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Chapter 28: Problem 70

You have 1.0 \(\mathrm{kg}\) of copper and want to make a practical solenoid that produces the greatest possible magnetic field for a given voltage. Should you make your copper wire long and thin, short and fat, or something else? Consider other variables, such as solenoid diameter, length, and so on.

Short Answer

Expert verified
Opt for a medium wire thickness, balancing current and number of turns.

Step by step solution

01

Understand the Problem

We need to determine the shape and dimensions of a copper solenoid that produces the maximum magnetic field when a given voltage is applied. This means we need to consider the length and diameter of the wire used to make the solenoid and other factors that influence its magnetic field strength.
02

Recall the Formula for Magnetic Field

The magnetic field inside a solenoid is given by \( B = \mu_0 \frac{IN}{L} \), where \( B \) is the magnetic field, \( \mu_0 \) is the permeability of free space, \( I \) is the current, \( N \) is the number of turns of wire, and \( L \) is the length of the solenoid.
03

Express Turns and Current Using Known Variables

Since the wire is made of copper, its mass determines its volume. The diameter of the wire (thin versus thick) is an important factor in determining how many turns \( N \) we can have since the volume of wire determines the length available. The current \( I \) is affected by the wire's resistance and the given voltage.
04

Determine Wire Resistance

The resistance \( R \) of the copper wire is given by \( R = \rho \frac{L_{\text{wire}}}{A} \), where \( \rho \) is the resistivity of copper, \( L_{\text{wire}} \) is its length, and \( A \) is its cross-sectional area. A smaller diameter results in a higher resistance.
05

Maximize the Magnetic Field Expression

To maximize \( B \), you want to balance \( N \) and \( I \). A thicker wire decreases resistance and increases current but reduces turns for a given length. Using the relation \( I = \frac{V}{R} \), the choice of wire diameter will affect \( I \). By making the wire neither too thick nor too thin, you maximize the product \( NI \).

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Key Concepts

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

Solenoid Design
A solenoid is essentially a coil of wire, where the wire is wound in a series of loops to create a helical shape. When current flows through the solenoid, it produces a magnetic field, making it a common choice in many electrical applications.
Creating an effective solenoid involves thoughtful design choices regarding its structure:
  • Diameter and Length: The diameter of the solenoid affects the size of the magnetic field generated. A larger diameter can produce a stronger field at the core, while the length of the solenoid can influence uniformity of the field.

  • Material: Using copper as the wire material is advantageous due to its excellent electrical conductivity, allowing efficient current flow with minimal resistance.

  • Portability and Application: The design also depends on whether the solenoid needs to be portable or fixed, and its application will dictate the optimal configuration.
Copper Wire Resistance
Copper is often selected for solenoid wires due to its low resistivity, which reduces the resistance over the length of the wire. Resistance in a wire is an essential factor because it influences how easily current flows through the solenoid.
Factors affecting copper wire resistance include:
  • Wire Diameter: Thicker wires provide less resistance, allowing more current flow, which is beneficial for generating a strong magnetic field.

  • Temperature: Resistance in copper will increase with temperature. Thus, maintaining optimal operating conditions is essential for maximal efficiency.

  • Length of the Wire: Longer wires have greater resistance. Therefore, finding the right balance between wire length and diameter is critical to optimizing solenoid performance.
Turns of Wire in Solenoid
The number of turns in a solenoid is pivotal in defining its magnetic field strength. More turns can enhance the strength of the field, but the design needs balancing with resistance and current.
Some key points include:
  • Balance Between Turns and Diameter: While more turns can increase field strength, using a thin wire to fit numerous turns can increase resistance and diminish current.

  • Layering: Coiling wires in multiple layers can compactly increase turns but may also lead to heat buildup, impacting performance.

