direction) at the following points in the \(x z\) -plane: (a) \(x=0, z=\) \(1.00 \mathrm{m} ;\) (b) \(x=1.00 \mathrm{m}, \quad z=0\) (c) \(x=0, z=-0.25 \mathrm{m} ?\)

A closed curve encircles several conductors. The line integral \(\oint \vec{B} \cdot d \vec{l}\) around this curve is \(3.83 \times 10^{-4} \mathrm{T} \cdot \mathrm{m}\) . (a) What is the net current in the conductors? (b) If you were to integrate around the curve in the opposite direction, what would be the value of the line integral? Explain.
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