An infinte wire place along z-axis has current I in positive z-direction A conducting rod placed in xy plane paralle to y-axis has current `I_(2)` in positive y-direction The ends of the rod subtend `+30^(@)` and `-60^(@)` at the origin with positive x direction The rod is at a distance a from the origin. Find net force on the rod .

Answer

`(mu_(0)I_(1)I_(2))/(4pi)` in (3) along -ve z direaction

Explanation

`(x)/(a) = tantheta`
`x = a tantheta`
`dx = asec^(2) thetadtheta`
`B =(mu_(0))/(4pi) (2I_(1))/ (R ) …(2)`
`dF = I_(2) dx B sintheta (-hatk)…(3)`
From 1,2& 3
`dF = I_(2) asec^(2)thetadtheta (mu_(0))/(4pi) (2I_(1))/(a sectheta). Sin theta = (mu_(0))/(4pi) 2I_(1)I_(2)`
`sin theta sec thetadtheta`
`F =(mu_(0))/(4pi) 2I_(1)I_(2)underset(-60^(@))overset(30^(@))inttan thetadtheta = (mu_(0))/(4pi) 2I_(1)I_(2)[-log_(e) cos0]_(60^(@))^(30^(@))`
`F = - (mu_(0))/(4pi) 2I_(1)I_(2) [-log_(e)cos30^(@)- log_(e) cos(-60^(@))]`
`=-(mu_(0))/(4pi) 2I_(1)I_(2)[log "sqrt3/(2) xx 2]`
`F =(mu_(0))/(4pi) 2I_(1)I_(2)ln_(3) (-hatk)`
`dF = I_(2) dxBsintheta`
`F=I_(2) dx (mu_(0))/(4pi) (2I_(1))/(sqrt(a^(2) + x^(2)))xx(xx)/(sqrt(a^(2)+x^(2)))`
`F =(mu_(0))/(4pi) 2I_(1)I_(2) underset(-sqrt3a)overset(a//sqrt3)int(2x)/((a^(2) +x^(1)))dx`
`F =mu_(0)/(4pi) I_(1)I_(2) [log_(e) (a^(2)+x^(2))]_(+sqrt3a)^(a//sqrt3)`
`F =(mu_(0))/(4pi) I_(1)I_(2)[log_(e)"(4a^(2))/(3)log_(e) 4a^(2)]=(mu_(0))/(4pi) I_(1)I_(2)ln3`
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