Apply Ampere's Circuital Law to find the magnetic field both inside and outside of a toroidal solenoid.

Explanation

A solenoid is bent in such a way its ends are joined together to form a closed ring shape , is called a toroid. The magnetic field has constant magnitude inside the toroid whereas in the interior region (say, at point P ) and exterior region (say, at point Q), the magnetic field is zero.
(a) Open space interior to the toroid : let us calculate the magnetic field `B_(p)` at point P. We construct an Amperian loop l of radius `r_(1)` around the point P. For simplicity, we take circular loop so that the lenth of the loop is its circumference .
`L_(1) = 2 pi_(1)`
Ampere.s circuital law for the loop l is
`oint_("loop 1") vec(B_(p)).vec(d)l = mu_(0) I_("enclosed")`
since, the loop l enecloses no current, `I_("enclosed")` = 0
`oint_("loop") vec(B_(P)). vec(d)l = 0`
this is possible only if the magnetic field at point P vanishes i.e.,
`vec(B_(p)) = 0 `
(b) Open space exterior to the toroid : let us calculate the magnetic field `B_(Q)` at point Q. We construct an Amperian loop 3 of radius `r_(3)` around the point Q. the length of the loop is `L_(3) = 2pi r_(3)`
Ampere. circuital law for the loop 3 is
`oint_("loop 3") vec(B_(Q)).vec(d)l = mu_(0) I_("enclosed")`
since, in each turn of the toroid loop , current coming out of the plane of paper is cancelled by the current going into the plane of paper . thus , `I_("encloed") = 0`
`oint_("loop 3") vec(B_(Q)).vec(d)l = 0`
This is possible only if the magnetic field at point Q vanishes i.e, `vec(B_(Q))` = 0
(c ) Inside the toroid : Let us calculate the magnetic field `B_(S)` at point S by constructing an Amperian loop 2 of radius `r_(2)` around the point S. the length of the loop is `L_(2) = 2 pi r_(2)` Ampere.s circuital law for the loop 2 is
`oint_("loop 2") vec(B_(s)).vec(d)l = mu_(0) I_("enclosed")`
Let I be the current passing through the toroid and N be the number of turns of the toroid then `I_("enclosed")` = NI
and `oint_("loop 1") vec(B_(s)).vec(d)l = oint_(loop 2) "B dl cos" theta = B 2 pi r_(2)`
`oint_("loop 1") vec(B_(s)).vec(d)l = mu_(0) NI`
`B_(s) = mu_(0) (NI)/(2pi r_(2))`
The number of turns per unit length is n = `(N)/(2pi r_(2))` then the magnetic field at point S is `B_(s) = mu_(0)` nI