A long straight wire of radius `a` carries a steady current `i`. The current is uniformly distributed across its cross section. The ratio of the magnetis field at `(a)//(2) and (2a)` is

Options

A. `(1)/(4)`
B. 4
C. 1
D. `(1)/(2)`

Answer

C

Explanation

(c ) Current density `J = (I)/(pi a^(2))`
From Ampere's circuital law
`oint B. dl = mu_(0) . I_("enclosed")`
For `r lt a`
`B xx 2 pi r = mu_(0) xx J xx pi r^(2)`
`implies B = (mu_(0) I)/(pi a^(2)) xx (r )/(2)`
At `r = a //2, B_(1) = (mu_(0) I)/(4 pi a)`
For `r gt a, B xx 2 pi r = mu_(0) I implies B = (mu_(0) I)/(2 pi r)`
At `r = 2 a, B_(2) = (mu_(0) i)/(4 pi a)`
So, `(B_(1))/(B_(2)) = 1`