Obtain an expression for the magnetic force on a current carrying conductor.

Explanation

Let l be the length of a conductor. Let A be its area of cross section. Let n be the number density of mobile charge carriers in the conductor.
The total number of mobile charge carries is nAl.
Let the averrage drift velocity be `'v_(d)'` corresponding to the current I.
Let B be an external magnetic field surrounding the conductor. Magnetic force on these carriers is,
`vec(F)=(nAlq)(vec(v_(d)) times vec(B))` where `q^(')=nalq`
By definition current density `j=I/A`
and q = e (charge on the electron)
`" "vec(F)="nAle "vec(v)_(d) timesvec(B)`
`" "=("ne"vec(v)_(d))Al times vec(B)`
`" "=vec(j)Al times vec(B)`
`" "=abs(vec(j))(hat(j)l)A times vec( B)`
`therefore" "vec(F)=Ivec(l) times vec(B)`
The direction of current is in the direction of `vec(l)` vector, `vec(l)=hat(j)l " and " jA=I=` current.