Problem 49 - University Physics with Modern Physics

Question

A long solenoid with 60 turns of wire per centimeter carries a current of 0.15 \(\mathrm{A}\) . The wire that makes up the solenoid is wrapped around a solid core of silicon steel \(\left(K_{\mathrm{m}}=5200\right) .\) (The wire of the solenoid is jacketed with an insulator so that none of the current flows into the core.) (a) For a point inside the core, find the magnitudes of (i) the magnetic field \(\overrightarrow{\boldsymbol{B}}_{0}\) due to the solenoid current; (ii) the magnetization \(\vec{M} ;\) (iii) the total magnetic field \(\overrightarrow{\boldsymbol{B}}\) . (b) In a sketch of the solenoid and core, show the directions of the vectors \(\overrightarrow{\boldsymbol{B}}, \overrightarrow{\boldsymbol{B}}_{0}\) , and \(\overrightarrow{\boldsymbol{M}}\) inside the core.

Step-by-Step Solution

Answer

(i) \( B_0 = 1.13 \times 10^{-3} \mathrm{T} \), (ii) \( M = 4.67 \times 10^6 \mathrm{A/m} \), (iii) \( B = 4.68 \times 10^{-3} \mathrm{T} \); vectors are aligned along the solenoid's axis.

Step 1: Determine Magnetic Field from Solenoid Current

The magnetic field inside a long solenoid can be calculated using the formula: \( B_0 = \mu_0 \cdot n \cdot I \), where \( \mu_0 = 4\pi \times 10^{-7} \, \mathrm{T\cdot m/A} \) is the permeability of free space, \( n \) is the number of turns per unit length, and \( I \) is the current. Given \( n = 60 \, \text{turns/cm} = 6000 \, \text{turns/m} \) and \( I = 0.15 \, \text{A} \), we calculate:\[ B_0 = 4\pi \times 10^{-7} \times 6000 \times 0.15 \approx 1.13 \times 10^{-3} \, \mathrm{T}. \]

Step 2: Calculate Magnetization

The magnetization \( M \) of the core material is related to the magnetic susceptibility \( \chi_m \) and the magnetic field \( B_0 \). The relation is \( M = \chi_m H_0 \), where \( \chi_m = K_m - 1 \) and \( H_0 = \frac{B_0}{\mu_0} \) is the magnetic field strength. For silicon steel, \( K_m = 5200 \), so \( \chi_m = 5199 \). Calculate \( H_0 \):\[ H_0 = \frac{B_0}{\mu_0} = \frac{1.13 \times 10^{-3}}{4\pi \times 10^{-7}} \approx 899.32 \, \mathrm{A/m}. \]Then, \( M = 5199 \times 899.32 \approx 4.67 \times 10^6 \, \mathrm{A/m}. \)

Step 3: Find Total Magnetic Field

The total magnetic field \( B \) inside a material is the sum of the external magnetic field \( B_0 \) and the field from the magnetic response of the material, given by \( B = \mu \cdot H_0 = (\mu_0 \cdot K_m) \cdot H_0 \). Calculate \( B \):\[ B = 4\pi \times 10^{-7} \times 5200 \times 899.32 \approx 4.68 \times 10^{-3} \, \mathrm{T}. \]

Step 4: Sketch the Vector Directions

In a sketch showing the cross-section of the solenoid and core, the vectors \( \overrightarrow{B} \), \( \overrightarrow{B}_0 \), and \( \overrightarrow{M} \) should be drawn inside the core pointing in the same direction along the axis of the solenoid. Each of these vectors represents the magnetic effects within the core, with the magnitude of \( \overrightarrow{B} \) being the largest, followed by \( \overrightarrow{M} \) and then \( \overrightarrow{B}_0 \).

Key Concepts

Magnetic Field

Magnetic fields are an invisible force that arise due to the movement of electric charges, such as a current flowing through a wire. Inside a solenoid—a coil of wire—the magnetic fields from individual loops add together to create a strong magnetic field along the center of the coil. This is why solenoids are often used for generating controllable magnetic fields.

The magnetic field inside a long solenoid can be calculated using the formula:

• \( B_0 = \mu_0 \cdot n \cdot I \)

where:

• \( B_0 \) is the magnetic field due to the current.
• \( \mu_0 \) is the permeability of free space, with a value of \( 4\pi \times 10^{-7} \ \mathrm{T\cdot m/A} \).
• \( n \) represents the number of turns per unit length in the solenoid.
• \( I \) is the current flowing through the solenoid.

A higher number of turns or a stronger current results in a larger magnetic field. In this exercise, with 60 turns per centimeter and a current of 0.15 A, the magnetic field inside the solenoid was found to be approximately \( 1.13 \times 10^{-3} \ \mathrm{T} \). This magnetic field is crucial for various applications that require precise magnetic control, from electric motors to magnetic resonance imaging (MRI) machines.

Solenoid

A solenoid is a coil of wire that is designed to generate a magnetic field when an electric current passes through it. The wire is typically wound in a tightly spaced helix, which causes the magnetic fields from individual loops to overlap and reinforce one another, creating a strong uniform field inside the coil.

A solenoid's efficiency in producing a magnetic field depends on several factors:

• Number of turns: More turns in the wire coil will increase the magnetic field strength.
• Current: A higher current will also lead to a stronger magnetic field.
• Core material: Introducing a ferromagnetic core, such as silicon steel, enhances the magnetic field by concentrating the magnetic lines of force.

In this problem, the solenoid wire, coated with an insulator, is wrapped around a silicon steel core. The core material is crucial because its high magnetic permeability aids in focusing the magnetic lines of force more effectively than air or plastic alone. This results in a significantly stronger magnetic field, useful for applications ranging from electromagnetic actuators to magnetic locks.

Magnetization

Magnetization is a measure of how a material responds to an external magnetic field, representing the induced magnetic moment per unit volume of the material. It's the result of the alignment of small magnetic domains within the material in the presence of an external field.

The relationship between magnetization \( M \) and the applied magnetic field \( B_0 \) involves the material's magnetic susceptibility \( \chi_m \), calculated as \( \chi_m = K_m - 1 \) where \( K_m \) is the relative permeability of the material.When an external magnetic field \( H_0 \) is applied to a material, the magnetization is given by:

• \( M = \chi_m \cdot H_0 \)

In this exercise, with a silicon steel core having a relative permeability \( K_m \) of 5200, the susceptibility was calculated at 5199. Consequently, the magnetization \( M \) was found to be around \( 4.67 \times 10^6 \ \mathrm{A/m} \), indicating a strong magnetic response.

Understanding magnetization is essential in electromagnetism as it helps explain the behavior of materials in magnetic fields. It is vital for developing materials used in electronics, transformers, and other devices requiring specific magnetic characteristics.