Encyclopedia of Electrical Engineering
Encyclopedia of Electrical Engineering

Force on a Current Carrying Conductor

A current-carrying conductor produces a magnetic field around it. i.e. behaves like a magnet and exerts a force when a magnet is placed in its magnetic field. Similarly, a magnet also exerts equal and opposite force on the current-carrying conductor. The direction of this force can be determined using Fleming's left-hand rule.

Magnetic Force on a single Current-Carrying Conductor

Electric current is an ordered movement of charge. Because charges ordinarily cannot escape a conductor, the magnetic force on charges moving in a conductor is transmitted to the conductor itself. A current-carrying wire in a magnetic field must, therefore, experience a force due to the field.
Force on a current-carrying conductor Force on a current-carrying conductor.
We can derive an expression for the magnetic force on a current by taking a sum of the magnetic forces on individual charges. The force on an individual charge moving at the drift velocity $v_d = l/t$ in a length $l$ of the wire is given by $$ F = q v_d B sin \phi$$
Taking B to be uniform over a length of wire $l$ and zero elsewhere, the total magnetic force on the wire is then $$ F = (q v_d B sin \phi) (n) $$ where $n$ is the number of charge carriers in the section of wire of length $l$. Gathering terms, $$ F=( nq v_d )Bsin\phi$$ Now, we are going to involve current $I$ and length $l$ of the wire instead of charge $q$ and velocity $v_d$. hence $nq (v_d) = nq (l/t) = nq/t (l) = I l$, the above equation will become
$$\bbox[5px,border:1px solid red] {\color{blue}{F=I.l.Bsin\phi}}$$Eq.(1)
Where F is force acting on a current carrying conductor, B is magnetic flux density (magnetic field strength), I is magnitude of current flowing through the conductor, l is length of conductor, $\phi$ is angle that conductor makes with the magnetic field.
When the conductor is perpendicular to the magnetic field, the force will be maximum. When it is parallel to the magnetic field, the force will be zero.
The direction of this force is always right angles to the plane containing both the conductor and the magnetic field and is predicted by Fleming's Left-Hand Rule.