Wire 1 in Fig. is oriented along the y-axis and carries a steady current `I_1`. A rectangular loop located to right of the wire and in the x-y plane carries a current `I_2`. Find the magnetic force exerted by wire (1) on the top wire (1) on the top wire of length b in the loop, labeled "wire (2)" in the figure.

Options

A. `[ (mu_0 I_1 I_2)/(2pi) ln (1 + b/a) hatj ]`
B. `[(mu_0 I_1 I_2)/(2pi) ln (1+ b/a) (-hatj) ]`
C. `[ (mu_0I_1I_2)/(2pi) ln (1 + b/a) (-hati) ]`
D. `[ (mu_0I_1 I_2)/(2pi) ln ( 1+ b/a) (hati) ]`

Answer

A

Explanation

`[ (mu_0 I_1 I_2)/(2pi) ln (1+ b/a) hatj ]`
magnetic field `(vecB)` created by wire -I at a distance x from it ,ie
` vecB = (mu_0 I_1)/(2pi x) (-k)`
Force acting on a smalll element dx of wire 2, i.e.,
` vecdF = I_2 vecdx xx vecB " or " vecdF = I_2 vecdx xx ( (mu_0 I_1)/(2pi x) (-k) ) `
` = (mu_0 I_1 I_2)/(2pi x) dx [ hati xx (-hatk) ] (as vecdx = dx hati)`
` = ( (mu_0 I_1 I_2)/(2pi x) dx) hatj , vecF = int vecdF = [(mu_0 I_1 I_2)/(2pi) int_(a)^(1+b) 1/x dx]`
`hatj = [ (mu_0I_1 I_2)/(2pi) ln (1 + b/a) ] hatj`
`vecF` point upward as indicated by `hatj `