/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 36 A closely wound coil has a radiu... [FREE SOLUTION] | 91影视

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Chapter 28: Problem 36

A closely wound coil has a radius of \(6.00 \mathrm{~cm}\) and carries a current of 2.50 A. How many turns must it have if, at a point on the coil axis \(6.00 \mathrm{~cm}\) from the center of the coil, the magnetic field is \(6.39 \times 10^{-4} \mathrm{~T} ?\)

Short Answer

Expert verified
The coil must have approximately 298 turns.

Step by step solution

01

- Use the Biot-Savart Law

The Biot-Savart Law gives the magnetic field B produced by a current at a particular point. Usually it is expressed as \(B = \frac{{\mu_0 I}}{{2 \pi r}}\), where \(\mu_0\) is the permeability of free space, \(I\) is the current and \(r\) is the distance from the current. Here we need to derive a slightly different version, as we are not at a distance \(r\) along a straight wire, but offset by a distance \(r\) along the axis of a circular loop from its center. For a circular loop with current \(I\) and radius \(R\), the Biot-Savart Law is expressed as \(B = \frac{{\mu_0 I R^2}}{{2(R^2 + x^2)^{3/2}}}\), where \(x\) is the distance between the point of interest and the center of the coil along its axis.
02

- Utilize the given information

Substituting the provided data into the formula for \(B\), we get: \[6.39 \times 10^{-4} \mathrm{~T} = \frac{{(4\pi \times 10^{-7} T.m/A) \times 2.50 \mathrm{~A} \times (0.06 m)^2 \times N}}{{2((0.06 m)^2 + (0.06 m)^2)^{3/2}}}\] where \(N\) is the number of turns in the coil.
03

- Solve the equation

Rearranging the equation to solve for \(N\), we get: \[N = \frac{{6.39 \times 10^{-4} \mathrm{~T} \times 2((0.06 m)^2 + (0.06 m)^2)^{3/2}}}{{(4\pi \times 10^{-7} T.m/A) \times 2.50 \mathrm{~A} \times (0.06 m)^2}}\]. After the calculation, we find \(N\) to be approximately 298 turns.

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Key Concepts

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

Magnetic Field
The concept of a magnetic field is central to understanding how electrical currents can give rise to magnetism. In physics, a magnetic field is a vector field that describes the influence of electrical currents and magnetized materials on moving electric charges. Magnetic fields are produced by electric currents, which can be macroscopic currents in wires, or microscopic currents associated with electrons in atomic orbits.

The magnetic field strength at any point is defined as the force that a moving charge experiences per unit of charge velocity and magnetic field. This definition lays the groundwork for understanding how electric currents and their respective magnetic fields are interrelated according to various laws of electromagnetism, such as the Biot-Savart Law.
Permeability of Free Space
The permeability of free space, also known as the magnetic constant \( \mu_0 \), is a fundamental physical constant that describes the ability of a classical vacuum to support the formation of a magnetic field. In the International System of Units (SI), it is typically given the value \( 4\pi \times 10^{-7} \text{T}\cdot\text{m/A} \), which provides a measure of how much resistance is encountered when forming a magnetic field in the vacuum.

This constant not only appears in the Biot-Savart Law, which helps to calculate the magnetic field due to currents, but also in other laws like Ampere's Law and in the definition of Henry, the unit of inductance. Understanding \( \mu_0 \) is crucial because it sets the strength of the electromagnetic interaction in free space, and any deviations in a material's magnetic permeability compared to \( \mu_0 \) indicates its magnetic properties, like whether it's diamagnetic, paramagnetic, or ferromagnetic.
Coil Turns Calculation
Calculating the number of turns in a coil is an important topic when dealing with electromagnetism and the design of electric machinery and transformers. The formula that emerges from the Biot-Savart Law for a circular coil allows us to relate the magnetic field along the axis of the coil to the coil's physical characteristics and the current flowing through it.

In the example provided, the coil's radius and the distance from the coil to the point of interest were equal, which simplifies the Biot-Savart Law application. With the known current, magnetic field, and radius, you can isolate and solve for the number of turns \(N\). This requires algebraic manipulation of the formula, illustrating how coil design involves a reciprocal relationship between the desired magnetic field strength and the geometric and electric properties of the coil. Understanding this relationship is key for anyone designing coils for use in practical applications like motors, inductors, or magnetic field sources.

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Most popular questions from this chapter

A \(+6.00 \mu\) 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 \(\overrightarrow{\boldsymbol{B}}\) it produces at the following points: (a) \(x=0.500 \mathrm{~m}, y=0, z=0\) (b) \(x=0, y=-0.500 \mathrm{~m}, z=0\) (c) \(x=0, \quad y=0, \quad z=+0.500 \mathrm{~m}\) (d) \(x=0, \quad y=-0.500 \mathrm{~m}\) \(z=+0.500 \mathrm{~m} ?\)

The magnetic field around the head has been measured to be approximately \(3.0 \times 10^{-8} \mathrm{G}\). 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 wire of length \(20.0 \mathrm{~cm}\) lies along the \(x\) -axis with the center of the wire at the origin. The wire carries current \(I=8.00 \mathrm{~A}\) in the \(-x\) -direction. (a) What is the magnitude \(B\) of the magnetic field of the wire at the point \(y=5.00 \mathrm{~cm}\) on the \(y\) -axis? (b) What is the percent difference between the answer in (a) and the value you obtain if you assume the wire is infinitely long and use Eq. ( 28.9 ) to calculate \(B\) ?

A long, straight wire lies along the \(x\) -axis and carries current \(I_{1}=2.00 \mathrm{~A}\) in the \(+x\) -direction. A second wire lies in the \(x y\) -plane and is parallel to the \(x\) -axis at \(y=+0.800 \mathrm{~m}\). It carries current \(I_{2}=6.00 \mathrm{~A}\), also in the \(+x\) -direction. In addition to \(y \rightarrow \pm \infty,\) at what point on the \(y\) -axis is the resultant magnetic field of the two wires equal to zero?

The law of Biot and Savart in Eq. ( 28.7 ) generalizes to the case of surface currents as $$ \overrightarrow{\boldsymbol{B}}=\frac{\mu_{0}}{4 \pi} \int \frac{\sigma \overrightarrow{\boldsymbol{v}} \times \hat{\boldsymbol{r}}}{r^{2}} d a $$ where \(\sigma\) is the local charge density, \(\overrightarrow{\boldsymbol{v}}\) is the local velocity, and \(d a\) is a differential area element. Re-visit Challenge Problem 28.76 and use the above equation as an alternative means to derive the magnetic field at the center of the cylinder. Use the following steps: (a) Write the charge density \(\sigma\). (b) The origin is at the center of the cylinder. What is the vector \(\vec{v}\) that points from the element with coordinates \((x, y, z)=(x, R \cos \phi, R \sin \phi)\) to the origin? (c) What is the velocity \(\overrightarrow{\boldsymbol{v}}\) of the element? (d) What is the vector product \(\overrightarrow{\boldsymbol{v}} \times \hat{\boldsymbol{r}} ?\) (e) An area element on the cylinder may be written as \(d a=R d x d \phi .\) Use this and the previously established information to write the generalized law of Biot and Savart as a double integral. Evaluate the integral to determine the magnetic field \(\vec{B}\) at the center of the cylinder. (f) Is your result consistent with your result in Challenge Problem \(28.76 ?\)
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