A uniform magnetic field `vec(B) = B_(0) hat(k)` exists in a region. A current carrying wire is placed in x-y plane as show. Find the force acting on the wire AB, if each section of the wire is of length 'a'.

Explanation

Method (1)
The conductor consists of 5 straight sections viz AC, CD, DE, EF and FB as shown, each of length a
`vec(F)_(AC) = l (a hat(i) xx B_(0) hat(k)) = - B_(0) l a hat(j)`
`vec(F)_(CD) = l (a hat(j) xx B_(0) hat(k)) = B_(0) l a hat(i)`
`vec(F)_(DE) = l (a hat(i) xx B_(0) hat(k)) = - B_(0) l a hat(j)`
`vec(F)_(EF) = l (- a hat(j) xx B_(0) hat(k)) = - B_(0) l a hat(i)`
`vec(F)_(FB) = l (a hat(i) xx B_(0) hat(k)) = - B_(0) l a hat(j)`
Net force, `vec(F) = vec(F)_(AC) + vec(F)_(CD) + vec(F)_(DE) + vec(F)_(EF) + vec(F)_(FB)`
`=-3 B_(0) l a hat(j)`
Method (2)
Instead of finding the force on individual wires, we can directly find the net force by finding effective length of the wire. The line joining A and B has length 3a thus net force equals
`vec(F) = l [3 a hat(i) xx B_(0) hat(k)]`
`rArr vec(F) = 3a B_(0) l (-hat(j))`
`rArr vec(F) = - 3a B_(0) l hat(j)`