A square frame of side 1 ~m carrying a current I produces a magnetic field B at its centre. The same current is passed through a circular coil having the same perimeter as the square. The magnetic field at the centre of the circular coil is B^ . The ratio of B B^ is

Step-by-step solution

Given, Side of square frame, a=1 ~m Electric current =I For square, Let r be the distance of centre O from all sides, r= a 2 Since, all sides of square produces equal field in the same direction. Using, B=4 [ ₀ I 4 r ( 45^ + 45^ ) ] B= 4 ₀ I 4 ( a 2 ) ( 1 2 + 1 2 ) B= 2 ₀ I a 2 2 B= 2 2 ₀ I a ...(i) For circle, Let R be the radius of circular coil. Then, 2 R=4 a R= 4 a 2 = 2 a Using the expression for magnetic field at centre of circular coil, B^ = ₀ I 2 R B^ = ₀ I 2 ( 2 a ) = ₀ I 4 a ...(ii) From Eqs. (i) and (ii)