Question

A toroidal magnet has an inner radius of \(1.351 \mathrm{~m}\) and an outer radius of \(1.541 \mathrm{~m}\). The wire carries a 49.13 -A current, and there are 24,945 turns in the toroid. What is the magnetic field at a distance of \(1.446 \mathrm{~m}\) from the center of the toroid?

Short answer

Answer: The magnetic field at a distance of 1.446 m from the center of the toroidal magnet is approximately 0.0263 T.

Step-by-step solution

Understand the given information and find the relevant formula

We are given the inner radius (\(R_1 = 1.351\mathrm{~m}\)), the outer radius (\(R_2 = 1.541\mathrm{~m}\)), the current in the wire (\(I = 49.13\mathrm{~A}\)), and the number of turns in the magnet (\(N = 24,945\)). We need to find the magnetic field (\(B\)) at a distance \(r = 1.446\mathrm{~m}\) from the center of the toroid. Using Ampere's law, the formula for the magnetic field at any point inside the toroidal magnet is: \(B = \frac{\mu_0 \cdot N \cdot I}{2 \pi r}\), where \(\mu_0\) is the permeability of free space, which is approximately \(4 \pi \times 10^{-7} \mathrm{T \cdot m/A}\).

Check if the distance is within the magnet boundary

Since we need to calculate the magnetic field at a distance \(r = 1.446\mathrm{~m}\) from the center, we must first check if this distance lies within the boundary of the toroidal magnet. As it falls between the inner radius (\(1.351\mathrm{~m}\)) and the outer radius (\(1.541\mathrm{~m}\)), the magnetic field can be found using the above formula.

Calculate the magnetic field

Now, we can calculate the magnetic field using the given data and the formula from Step 1. \(B = \frac{\mu_0 \cdot N \cdot I}{2 \pi r} = \frac{(4 \pi \times 10^{-7} \mathrm{T \cdot m/A}) \cdot 24,945 \cdot 49.13\mathrm{~A}}{2 \pi \cdot 1.446\mathrm{~m}}\) Solving this equation, we get, \(B \approx 0.0263\mathrm{~T}\) So, the magnetic field at a distance of \(1.446\mathrm{~m}\) from the center of the toroid is approximately \(0.0263\mathrm{~T}\).

Key concepts

Ampere's Law

Ampere's Law is a fundamental rule in electromagnetism that relates the magnetic field around a closed loop to the electric current passing through it. The beauty of Ampere's Law is in its simplicity; it states that the line integral of the magnetic field along a closed path is directly proportional to the net electric current enclosed by the path. This can be mathematically expressed as:

Toroidal Magnet

A toroidal magnet, often simply termed a 'toroid,' is a donut-shaped magnet characterized by its continuous, ring-like form, which creates a closed magnetic circuit. The toroid's shape is essential because it confines most of the magnetic field within the doughnut, with very little magnetic field escaping outside it. As a result, toroidal magnets are prevalent in scientific and medical equipment, where magnetic fields need to be precise and concentrated.

Permeability of Free Space

The permeability of free space, denoted as \( \mu_0 \), is a fundamental physical constant which represents the capability of the vacuum to permit the passage of magnetic field lines. It is a crucial parameter in Ampere's Law and is given the value of approximately \( 4 \pi \times 10^{-7} \) Tesla meter per ampere (T·m/A). When determining the strength of the magnetic field in a vacuum, or in our example, inside a toroid with air as the medium, it is this permeability that we consider.

Magnetic Field Calculation

The calculation of the magnetic field within a toroidal magnet demands attention to the symmetry of the toroid and the uniformity of the current distribution. Using Ampere's Law, the calculation is straightforward for points inside the toroid. Matters become more complex at points at the edge or outside the toroidal structure, as these locations involve different field considerations. In the given exercise, the expressions shine due to the idealistic scenario of a well-defined, symmetrical toroid.