Ch 29: The Magnetic Field

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 29: The Magnetic Field

Problem 54c

Chapter 29, Problem 54c

The toroid of FIGURE P29.54 is a coil of wire wrapped around a doughnut-shaped ring (a torus). Toroidal magnetic fields are used to confine fusion plasmas. Is a toroidal magnetic field a uniform field? Explain.

Verified step by step guidance

Step 1: Begin by understanding the structure of the toroid shown in the image. A toroid is a coil of wire wrapped around a doughnut-shaped ring (torus). The wire carries a current, denoted as 'I', and the toroid has 'N' turns of wire. The magnetic field is generated within the toroidal ring due to the current flowing through the wire.

Step 2: Recall Ampere's Law, which is used to calculate the magnetic field inside a toroid. Ampere's Law states: ∮𝐵⋅𝑑𝑙 = μ₀𝐼ₑ𝑛𝑐, where 𝐵 is the magnetic field, 𝑑𝑙 is the path length, μ₀ is the permeability of free space, and 𝐼ₑ𝑛𝑐 is the enclosed current. For a toroid, the magnetic field is confined within the ring and circulates along the circular path inside the torus.

Step 3: Analyze the uniformity of the magnetic field. The magnetic field inside a toroid depends on the radius 'r' of the circular path. The formula for the magnetic field inside a toroid is: 𝐵 = (μ₀𝑁𝐼)/(2π𝑟), where 𝑟 is the distance from the center of the toroid to the point of interest. This indicates that the magnetic field strength decreases as the radius increases.

Step 4: Consider the implications of the formula. Since the magnetic field strength varies with the radius 'r', the field is not uniform throughout the toroid. It is stronger closer to the inner radius and weaker near the outer radius of the toroidal ring.

Step 5: Conclude that a toroidal magnetic field is not a uniform field. The variation in the magnetic field strength with radius makes it non-uniform. However, the field is confined within the toroid and does not extend outside the ring, which is a key characteristic of toroidal magnetic fields.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Toroidal Magnetic Field

A toroidal magnetic field is generated by a current flowing through a wire wound in a circular shape, forming a doughnut-like structure. The magnetic field lines are confined within the torus, creating a closed loop. This configuration is particularly useful in applications like magnetic confinement in fusion reactors, as it helps to contain plasma effectively.

Uniform Magnetic Field

A uniform magnetic field is one where the magnetic field strength and direction are constant throughout a given region. In contrast, a non-uniform field varies in strength and direction at different points. Understanding whether a toroidal magnetic field is uniform involves analyzing how the field strength changes with distance from the center of the torus.

Magnetic Field Strength Variation

In a toroidal configuration, the magnetic field strength is not uniform; it varies with the radial distance from the center of the torus. The field is strongest near the center and decreases as one moves outward. This variation is crucial for applications in plasma confinement, as it affects the stability and behavior of the plasma within the toroidal magnetic field.

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