Problem 54 - University Physics with Modern Physics

Question

The current in the windings of a toroidal solenoid is 2.400 A. There are 500 turns, and the mean radius is 25.00 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.

Step-by-Step Solution

Answer

The relative permeability is approximately 3221, and the magnetic susceptibility is approximately 3220.

Step 1: Understand the Magnetic Field in a Toroid

The magnetic field inside a toroid is given by the formula \( B = \mu N I / (2\pi r) \), where \( B \) is the magnetic field, \( \mu \) is the permeability of the material, \( N \) is the number of turns, \( I \) is the current, and \( r \) is the mean radius of the toroid. We know \( B = 1.940 \, \text{T} \), \( N = 500 \), \( I = 2.400 \, \text{A} \), and \( r = 0.25 \, \text{m} \). We need to solve for \( \mu \).

Step 2: Calculate Permeability

Rearrange the formula to solve for \( \mu \): \( \mu = \frac{B \cdot 2\pi r}{N I} \). Substitute the known values into this formula: \( \mu = \frac{1.940 \times 2\pi \times 0.25}{500 \times 2.400} \). Calculate to find \( \mu \approx 4.058 \times 10^{-3} \, \text{T}\cdot\text{m/A} \).

Step 3: Determine the Relative Permeability

The relative permeability \( \mu_r \) is given by \( \mu_r = \frac{\mu}{\mu_0} \), where \( \mu_0 = 4\pi \times 10^{-7} \, \text{T}\cdot\text{m/A} \) is the permeability of free space. Substitute the values: \( \mu_r = \frac{4.058 \times 10^{-3}}{4\pi \times 10^{-7}} \). Calculate to find \( \mu_r \approx 3221 \).

Step 4: Calculate Magnetic Susceptibility

The magnetic susceptibility \( \chi_m \) is related to the relative permeability by the equation \( \chi_m = \mu_r - 1 \). Using \( \mu_r = 3221 \), we find \( \chi_m = 3221 - 1 = 3220 \).

Step 5: Review and Verify

Double-check your calculations to ensure accuracy. Review each step to ensure all units are consistent and that you correctly applied the formulas.

Key Concepts

Toroidal Solenoid

A toroidal solenoid is a coil of wire shaped like a donut with multiple closely wound turns. When an electric current flows through this coil, it creates a magnetic field within its center. This magnetic field is crucial in electrical engineering and physics applications, as it helps contain and guide magnetic fields more effectively.

The shape of a toroidal solenoid makes it different from a straight solenoid. Because it's circular and closed, the magnetic field remains inside the core, minimizing magnetic interference with the surrounding space. The mathematical expression for the magnetic field inside a toroid is given by:

• \[B = \frac{\mu N I}{2 \pi r}\]

where:

• \(B\) = magnetic field inside the toroid
• \(\mu\) = magnetic permeability of the core material
• \(N\) = number of turns in the toroid
• \(I\) = current flowing through the coil
• \(r\) = mean radius of the toroid

Understanding these variables is essential to design efficient electromagnetic devices that use toroidal solenoids effectively.

Magnetic Susceptibility

Magnetic susceptibility measures how easily a material can be magnetized when exposed to an external magnetic field. It quantifies how much a material will become magnetized and is denoted by the Greek letter \(\chi_m\).

It connects closely to the relative permeability of materials, which indicates how a medium can support the formation of a magnetic field in comparison to a vacuum. Importantly, the relationship between magnetic susceptibility and relative permeability is given by the formula:

• \[\chi_m = \mu_r - 1\]

This equation reveals that a higher magnetic susceptibility means a stronger capacity for magnetization, signifying a greater enhancement of the external magnetic fields by the material. Being familiar with magnetic susceptibility is crucial for understanding a material's response in applications such as inductors, transformers, and magnetic shielding devices.

Relative Permeability

Relative permeability \(\mu_r\) is a dimensionless quantity that compares a material's permeability to the permeability of free space, \(\mu_0\). It shows how much more a substance can be magnetized than a vacuum under a given magnetic field.

Calculating relative permeability helps to understand how well a material can "conduct" magnetic flux. You find it using the formula:

• \[\mu_r = \frac{\mu}{\mu_0}\]

where:

• \(\mu\) is the magnetic permeability of the material
• \(\mu_0 = 4\pi \times 10^{-7} \text{ T}\cdot\text{m/A}\) is the permeability of free space

High relative permeability indicates that a material is more prone to channeling magnetic field lines effectively, making it a good candidate for applications requiring efficient magnetic field transmission, such as in the cores of inductors and transformers. Understanding this concept is essential for designing systems that manage electromagnetic fields efficiently.