Question

The magnetic field inside a toroidal solenoid is

Answer 1

Toroidal Solenoid:

1. A toroidal solenoid is also known as simply toroid or endless solenoid.
2. A long wire wound closely around a circular ring forms a toroid.
3. Considering the symmetry of the problem, it can be argued that the lines of force would be circles whose centers lie on the axis of the toroid passing through $O$.
4. To apply Ampere's law, we consider one such circle as an Amperian loop.
5. The magnetic field will be of the same magnitude and directed tangent to the loop at every point.

Derivation:

So for loop $1$ lying inside the solenoid, if $B$ is the magnetic field, $dl$ is the area element, $R$ is the radius of the Amperian loop $1$, $N$ is the total number of turns on the toroid and $I$ is the steady current through each turn,

$\oint \overrightarrow{B}.d\overrightarrow{l}=B\oint dl=B.2\mathrm{\pi R}={\mathrm{\mu }}_0\mathrm{NI}$

Here $NI$ is the current enclosed by the loop $1$.

Thus,

$B=\frac{{\mu }_{0}NI}{2\mathrm{\pi R}}$

If $\frac{N}{2\mathrm{\pi R}}=n$ be the number of turns per unit length then

$B={\mu }_{0}nI$.

For Amperian loop $2$ chosen inside or outside the toroid total enclosed current is $NI-NI=0$ and hence,

$$\begin{gathered} \oint \overrightarrow{B}.d\overrightarrow{l}=0 \\ \Rightarrow B.2\mathrm{\pi r}=0 \\ \Rightarrow \mathrm{B}=0 \end{gathered}$$

This means that inside or outside the toroid magnetic field is zero.

Hence, the magnetic field inside a toroidal solenoid is zero.