(III) A toroid is a solenoid in the shape of a donut (Fig. 20-60). Use Ampére's law along the ci

step 1 All right, so we use Amper's Law, which, so for the first one, first part, path 1, we use Amper'

(III) A toroid is a solenoid in the shape of a donut (Fig....

(III) A $toroid$ is a solenoid in the shape of a donut (Fig. 20-60). Use Amp$\acute{e}$re's law along the circular paths, shown dashed in Fig. 20-60a, to determine that the magnetic field ($a$) inside the toroid is $B = \mu_0 NI/2 \pi R$, 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? FIGURE 20-60 Problem 49. (a) A toroid or torus. (b) A section of the toroid showing direction of the current for three loops: $\odot$ means current toward you, and $\otimes$ means current away from you. (CANT COPY FIGURE)

(III) A $toroid$ is a solenoid in the shape of a donut (Fig. 20-60). Use Amp$\acute{e}$re's law along the circular paths, shown dashed in Fig. 20-60a, to determine that the magnetic field ($a$) inside the toroid is $B = \mu_0 NI/2 \pi R$, 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?
FIGURE 20-60 Problem 49. (a) A toroid or torus. (b) A section of the toroid showing direction of the current for three loops: $\odot$ means current toward you, and $\otimes$ means current away from you. (CANT COPY FIGURE)

Key Concepts

Non-uniformity of the Toroidal Magnetic Field

Unlike the ideal solenoid where the magnetic field inside is considered uniform, the magnetic field within a toroid is not uniform if the toroid has appreciable radial thickness. The field strength varies inversely with the radial distance from the center of the toroid (i.e., it varies as 1/R), meaning that points closer to the inner edge of the winding experience a slightly different field compared to points near the outer edge.

Symmetry in Magnetostatics

Symmetry is an essential concept when applying Ampere's Law. In the case of the toroid, the circular symmetry allows us to assume that the magnetic field has the same magnitude along a circular path at a fixed radius within the toroid. This symmetry simplifies the integral in Ampere's Law and is key to arriving at a tractable expression for the magnetic field.

Ampere's Law

Ampere's Law is a fundamental law of magnetism that relates the integrated magnetic field around a closed loop to the net current passing through the loop. It is expressed mathematically as ?B·dl = ??I_enclosed. This principle is crucial in determining magnetic fields in systems with high symmetry and is the basis for deriving the magnetic field in a toroid.

Magnetic Field in Toroidal Structures

A toroid is a coil of wire wound in a doughnut shape, and its curved geometry creates a magnetic field that loops through its center. By using Ampere's Law along a circular path concentric with the toroid, one finds that the field inside the toroid is given by B = ??NI/(2?R), where R is the distance from the center of the toroid to the chosen path. Outside the toroid, the contributions cancel exactly, yielding zero net magnetic field.

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