A polygon of n equal sides is formed by bending a current carrying wire to total length 2πr which carries a current i.
(a) Find the magnetic field B at the center of the n-sided polygon.
(b) By letting n → ∞, derive the expression for the induced magnetic field at the center of a circular current carrying wire.
a) Using the diagram,
For a polygon of n equal sides, the angle at the centre is .
⇒
Considering angle to be very small
Using Biot–Savart’s law for one side of the n-sided polygon,
⇒
(substituting the value of r)
(substituting the value of l)
For n-sided polygon
B’ =nB
⇒
b) When n → ∞, polygon tends to a circle with radius r and magnetic field will tend toward