Two circular coils X and Y, having equal number of turns and carrying currents in the same sense, subtend same solid angle at point O. If the smaller coil X is midway between O and Y and if we represent the magnetic induction due to bigger coil Y at O as `B_y` and the due to smaller coil X at O as `B_x`,then find the ratio `B_x//B_y`.

Options

A. `(B_(Y))/(B_(X))=1`
B. `(B_(Y))/(B_(X))=2`
C. `(B_(Y))/(B_(X))=(1)/(2)`
D. `(B_(Y))/(B_(X))=(1)/(4)`

Answer

C

Explanation

As two coils subtedn the same solid angle at O, hence area of coi, Y=4`xx` area of coil X
`["solid angel"=("area")/(bot"distance")^(2)]`
i.e. radius of coil Y=2 `xx` radius of coul X
`:. B_(Y)=(mu_(0))/(4pi)xx(2piI(2r)^(2))/([(2r)^(2)+(d^(2))]^(3//2))`
`B_(x)=(mu_(0))/(4pi)xx(2piI(r)^(2))/([r^(2)+((d)/(2))^(2)]^(2//3))`
`(B_(y))/(B_(X))=(4)/((4r^(2)+d^(2))^(3//2))xx[(4r^(2+d^(2)))/(4)]^(3//2)`
`" " (4)/((4)^(3//2))=(4)/(8)=(1)/(2)`