The figure shows top view of a conducting rob AB, bent to form a semicircle of radius R, carrying current I, placed on a smooth horizontal table. It is subjected to a uniform magnetic field B perpendicular to its plane as shown. The rod is held at rest in its position by two light strings attached to its ends. Calculate the magnetic force acting on the conductor AB and the tension in the strings.
Strategy: To find the force conductor, consider a small arc of length `dl = Rd theta`. This arc can be consider as a straight conductor of length dl.

Explanation

`dF = Bidl`
`rArr dvec(F) = Bidl [-cos theta hat(i) + sin theta hat(j)]`
`rArr dvec(F) = BiRd theta [-cos theta hat(i) + sin theta hat(j)]`
Net force on the rod is given by
`vec(F)_(m) = int_(0)^(pi) dvec(F)`

`rArr vec(F)_(m) = BiR [-int_(0)^(pi) cos theta d theta hat(i) + int_(0)^(pi) sin theta d theta hat(j)]`
`rArr vec(F)_(m) = BiR [theta + 2hat(j)]`
`= 2BiR hat(j)`
`rArr` The force experienced by AB is 2BiR, perpendicular to its length AB
For equilibrium, `2T = F_(m) rArr T = BiR`