(II) A toroid is a solenoid in the shape of a circle (Fig....

(II) A toroid is a solenoid in the shape of a circle (Fig. $20-62$ ). Use Ampere's law along the circular path, shown dashed in Fig. $20-62$ a, to determine that the magnetic field $(a)$ inside the toroid is $B=\mu_{0} N I / 2 \pi R,$ where $N$ is the total number of turns, and $(b)$ outside the toroid is $B=0 .(c)$ Is the field inside a toroid uniform like a solenoid's? If not, how does it vary?

Key Concepts

Solenoids

A solenoid is a coil of wire designed to create a magnetic field when an electric current passes through it. In the ideal case of an infinitely long solenoid, the magnetic field inside is uniform and parallel to the axis of the solenoid, while the field outside is negligibly small. Solenoids serve as a comparison to toroids in the analysis of magnetic field distribution and uniformity.

Magnetic Field Uniformity in Toroids

While a long solenoid produces a nearly uniform magnetic field along its axis, the field within a toroid is not uniform. Instead, the magnetic field in a toroid decreases inversely with the distance from the center of the circle (i.e., it is proportional to 1/R). This variation arises due to the curvature of the toroid and the resulting change in the path length over which the magnetic field is integrated.

Ampere's Law

Ampere’s law relates the line integral of the magnetic field around a closed loop to the net current enclosed by the loop. It is a fundamental tool for calculating magnetic fields in situations with high symmetry, such as in toroids and solenoids. The law is expressed mathematically as ?B·dl = ??I_enc, and its application simplifies the derivation of the magnetic field inside a toroid.

Toroidal Coil

A toroidal coil is a solenoid bent into a circular shape, forming a donut-like configuration. The circular geometry causes the magnetic field lines to be confined within the core of the toroid, minimizing stray fields outside of it. The geometry also leads to a magnetic field that varies with the radial distance from the center of the toroid, unlike the nearly uniform field typically found in an ideal solenoid.

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