A wire of total length 2 r carrying a current i is bent to form a regular polygon of n sides. By taking the limit as n , determine the expression for the magnetic field at the centre of the resulting circular current.

Step-by-step solution

Let the regular polygon have n sides. The total length of the wire is 2 r , so the length of each side is a = 2 r n . The perpendicular distance from the center to any side is d = a 2 ( n ) = r n ( n ) . The magnetic field at the center due to one side of the polygon is given by: B₁ = ₀ i 4 d [ ( n ) + ( n ) ] = ₀ i 2 d ( n ) The total magnetic field due to all n sides is: B = n B₁ = n ₀ i 2 d ( n ) Substituting the value of d : B = n ₀ i 2 [ r n ( n ) ] ( n ) = ₀ i 2r ( n )^2 ( n ) ( n ) Taking the limit as n , we