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Chapter 20: Problem 58

A closely wound circular coil with a diameter of 4.00 \(\mathrm{cm}\) has 600 turns and carries a current of 0.500 A. What is the magnetic field at the center of the coil?

Short Answer

Expert verified
The magnetic field at the center of the coil is approximately 9.42 mT.

Step by step solution

01

Identification of Given Values

We need to determine the magnetic field at the center of the coil. We are given the diameter of the coil as 4.00 cm. The current through the coil is 0.500 A, and the coil has 600 turns.
02

Calculate the Radius of the Coil

First, we convert the diameter from centimeters to meters and then determine the radius by dividing the diameter by 2.\[\text{Diameter} = 4.00 \text{ cm} = 0.04 \text{ m}\]\[ ext{Radius} = \frac{0.04 \text{ m}}{2} = 0.02 \text{ m}\]
03

Use the Magnetic Field Formula for a Circular Coil

The magnitude of the magnetic field at the center of a circular coil can be found using the formula:\[B = \frac{{\mu_0 \, n \, I}}{{2 \, R}}\]where \(B\) is the magnetic field, \(\mu_0 = 4\pi \times 10^{-7} \, \text{T m/A}\) is the permeability of free space, \(n\) is the number of turns, \(I\) is the current, and \(R\) is the radius of the coil.
04

Substitute Values into the Formula

Substitute the known values into the formula:\[B = \frac{{4\pi \times 10^{-7} \, \text{T m/A} \times 600 \times 0.500 \, \text{A}}}{{2 \times 0.02 \, \text{m}}}\]
05

Simplify and Calculate

Simplify the equation:\[B = \frac{{4\pi \times 10^{-7} \times 600 \times 0.500}}{{0.04}}\]Calculate:\[B = \frac{{4\pi \times 10^{-7} \times 300}}{{0.04}}\]Finally, calculate:\[B \approx 9.42 \times 10^{-3} \, \text{T}\]

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Key Concepts

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

Circular Coil
A circular coil is an important component in electromagnetism and electronics. It is essentially a loop or series of loops made of wire, and its shape is circular. This coil generates a magnetic field when an electric current passes through it.
Circular coils are used in many electrical devices such as transformers, motors, and inductors, due to their ability to generate magnetic fields efficiently. The circular shape ensures that the magnetic fields produced are concentrated and uniform within the loop, especially at the center.
Understanding how a circular coil works will help you comprehend how such coils create and maintain magnetic fields, which is vital for various applications.
Key points about circular coils:
  • Circular shape allows for efficient magnetic field generation.
  • Typically made from conductive materials like copper.
  • Commonly used in electrical and electronic devices.
Number of Turns
The number of turns in a coil refers to how many loops the wire is wound into. This number plays a crucial role in determining the strength of the magnetic field generated by the coil.
The more turns a coil has, the stronger the magnetic field it can produce. This is because each loop contributes to the overall magnetic field, and having more loops means that the currents add up to create a more substantial field.
In the given exercise, the coil has 600 turns, which significantly enhances its ability to produce a strong magnetic field.
Important aspects of the number of turns:
  • Directly affects the strength of the generated magnetic field.
  • Increasing the number of turns enhances the coil's magnetic capability.
  • Highly applicable in designing electromagnetic devices.
Permeability of Free Space
The permeability of free space, denoted as \( \mu_0 \), is a constant value that describes how a magnetic field is distributed in a vacuum. This value is fundamental in magnetic field calculations and is utilized in various electromagnetic equations.
The standard value for the permeability of free space is \( \mu_0 = 4\pi \times 10^{-7} \, \text{T m/A} \). This constant helps us to calculate how easily the magnetic field lines can pass through a given space.
Understanding \( \mu_0 \) is essential for comprehending the behavior of magnetic fields in space, especially when dealing with calculations involving coils and current.
Core insights about the permeability of free space:
  • A constant that affects the calculation of magnetic fields in a vacuum.
  • Has a fixed numeric value used in formulas for consistency.
  • Plays a crucial role in equations related to electromagnetism.
Current and Magnetic Field
The relationship between electric current and magnetic field is a foundational concept in electromagnetism. When a current flows through a wire, a magnetic field is generated around the wire. This phenomenon is used extensively in the workings of coils.
In the case of a circular coil, the generated magnetic field is more concentrated and uniform at the center of the loop due to the circular arrangement of the wire.
The strength of the magnetic field depends on the amount of current flowing through the coil and the number of turns in the coil. Crucial points about current and magnetic field:
  • Electric current creates a magnetic field around conductors.
  • More current results in a stronger magnetic field.
  • Critical for the functioning of inductors, transformers, and other electrical components.

