Problem 44 - University Physics with Modern Physics

Question

A toroidal solenoid (see Example 28.10 ) has inner radius \(r_{1}=15.0 \mathrm{cm}\) and ourer radius \(r_{2}=18.0 \mathrm{cm} .\) The solenoid has 250 turns and carries a current of 8.50 A. What is the magnitude of the magnetic field at the following distances from the center of the torus: (a) \(12.0 \mathrm{cm} ;(\mathrm{b}) 16.0 \mathrm{cm} ;(\mathrm{c}) 20.0 \mathrm{cm} ?\)

Step-by-Step Solution

Answer

(a) 0 T; (b) 4.43 mT; (c) 0 T.

Step 1: Understand the Problem

We need to find the magnitude of the magnetic field at different distances from the center of a toroidal solenoid. The toroidal solenoid has an inner radius of 15.0 cm, an outer radius of 18.0 cm, 250 turns, and carries a current of 8.50 A.

Step 2: Identify Key Formula

Inside a toroidal solenoid, the magnetic field at a distance \( r \) from the center is given by the formula: \( B = \frac{\mu_0 n I}{2 \pi r} \), where \( n \) is the number of turns per unit length, \( I \) is the current, and \( \mu_0 \) is the permeability of free space \( (4\pi \times 10^{-7} \, \text{T} \cdot \text{m/A}) \). Here, \( n = \frac{N}{2\pi (r_2-r_1)} \), where \( N \) is the total number of turns.

Step 3: Calculate Magnetic Field for Part (a)

For distance \( r = 12.0 \text{ cm} = 0.12 \text{ m} \), it is below the inner radius, therefore there is no magnetic field inside the toroidal solenoid. Thus, \( B = 0 \text{ T} \).

Step 4: Calculate Magnetic Field for Part (b)

For distance \( r = 16.0 \text{ cm} = 0.16 \text{ m} \), which lies between inner and outer radii:1. Calculate \( n = \frac{250}{2\pi (0.18 - 0.15)} \approx \frac{250}{2\pi \times 0.03} \approx 1326.914 \, \text{turns/m} \).2. Use the formula: \[ B = \frac{4\pi \times 10^{-7} \times 1326.914 \times 8.5}{2 \pi \times 0.16} \approx 4.43 \times 10^{-3} \, \text{T} \approx 4.43 \, \text{mT} \].

Step 5: Calculate Magnetic Field for Part (c)

For distance \( r = 20.0 \text{ cm} = 0.20 \text{ m} \), which is outside the toroidal solenoid, the magnetic field is zero, as there is no magnetic field outside the solenoid. Thus, \( B = 0 \text{ T} \).

Key Concepts

Magnetic Field Calculation

The magnetic field generated by a toroidal solenoid, such as the one described in this exercise, is a function of several key parameters. Understanding how to calculate this field is essential in electromagnetism.
To calculate the magnetic field at a point, the distance from the solenoid's center, or radius, is crucial. The formula for the magnetic field inside a toroidal solenoid is given by:

• \[ B = \frac{\mu_0 n I}{2 \pi r} \]

Here, \(\mu_0\) is the permeability of free space, \(n\) is the number of turns per unit length, \(I\) is the current passing through the solenoid, and \(r\) is the radial distance from the center of the torus.
This formula reveals that the magnetic field is inversely proportional to the radius, meaning the field strength decreases as you move away from the center of the solenoid.

Ampere's Law

Ampere's Law is a fundamental principle of electromagnetism that provides significant insight when solving problems related to magnetic fields. It relates the integrated magnetic field around a closed loop to the electric current passing through the loop.
The general form of Ampere's Law can be expressed as:

• \[ \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{enc} \]

In this equation, the left side is the line integral of the magnetic field \(\mathbf{B}\) along a closed loop, and \(I_{enc}\) denotes the current enclosed by the loop.
When applied to a toroidal solenoid, the law simplifies the analysis of the magnetic field inside the solenoid itself, allowing the derivation of the practical formula used to compute \(B\) for any point within its bounds.

Electromagnetism

Electromagnetism is a branch of physics that studies the interactions between electric charges and currents through magnetic fields. It combines two essential concepts: electricity and magnetism.
In the context of a toroidal solenoid, electromagnetism helps us understand how a current-carrying coil generates a magnetic field. The current passing through the solenoid creates concentric loops of magnetic field lines within the solenoid, impacting points both within the coil and beyond its physical boundaries.
Essentially, electromagnetism describes how the movement of electric charges (current) through a conductor (like a solenoid) results in the generation of magnetic forces and fields, embodying the dynamic relationship between these two phenomena.