## Magnetic Induction Due To Toroid

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# 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

ф

**B**. dl = μ

_{0}i, …(1)

where i is the net current enclosed by the circle. Now,

ф

**B**. d

**l**= B (2πr)

and i = Ni0, where N is the total number of turns in the toroid.

. : Eq. (1) becomes B (2πr) = μ

_{0}Ni

_{0}

or B =

__μ__

_{0}__Ni__

_{0 }_ 2π _r

Thus the field B varies with r.

If l be the mean circumference of the toroid, then l = 2πr, so that

B =

__μ__

_{0}Ni_{0 }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.

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