Shown in the figure is a rectangular loop of conductor carrying a current i. The length and breath of the loop are respectively a and b. The magnetic field at the centre of loop is -

Options

A. `(mu_(0)i(a+b))/(2pisqrt(a^(2) + b^(2)))`
B. `(mu_(0)iab)/(2pisqrt(a^(2) + b^(2)))`
C. `(mu_(0)i(a+b))/(pisqrt(a^(2) + b^(2)))`
D. `(2mu_(0)isqrt(a^(2) + b^(2)))/(piab)`

Answer

D

Explanation

`B_(AB) = (mu_(0)I)/(4pib//2)[2xx(a)/(2sqrt((a^(2) + b^(2))/(4)))]`
`B_(AB) = (mu_(0)I)/(2pib)(2a)/sqrt(a^(2) + b^(2))`
`B_(BC) = (mu_(0)I)/(4pi(a)/(2)) xx 2 xx (b)/(2sqrt((b^(2) + a^(2))/(4))) = (mu_(0)Ib)/(piasqrt(a^(2) + b^(2)))`
`B_("net") = 2B_(AB) + 2B_(BC)`
`= (2mu_(0)Ia)/(bpisqrt(a^(2) + b^(2))) + (2mu_(0)Ib)/(piasqrt(a^(2) + b^(2)))`
` = (2mu_(0)I)/(pisqrt(a^(2) + b^(2))) ((a)/(b) + (b)/(a))`
`= (2mu_(0)I)/(pisqrt(a^(2) + b^(2))) (a^(2) + b^(2))/(ab) = (2mu_(0)I)/(pi) (sqrt(a^(2) + b^(2)))/(ab)`