A current I flows through a thin wire shaped as regular polygon of n sides which can be inscribed in a circle of radius R. The magnetic fiedl induction at the center of polygon due to one side of the polygon is

Explanation

Angle subtended by one side at centre `=2pi//n`
`d=R cos (pi//n)`
Magnetic field at `O` due to one side
`B_(1)=(mu_(0)i)/(4pid)(sin.(pi)/(n)+sin.(pi)/(n))`
`=(mu_(0)i)/(4pi.R.cos(pi/n)).2sin(pi/n)`
`=(mu_(0)tan(pi//n))/(2piR)`
Magnetic field due to polygon
`B_(O)=nB_(1)=(mu_(0)ni tan(pi//n))/(2piR)`
when `ntooo`, polygon becomes circle
`B_(O)=(mu_(0)i)/(2R)`