Magnetic Induction due to toroid

Consider a toroid carrying a current i0. Point P is within the toroid while Q is inside and point r outside.

By symmetry, direction of B at any point is tangential to a circle drawn through that point with same centre as that of toroid. The magnitude of B on any point of such a circle will be constant. Let us consider a point P within the toroid. Let us draw a circle of radius r through it. Applying Ampere’s law to this circle, we have

    Ñ„ B . dl = μ₀i,                     …(1)
where i is the net current enclosed by the circle. Now,
    Ñ„ B. dl = B (2πr)
and    i = Ni0, where N is the total number of turns in the toroid.
. :    Eq. (1) becomes B (2πr) = μ₀ Ni₀
or        B = μ₀     Ni₀
             _  2π     _r

Thus the field B varies with r.
If l be the mean circumference of the toroid, then l = 2πr, so that
        B = μ₀Ni₀
                  l
The field B at an inside point such as Q is zero because there is no current enclosed by the circle through Q.
The field B at an outside point such as R is also zero because there is no current enclosed by the circle through R will be zero. This is because each turn of the winding passes twice through this area enclosed by the circle, carrying equal currents in opposite directions.
The field of a toroid is thus zero at all point except within the core.