Induced Magnetic Fields

Magnetic Induction: Center vs. Axis of a Coil

We want to find the magnetic induction at a point on the axis of a circular coil, a distance (the radius) away from the center, in terms of the magnetic induction at the center of the coil.

Step 1: Magnetic Induction at the Center of a Circular Coil

The magnetic induction ( ) at the center of a circular coil of radius with turns carrying a current is given by:

Where is the permeability of free space.

Step 2: Magnetic Induction on the Axis of a Circular Coil

The magnetic induction ( ) at a point on the axis of the same circular coil, at a distance from the center, is given by:

Step 3: Setting

We are given that the distance from the center is equal to the radius, so we set :

Step 4: Relate to

We want to express in terms of . Notice that :

Conclusion

The magnetic induction at a point on the axis of the coil at a distance equal to the radius of the coil is not any of the solution set but follows to be