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Chapter 28: Problem 53

A wooden ring whose mean diameter is 14.0 \(\mathrm{cm}\) is wound with a closely spaced toroidal winding of 600 turns. Compute the magnitude of the magnetic field at the center of the cross section of the windings when the current in the windings is 0.650 \(\mathrm{A}\) .

Short Answer

Expert verified
The magnetic field is approximately \(2.23 \times 10^{-3} \mathrm{T}\).

Step by step solution

01

Understanding the Problem

We need to find the magnetic field at the center of a toroidal winding, which is shaped like a donut and consists of 600 turns of wire with a current of 0.650 A. The mean diameter of the toroid is given as 14.0 cm.
02

Calculate the Mean Radius

First, we convert the mean diameter of the toroid to radius since the formula for the magnetic field in a toroid uses radius. The mean radius \( r \) is half the diameter: \( r = \frac{14.0\, \mathrm{cm}}{2} = 7.0 \; \mathrm{cm} = 0.070 \, \mathrm{m} \).
03

Apply the Toroid Magnetic Field Formula

The formula for the magnetic field \( B \) inside a toroid is given by: \[ B = \frac{\mu_0 N I}{2\pi r} \] where \( \mu_0 \) is the permeability of free space \( 4\pi \times 10^{-7} \frac{\mathrm{T}\cdot\mathrm{m}}{\mathrm{A}} \), \( N \) is the number of turns, \( I \) is the current, and \( r \) is the mean radius.
04

Substitute Known Values

Substitute the known values into the formula: \( N = 600 \), \( I = 0.650 \mathrm{A} \), \( r = 0.070 \mathrm{m} \). Thus, the magnetic field \( B \) becomes:\[ B = \frac{(4\pi \times 10^{-7} \mathrm{T}\cdot\mathrm{m/A}) \times 600 \times 0.650}{2 \pi \times 0.070} \]
05

Simplify and Calculate

Simplify and calculate the expression:\( B = \frac{4 \times 10^{-7} \times 600 \times 0.650}{0.070} \)Calculate this using the given values:\[ B = \frac{1.56 \times 10^{-4}}{0.070} \approx 2.23 \times 10^{-3} \mathrm{T} \]

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Key Concepts

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

Toroidal Winding
A toroid is often described as a doughnut-shaped object covered with multiple wire windings. These tightly wound coils create a loop where electric current can flow. This configuration of wire is referred to as a toroidal winding. The design of a toroid is beneficial for creating strong magnetic fields confined within the core.
  • In a toroid, wires are wound closely in a circular path around a central core.
  • The design allows for minimal magnetic field leakage outside the toroid, making it efficient for electromagnetic applications.
  • The winding pattern establishes a clear, concentrated magnetic field inside the core due to the symmetry of the turns.
For the exercise, we have a wooden toroid with a winding of 600 turns. The number of turns directly affects the strength of the magnetic field produced.
Magnetostatics
Magnetostatics involves the study of magnetic fields in systems where currents are steady, meaning they do not change with time. In this context, it acts as the magnetic counterpart to electrostatics, focusing on the behavior and interactions of magnetic fields under constant conditions.
  • Magnetostatics handles situations where currents, such as in a toroid's winding, are constant.
  • It simplifies the analysis of magnetic fields because it disregards changing electric fields or electromagnetic waves.
  • The approach allows us to utilize principles and laws, like Ampere's Law, to predict the behavior and strength of magnetic fields around steady currents.
In the problem, we consider a consistent current of 0.650 A flowing through 600 turns in the toroidal winding, fitting perfectly into the magnetostatics framework.
Ampere's Law
Ampere's Law plays a pivotal role in understanding magnetic fields, especially in closed loop configurations like a toroid. The law states that the magnetic field created by a current-carrying conductor is proportional to the current times the number of turns in the path.
  • Mathematically, Ampere's Law is expressed as \( \oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc} \), where \( \oint \vec{B} \cdot d\vec{l} \) represents the integral of the magnetic field B along a closed path.
  • In simpler terms, the magnetic field along any closed path surrounding the current is determined by the product of the current and the number of loops it traverses.
  • It is particularly useful in symmetrical arrangements like toroids, helping to calculate the magnetic field inside the loop accurately.
For the toroid in our exercise, Ampere's Law simplifies the calculation of the magnetic field to a simple function of the current, number of turns, and radius.
Permeability of Free Space
The concept of permeability of free space \( \mu_0 \) is essential in describing how a magnetic field distributes within and around conductors under steady current conditions. It is a constant representing the capability of a vacuum to support magnetic fields.
  • Typically, \( \mu_0 \) is defined as \( 4\pi \times 10^{-7} \frac{\mathrm{T \cdot m}}{\mathrm{A}} \).
  • This constant is an intrinsic property of free space that amplifies the magnetic effect of currents in materials not influencing the magnetic field.
  • Understanding \( \mu_0 \) ensures precise magnetic field calculations, as seen in our toroid example.
In the given exercise, using \( \mu_0 \) allows us to compute the magnetic field generated by the toroidal winding accurately.

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Most popular questions from this chapter

The current in the windings of a toroidal solenoid is 2.400 A. There are 500 turns, and the mean radius is 25.00 \(\mathrm{cm} .\) The toroidal solenoid is filled with a magnetic material. The magnetic field inside the windings is found to be 1.940 T. Calculate (a) the relative permeability and (b) the magnetic susceptibility of the material that fills the toroid.

Two long, straight conducting wires with linear mass density \(\lambda\) are suspended from cords so that they are each horizontal, parallel to each other, and a distance \(d\) apart. The back ends of the wires are connected to each other by a slack, low-resistance connecting wire. A charged capacitor (capacitance \(C )\) is now added to the system; the positive plate of the capacitor (initial charge \(+Q_{0}\) ) is connected to the front end of one of the wires, and the negative plate of the capacitor (initial charge \(-Q_{0} )\) is connected to the front end of the other wire (Fig. P28.87). Both of these connections are also made by slack, low-resistance wires. When the connection is made, the wires are pushed aside by the repulsive force between the wires, and each wire has an initial horizontal velocity of magnitude \(v_{0} .\) Assume that the time constant for the capacitor to discharge is negligible compared to the time it takes for any appreciable displacement in the position of the wires to occur. (a) Show that the initial speed \(v_{0}\) of either wire is given by $$ v_{0}=\frac{\mu_{0} Q_{0}^{2}}{4 \pi \lambda R C d} $$ where \(R\) is the total resistance of the circuit. (b) To what height \(h\) will each wire rise as a result of the circuit connection?

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 solenoid is designed to produce a magnetic field of 0.0270 T at its center. It has radius 1.40 \(\mathrm{cm}\) and length \(40.0 \mathrm{cm},\) and the wire can carry a maximum current of 12.0 A. (a) What minimum number of turns per unit length must the solenoid have? (b) What total length of wire is required?

A long solenoid with 60 turns of wire per centimeter carries a current of 0.15 A. The wire that makes up the solenoid is wrapped around a solid core of silicon steel \(\left(K_{\mathrm{m}}=5200\right) .\) (The wire of the solenoid is jacketed with an insulator so that none of the current flows into the core.) (a) For a point inside the core, find the magnitudes of (i) the magnetic field \(\vec{B}_{0}\) due to the solenoid current; (ii) the magnetization \(\vec{M} ;\) (iii) the total magnetic field \(\vec{\boldsymbol{B}}\) . (b) In a sketch of the solenoid and core, show the directions of the vectors \(\vec{\boldsymbol{B}}, \vec{\boldsymbol{B}}_{0},\) and \(\vec{M}\) inside the core.
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