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 consistsof 5 straight sections viz AC, CD, DE, EF and FB as shown, each of length a
`vecF_(AC)=l(ahatixxB_(0)hatk)=-B_(0)Iahatj`
`vecF_(CD)=I(ahatjxxB_(0)hatk)=B_(0)Iahatj`
`vecF_(DE)=I(ahatixxB_(0)hatk)=-B_(0)Iahatj`
`vecF_(EF)=I(-hatjxxB_(0)hatk)=-B_(0)Iahati`
`vecF_(FB)=I(ahatixxB_(0)hatk)=-B_(0)Iahatj`
Net force `vecF=vecF_(AC)+vecF_(CD)+vecF_(EF)+vecF_(FB)=-3B_(0)Iahatj`
Method (2) Instead of finding the force on individual wires, we can directly find the net force finding effective length of the wire. The line joining A and B has length 3a thus net force equals,
`vecF=I[3hatixxB_(0)hatk]`
`impliesvecF=-3B_(0)(-hatj)`
`impliesvecF=-3aB_(0)Ihatj`