91Ó°ÊÓ

A toroidal solenoid is essentially a coil of wire shaped like a donut. It creates a magnetic field when an electric current passes through it. The magnetic field inside a toroidal solenoid is determined by the current flowing through the wire and the characteristics of the coil itself, such as the number of turns and the radius. For a toroid filled with vacuum, the magnetic field can be calculated using the formula:
\[ B = \frac{\mu_0 \cdot n \cdot I}{2\pi r} \]where:

• \( \mu_0 \) is the permeability of free space, \(4\pi \times 10^{-7} \, \text{T} \cdot \text{m/A} \).
• \( n \) is the number of turns in the solenoid.
• \( I \) is the current passing through the windings.
• \( r \) is the mean radius of the toroid in meters.

The formula shows how these parameters affect the strength of the magnetic field. A larger current or more turns increases the magnetic field, while a larger radius decreases it.
In cases where the toroid is filled with a material other than vacuum, you need to consider the material's permeability, which is where the next two concepts come into play.

Relative permeability \( \mu_r \) is a measure of how much better a material can conduct magnetic lines of flux compared to a vacuum. It reflects the material's ability to enhance the magnetic field inside the solenoid. In simple terms, relative permeability indicates how a material increases the magnetic field compared to when the space inside the toroid is filled with vacuum.
To find the relative permeability, we use the magnetic field inside the toroid when it is filled with the material \( B_{\text{material}} \) and divide it by the magnetic field with a vacuum \( B \):
\[ \mu_r = \frac{B_{\text{material}}}{B} \]With this formula, you can calculate how much stronger the magnetic field is in a given material environment compared to a vacuum condition.
In our example, the magnetic field with the material is given as \(1.940 \, \text{T}\), while the field with a vacuum was calculated to be approximately \(0.0012 \, \text{T}\). This results in a relative permeability of about \(1616.67\), meaning the magnetic material enhances the magnetic field by this factor.

Magnetic susceptibility, \( \chi_m \), is closely related to relative permeability. It quantifies a material's tendency to become magnetized in response to an external magnetic field. Essentially, it tells us how readily a material will polarize in response to the magnetic influence.
The relationship between magnetic susceptibility and relative permeability is expressed by the equation:
\[ \chi_m = \mu_r - 1 \]This formula signifies that susceptibility is one less than the relative permeability. This subtraction occurs because susceptibility zeroes in on the increase in the field inside the material compared to the field it would have without the material.
In the exercise example, with a calculated relative permeability of \(1616.67\), the magnetic susceptibility is \(1615.67\). This sizable number indicates that the material significantly enhances the magnetic field, showing a strong response to the external magnetic environment within the toroid.