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Most popular questions from this chapter

\(\bullet\) Atom smashers! A cyclotron particle accelerator (sometimes called an "atom smasher" in the popular press) is a device for accelerating charged particles, such as electrons and protons, to speeds close to the speed of light. The basic design is quite simple. The particle is bent in a circular path by a uniform magnetic field. An electric field is pulsed periodically to increase the speed of the particle. The charged particle (or ion) of mass \(m\) and charge \(q\) is introduced into the cyclotron so that it is moving perpendicular to a uniform magnetic field \(\vec{B}\) (a) Starting with the radius of the circular path of a charge moving in a uniform magnetic field, show that the time \(T\) for this particle to make one complete circle is \(T=\frac{2 \pi m}{|q| B}\) . (Hint: You can express the speed \(v\) in terms of \(R\) and \(T\) because the particle travels through one circumference of the circle in time \(T\) . (b) Which would take longer to complete one circle, an ion moving in a large circle or one moving in a small circle? Explain.

A closely wound, circular coil with radius 2.40 \(\mathrm{cm}\) has 800 turns. What must the current in the coil be if the magnetic field at the center of the coil is 0.0580 \(\mathrm{T}\) ?

A rectangular 10.0 \(\mathrm{cm}\) by 20.0 \(\mathrm{cm}\) circuit carrying an 8.00 \(\mathrm{A}\) current is oriented with its plane parallel to a uniform 0.750 T magnetic field (Figure 20.62\()\) . (a) Find the magnitude and direction of the magnetic force on each segment \((a b, b c,\) etc. \()\) of this circuit. Illustrate your answers with clear diagrams. (b) Find the magnitude of the net force on the entire circuit.

\(\bullet\) Determining diet. One method for determining the amount of corn in early Native American diets is the stable isotope ratio analysis (SIRA) technique. As corn photosynthesizes, it concentrates the isotope carbon-13, whereas most other plants concentrate carbon-12. Overreliance on corn consumption can then be correlated with certain diseases, because corn lacks the essential amino acid lysine. Archaeologists use a mass spectrometer to separate the \(^{12} \mathrm{C}\) and \(^{13} \mathrm{C}\) isotopes in samples of human remains. Suppose you use a velocity selector to obtain singly ionized (missing one electron) atoms of speed 8.50 \(\mathrm{km} / \mathrm{s}\) and want to bend them within a uniform magnetic field in a semicircle of diameter 25.0 \(\mathrm{cm}\) for the 12 \(\mathrm{C}\) . The measured masses of these isotopes are \(1.99 \times 10^{-26} \mathrm{kg}(12 \mathrm{C})\) and \(2.16 \times 10^{-26} \mathrm{kg}\left(^{13} \mathrm{C}\right) .\) (a) What strength of magnetic field is required? (b) What is the diameter of the \(^{13} \mathrm{C}\) semicircle? (c) What is the separation of the \(^{12} \mathrm{C}\) and \(^{13} \mathrm{Cions}\) at the detec- tor at the end of the semicircle? Is this distance large enough to be easily observed?

A long, straight horizontal wire carries a current of 2.50 \(\mathrm{A}\) directed toward the right. An electron is traveling in the vicinity of this wire. (a) At the instant the electron is 4.50 \(\mathrm{cm}\) above the wire's center and moving with a speed of \(6.00 \times 10^{4} \mathrm{m} / \mathrm{s}\) directly toward it, what are the magnitude and direction of the force that the magnetic field of the current exerts on the electron? (b) What would be the magnitude and direction of the magnetic force if the electron were instead moving parallel to the wire in the same direction as the current?
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