Question

Derive an expression for the force `vec F` acting on a conductor of length L and area of cross-section A carrying current I and placed in a magnetic field `vec B`.

Answer 1

A current-carrying conductor in a magnetic field experiences a magnetic force due to the motion of charge carriers.

Force on a moving charge `(vec f) = q(vec v xx vec B)`

Let:

Number density of electrons = n

Drift velocity = `vec v_d`

Volume of conductor = AL

Total number of electrons (N) = nAL

Force on one electron `(vec f) = -e(vec v_d xx vec B)`

Total force on all electron `(vec F) = N vec f`    ...(i)

= `n A L(-e)(vec v_d xx vec B)`

= `-n e A L(vec v_d xx vec B)`

Current (I) = neAv_d

⇒ `n e A vec v_d = I vec l`

Substitute:

`vec F = -L(I hat l xx vec B)`

Force direction is defined using conventional current direction (opposite electron motion). Thus:

`vec F = I(vec L xx vec B)`

Magnitude (F) = ILB sin θ