Question

Obtain an expression for magnetic induction of a toroid of \( N \) turns about an axis passing through its centre and perpendicular to its plane.

Hint:

The magnetic field inside a toroid depends on the number of turns, the current, and the radius. It is strongest at the center and weakest at the edges.

Answer 1

Step 1: Magnetic field in a toroid.
A toroid is a solenoid bent into a circular shape. The magnetic field inside the toroid can be derived using Ampère's law, which states: \[ \oint B \cdot dl = \mu_0 I_{\text{enc}} \] where \( B \) is the magnetic field, \( dl \) is an infinitesimal length element along the path of integration, and \( I_{\text{enc}} \) is the total current enclosed by the path.
Step 2: Apply Ampère's Law.
For a toroid, the path of integration is a circle of radius \( r \), and the current enclosed is \( N I \), where \( N \) is the number of turns and \( I \) is the current through each turn. \[ B \cdot (2 \pi r) = \mu_0 N I \] Solving for \( B \), we get: \[ B = \frac{\mu_0 N I}{2 \pi r} \]
Step 3: Conclusion.
The magnetic induction inside the toroid is \( B = \frac{\mu_0 N I}{2 \pi r} \), where \( r \) is the radius of the toroid.