An equilateral triangular frame with side a carrying a current `I` is placed at a distance a from an infinitely long straight wire carrying a current `I` as shown in the figure. One side of the frame isparallel to the wire. The whole system lies in the xy-plane. Find the magnetic force F acting on the frame.

Answer

A

Explanation

Force on wire 12 will be `(mu_0/(2pi) I/a)(Ia)` i positive x-direction.

Force on 2-3 and 3-1
`r=axsin60^@a+sqrt3x/2`
`B_r=mu_0/(2pi) I/r=mu_0/(2pi)1/(a+sqrt3x/2)`
`mu_0/pi I/((2a+sqrt3x)`
`:. F_(231) [2int_(x=0)^(x=a) (dFcos60^@)](-hati)`
`=[2int_0^amu_0/(2pi). I/(2a+sqrt3x).I.(dx) 1/2](-hati)`
`=((mu_0I^2)/pi) .1/sqrt3 [ln (2a+sqrt3x)]_0^a(-hati)`
`=(mu_0I^2)/(sqrt3pi) ln ((2+sqrt3)/2) (-hati)`
`and F_12=(mu_0I^2)/(2pi)(hati)`
`:. F_(Total)=(mu_0I^2)/pi[1/2-1/sqrt3 ln ((2+sqrt3)/2)](hati)`