  • Wire Length: Given a fixed mass of copper, longer wire translates to thinner wire, more turns, but higher resistance, which is a trade-off to consider.
Factors Affecting Magnetic Field Strength
Several factors determine the intensity and reach of a magnetic field generated by a solenoid. Understanding these is crucial in designing a solenoid for specific tasks and efficiency.
Important influences include:
  • Current Through the Solenoid: Higher current usually translates to a stronger magnetic field, but is limited by wire resistance and potential overheating.

  • Number of Wire Turns: As previously mentioned, more turns generally enhance the magnetic field, provided the current can be maintained.

  • Core Material: Some solenoids feature a core, such as iron, which can significantly intensify the magnetic field due to its high magnetic permeability compared to air.
  • Uniformity of Turns: Even spacing and consistent layering of the wire ensure a uniform magnetic field, reducing potential imbalances or weaker sections.

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Most popular questions from this chapter

Four long straight parallel wires located at the corners of a square of side \(d\) carry equal currents \(I_{0}\) perpendicular to the page as shown in Fig. \(28-59 .\) Determine the magnitude and direction of \(\overrightarrow{\mathbf{B}}\) at the center \(\mathrm{C}\) of the square.

The design of a magneto-optical atom trap requires a magnetic field \(B\) that is directly proportional to position \(x\) along an axis. Such a field perturbs the absorption of laser light by atoms in the manner needed to spatially confine atoms in the trap. Let us demonstrate that "anti-Helmholtz" coils will provide the required field \(B=C x,\) where \(C\) is a constant. AntiHelmholtz coils consist of two identical circular wire coils, each with radius \(R\) and \(N\) turns, carrying current \(I\) in opposite directions (Fig. \(28-62\) ). The coils share a common axis (defined as the \(x\) axis with \(x=0\) at the midpoint (0) between the coils). Assume that the centers of the coils are separated by a distance equal to the radius \(R\) of the coils. ( \(a\) ) Show that the magnetic field at position \(x\) along the \(x\) axis is given by \(B(x)=\frac{4 \mu_{0} N I}{R}\left\\{\left[4+\left(1-\frac{2 x}{R}\right)^{2}\right]^{-\frac{3}{2}}-\left[4+\left(1+\frac{2 x}{R}\right)^{2}\right]^{-\frac{3}{2}}\right\\} .\) (b) For small excursions from the origin where \(|x| \ll R\), show that the magnetic field is given by \(B \approx C x,\) where the constant \(C=48 \mu_{0} N I / 25 \sqrt{5} R^{2} .\) (c) For optimal atom trapping, \(d B / d x\) should be about \(0.15 \mathrm{~T} / \mathrm{m} .\) Assume an atom trap uses anti-Helmholtz coils with \(R=4.0 \mathrm{~cm}\) and \(N=150 .\) What current should flow through the coils? [Coil separation equal to coil radius, as assumed in this problem, is not a strict requirement for antiHelmholtz coils.

A small solenoid (radius \(r_{\mathrm{a}}\) ) is inside a larger solenoid (radius \(\left.r_{\mathrm{b}}>r_{\mathrm{a}}\right)\). They are coaxial with \(n_{\mathrm{a}}\) and \(n_{\mathrm{b}}\) turns per unit length, respectively. The solenoids carry the same current, but in opposite directions. Let \(r\) be the radial distance from the common axis of the solenoids. If the magnetic field inside the inner solenoid \(\left(r

Three long parallel wires are 3.5 \(\mathrm{cm}\) from one another. (Looking along them, they are at three corners of an equilateral triangle.) The current in each wire is \(8.00 \mathrm{A},\) but its direction in wire \(\mathrm{M}\) is opposite to that in wires \(\mathrm{N}\) and \(\mathrm{P}\) (Fig. 54 ). Determine the magnetic force per unit length on each wire due to the other two.

(II) A small loop of wire of radius \(1.8 \mathrm{~cm}\) is placed at the center of a wire loop with radius \(25.0 \mathrm{~cm} .\) The planes of the loops are perpendicular to each other, and a 7.0 -A current flows in each. Estimate the torque the large loop exerts on the smaller one. What simplifying assumption did you make?
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