A square conducting loop of side length L carries a current I.The magnetic field at the centre of the loop is (dependence on L)

Options

A. Independent of L
B. Proportional to `L^(2)`
C. Inversely proportional to L
D. Linearly proportional to L

Answer

C

Explanation

For one side
`B_(1) = (mu_(0) I)/(4pi r) [sin 45^(@) + sin 45^(@)]`
`= (mu_(0) xx 2l)/(4pi (L)) [ (1)/(sqrt2) + (1)/(sqrt2)]`
`= (2 sqrt2 mu_(0) I)/(4pi L)`

B due to square
`B = 4B_(1) = (mu_(0))/(pi ) (2sqrt2 I)/(L) , { r = sqrt(((L)/(sqrt2))^(2) - ((L)/(2))^(2))= sqrt((L^2)/(2) - (L^2)/(4)) = (L)/(2)}`
`therefore B prop (1)/(L)`