A toroid is a solenoid bent into the shape of a doughnut. It looks similar to a toy Slinky® with…

A toroid is a solenoid bent into the shape of a doughnut. It looks similar to a toy Slinky® with ends joined to make a circle. Consider a toroid consisting of N turns of a single wire with current I flowing through it. (Figure 1) Consider the toroid to be lying in the r ? plane of a cylindrical coordinate system, with the z axis along the axis of the toroid (pointing out of the screen). Let ? represent the angular position around the toroid, and let r be the distance from the axis of the toroid. For now, treat the toroid as ideal; that is, ignore the component of the current in the ?^

»A toroid is a solenoid bent into the shape of a doughnut. It looks similar to a toy Slinky® with ends joined to make a circle. Consider a toroid consisting of N turns of a single wire with current I flowing through it. (Figure 1) Consider the toroid to be lying in the r ? plane of a cylindrical coordinate system, with the z axis along the axis of the toroid (pointing out of the screen). Let ? represent the angular position around the toroid, and let r be the distance from the axis of the toroid. For now, treat the toroid as ideal; that is, ignore the component of the current in the ?^ direction. Part C What is B ( r ), the magnitude of the magnetic field inside the toroid and at a distance r from the axis of the toroid? Express the magnetic field in terms of I , ? 0 (the permeability of free space), N , and r . Part D In an ideal toroid, current would flow only in the r ^ and k ^ directions. The magnetic field in the central plane, outside of the coils of such a toroid, is zero. For the toroid shown in the figures however, this field is not quite zero. This is because in this problem, there is a single wire that is wrapped around a doughnut shape. This wire must point somewhat in the ? ? ^ direction, and thus the current must actually have a component in the ? ? ^ direction. Compute B , the magnitude of the magnetic field in the center of the toroid, that is, on the z axis in the plane of the toroid. Assume that the toroid has an overall radius of R (the distance from the center of the toroid to the middle of the wire loops) and that R is large compared to the diameter d of the individual turns of the toroid coils. Note that whether the field points upward or downward depends on the direction of the current, that is, on whether the coil is wound clockwise or counterclockwise. Express B in terms of ? 0, R , I , N , and the local diameter d of the coils.

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