Question

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.

A. `[(mu_(0)I_(1)I_(2))/(2pi)ln(1+(b)/(a))hat(j)]`
B. `[(mu_(0)I_(1)I_(2))/(2pi)ln(1+(b)/(a))(-hat( j))]`
C. `[(mu_(0)I_(1)I_(2))/(2pi)ln(1+(b)/(a))(-hat( i))]`
D. `[(mu_(0)I_(1)I_(2))/(2pi)ln(1+(b)/(a))(hat( i))]`

Answer 1

Correct Answer - 1
`[(mu_(0)I_(1)I_(2))/(2pi)ln (1+(b)/(a))hat(j)]`
Magnetic field `(vec(B))` created by wire `-I` at a distancex from it, i.e.,
`vec(dF)=I_(2)vec(dx)xxvec(B)or vec(dF)=I_(2)vec(dx)xx((mu_(0)I_(1))/(2pix)(-k))`
`=(mu_(0)I_(1)I_(2))/(2pix)dx[hat(i)xx(-hat(k))](asvec(d)x=dxhat(i))`
`=((mu_(0)I_(1)I_(2))/(2pix)dx)hat(j),vec(F)=intvec(d)F=[(mu_(0)I_(1)I_(2))/(2pi)underset(a)overset(a+b)int(1)/(x)dx]`
`hat(j)=[(mu_(0)I_(1)I_(2))/(2pi)ln(1+(b)/(a))]hat(j)`
`vec(F)` point upward as indicated by `hat(j)`