Problem 51 - University Physics with Modern Physics

Question

A wooden ring whose mean diameter is 14.0 cm is wound with a closely spaced toroidal winding of 600 turns. Compute the magnitude of the magnetic field at the center of the cross section of the windings when the current in the windings is 0.650 A.

Step-by-Step Solution

Answer

The magnetic field is approximately \( 1.114 \times 10^{-3} \text{ T} \).

Step 1: Identify the Formula for Magnetic Field in a Toroid

We will use the formula for the magnetic field at the center of a toroid: \[ B = \frac{\mu_0 N I}{2 \pi r} \] where \( B \) is the magnetic field, \( \mu_0 \) is the permeability of free space \( (4\pi \times 10^{-7} \text{ T m/A}) \), \( N \) is the number of turns, \( I \) is the current, and \( r \) is the radius of the toroid.

Step 2: Convert Diameter to Radius

Given the mean diameter of the toroid is 14.0 cm, we convert it to meters and calculate the radius. \[ r = \frac{14.0 \text{ cm}}{2} = 7.0 \text{ cm} = 0.07 \text{ m} \]

Step 3: Plug Values into the Formula

Substitute the given values into the formula. \[ B = \frac{(4\pi \times 10^{-7} \text{ T m/A}) \times 600 \times 0.650 \text{ A}}{2 \pi \times 0.07 \text{ m}} \]

Step 4: Simplify and Calculate

Simplify the expression to find \( B \). \[ B = \frac{2.4 \times 10^{-4} \text{ T m/A} \times 0.650 \text{ A}}{0.14 \text{ m}} \] \[ B = \frac{1.56 \times 10^{-4}}{0.14} \] \[ B = 1.114 \times 10^{-3} \text{ T} \]

Key Concepts

Toroidal Winding

A toroid is a doughnut-shaped coil commonly used in electrical engineering, while a toroidal winding refers to the turns of wire wound tightly around this shape. The design of a toroidal winding is essential for creating a stable and controlled magnetic field. In a toroid, the winding is done continuously in a loop, which ensures that the magnetic field lines are confined within the core.
This containment reduces electromagnetic interference with surrounding devices, making toroids particularly advantageous for use in transformers and inductors. Due to their structure:

• The winding helps achieve uniform magnetic properties inside the loop.
• Losses are minimized compared to other core shapes, such as solenoids.
• They provide high inductance and lower losses relative to their volume.

When working with toroidal windings, it's essential to know the number of turns and the geometry of the toroid, as these factors critically influence the resultant magnetic field.

Magnetic Field Calculation

Calculating the magnetic field in a toroid involves using a specific formula that considers the number of turns, current, and radius. The formula is given by
\[B = \frac{\mu_0 N I}{2 \pi r}\]
Here, **B** represents the magnetic field strength, **\( \mu_0 \)** is the permeability of free space, **N** is the number of turns of the winding, **I** is the current flowing through the coil, and **r** is the central radius of the toroid. To successfully calculate the magnetic field, follow these steps:

• Convert all measurements to SI units (e.g., centimeters to meters for radius).
• Substitute each value into the formula accurately.
• Solve the equation to find the magnitude of the magnetic field, typically in teslas (T).

This formula highlights how both the current and the number of turns directly increase the magnetic field, while a larger radius reduces the field's intensity.

Permeability of Free Space

The permeability of free space, denoted by \( \mu_0 \), is a crucial physical constant in electromagnetism. It helps describe how much a magnetic field can penetrate free space, which is an idealized vacuum. The value of \( \mu_0 \) is approximately \( 4\pi \times 10^{-7} \, \text{T} \, \text{m/A} \). This constant is used in many equations, including those for calculating the magnetic field in a toroid.
Permeability reflects the ability of a material or space to support the formation of magnetic fields within it:

• A higher permeability indicates that a medium can easily support magnetic field lines.
• Free space has a defined permeability value, serving as the baseline against which other materials are compared.

Understanding permeability is critical when analyzing magnetic fields, as it determines how materials will influence these fields. In practical calculations, knowing \( \mu_0 \) allows for accurate predictions of field behavior in ideal conditions.