A toroid is like a solenoid that has been bent around in a...

A toroid is like a solenoid that has been bent around in a circle until its ends meet. The field lines are circular, as shown in the figure. What is the magnitude of the magnetic field inside a toroid of $N$ turns carrying current $I$ ? Apply Ampère's law, following a field line at a distance $r$ from the center of the toroid. Work in terms of the total number of turns $N$, rather than the number of turns per unit length (why?). Is the field uniform, as it is for a long solenoid? Explain.

Key Concepts

Ampère's Law

Ampère's Law is a fundamental law in electromagnetism that relates the integrated magnetic field around a closed loop to the electric current passing through the loop. It is expressed as ?B·dl = ??I_enc, and in this context, it is used to derive the magnetic field inside the toroid by considering a circular path centered on the toroid’s axis.

Toroidal Geometry

A toroid is a doughnut-shaped coil in which the coil is bent so that its ends meet, forming a closed loop. This geometry confines the magnetic field mainly within the core of the toroid, following circular paths, which is crucial for applying Ampère's law effectively and understanding the dependence of the magnetic field on the radial distance from the center.

Magnetic Field Distribution

Unlike a long solenoid where the magnetic field is approximately uniform inside, the magnetic field in a toroid varies with the radius. When applying Ampère’s law, the result shows that the field is proportional to I/(2?r), indicating that the field strength decreases with increasing radial distance from the center.

Total Number of Turns vs. Turns per Unit Length

In a toroid, the total number of turns N is used instead of turns per unit length because the geometry of the toroid does not have a uniform cross-sectional extent over an infinite length. The enclosed current in Ampère’s law is the sum of the currents through all the turns that the Amperian loop encloses, making the total number of turns a more natural and direct parameter in the analysis.

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