The formula for the magnitude of magnetic field inside a toroid is:

$$B = \frac{\mu_oNI}{2\pi r}$$

where $N$ is the number of turns threading a curve along, maybe, the axis of the toroid or along a field line inside the toroid. And $I$ is the current in each turn. Now, if there are $n$ turns per unit length of this curve, then for the whole curve, which is a circle, I can write:

$$B = \frac{\mu_o(2\pi rn)I}{2\pi r}$$ which is $$B = \mu_o nI,$$

but that's the magnitude of magnetic field inside a solenoid or should I say non-toroidal solenoid. Even more, it says that the magnetic field is constant inside the toroid? Or maybe not? Where am I going wrong? I know calculus only roughly. Thank you.


1 Answer 1


You made a mistake - the number of turns per unit length in a torus depends on where you are measuring. On the inner surface they are a little closer than on the other surface.

Because of this, the expression for the toroidal coil should be

$$B = \frac{\mu N I}{2\pi r}$$

This means that the field is a little stronger towards the inner radius of the toroid than towards the outer radius. For the derivation, see http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/toroid.html


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