A rectangular polygon of n side is formed by bending a wire of total length `2piR` which carries a current i. Find the magnetic field at the centre of the polygon

Explanation

One side of polygon `a = (2pir)/(n)`
`alpha = beta = ((2pi)/(n))//2 = (pi)/(n) , cot alpha = (d)/(a//2)`
` d = (a)/(2) cot alpha = (pi R)/(n) cot"" ((pi)/(n))`
All sides of polygen produce magnetic field at O in same direction.
so,, B = n (magnetic field due to one side)
` = n (mu_0i)/(4pi)(n)/(pi R cot ""(pi)/(n))(sin ""pi/n + sin "" pi/n)`
`= (mu_0i)/(4R) (n^2)/(pi^2)(2 sin pi//n)/(cos pi//n) . sin ""pi/n`
`B = (mu_0i)/(2R) xx ({sin pi//n//_(pi//n)}^2)/(cos pi//n)`