A toroid has a major radius R and a minor radius r, and is...

A toroid has a major radius $R$ and a minor radius $r,$ and is tightly wound with $N$ turns of wire, as shown in Figure P32.12. If $R>>r,$ the magnetic field in the region enclosed by the wire of the torus, of cross-sectional area $A=\pi r^{2},$ is essentially the same as the magnetic field of a solenoid that has been bent into a large circle of radius $R$ . Modeling the field as the uniform field of a long solenoid, show that the self-inductance of such a toroid is approximately $$ L \approx \frac{\mu_{0} N^{2} A}{2 \pi R} $$ (An exact expression of the inductance of a toroid with a rectangular cross section is derived in Problem $64 . )$

A toroid has a major radius $R$ and a minor radius $r,$ and is tightly wound with $N$ turns of wire, as shown in Figure P32.12. If $R>>r,$ the magnetic field in the region enclosed by the wire of the torus, of cross-sectional area $A=\pi r^{2},$ is essentially the same as the magnetic field of a solenoid that has been bent into a large circle of radius $R$ . Modeling the field as the uniform field of a long solenoid, show that the self-inductance of such a toroid is approximately
$$
L \approx \frac{\mu_{0} N^{2} A}{2 \pi R}
$$
(An exact expression of the inductance of a toroid with a rectangular cross section is derived in Problem $64 . )$

Key Concepts

Self-Inductance

Self-inductance is a property of an electrical circuit by which a change in current generates an electromotive force (EMF) in the same circuit. It is defined as the ratio of the magnetic flux linkage to the current that produces it. In the context of the toroid, the inductance is computed by summing the contributions of magnetic flux through each of the N turns of wire, leading to the relation L = N?/I.

Magnetic Flux

Magnetic flux is the measure of the quantity of magnetism, taking into account the strength and extent of a magnetic field. It is calculated as the product of the magnetic field and the area perpendicular to the field through which it passes. In calculating the self-inductance of the toroid, the magnetic flux through its cross-sectional area is key to determining the flux linkage over all turns.

Uniform Magnetic Field Approximation

In electromagnetism, assuming a uniform magnetic field simplifies analysis significantly. In this problem, the toroid is approximated as having a uniform magnetic field like that found inside a long solenoid. This approximation is valid because the major radius R is much larger than the minor radius r, which implies that within the cross-sectional area, the variation of the magnetic field is negligible.

Toroidal Geometry

A toroid is a donut-shaped coil characterized by a major radius (the distance from the center of the toroid to the center of its cross-section) and a minor radius (the radius of the cross-section). When the major radius is much larger than the minor radius, the toroid's magnetic behavior closely resembles that of a bent solenoid, allowing for simpler calculations of its magnetic field and self-inductance.

Permeability of Free Space (?0)

The permeability of free space, denoted ?0, is a fundamental physical constant that quantifies the ability of vacuum to support the formation of a magnetic field. It appears in the expression for the magnetic field in a solenoid and consequently in the derivation of self-inductance for the toroid, representing the influence of the medium on the electromagnetic field.

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