Magnetic Force On A Current-Carrying Conductor

The magnetic force, exerted on a current-carrying conductor by a magnetic field, is perpendicular to the plane, containing the current and the magnetic field. This force is the resultant of the forces acting on individual charged particles, which constitute the current. The direction of the force can be realized by the right-hand rule and the magnitude can be derived from Lorentz law of electromagnetic force.

Magnetic force on a moving charge
Lorentz force gives the expression of force on a charged particle due to an electromagnetic field.


Lorentz force on a particle of charge due to an electromagnetic field with electric field strength and magnetic field strength is given by,

overrightarrow {{F_L}}=q(overrightarrow E+overrightarrow vtimes overrightarrow B )



If only a uniform magnetic field is applied on the particle, the magnetic force is,

overrightarrow F=qoverrightarrow vtimes overrightarrow B


The magnetic force is perpendicular to both the magnetic field and the velocity of the charged particle.

Magnetic force on a current – carrying conductor
The free electrons flow with uniform velocity in a conductor. If the amount of charge flowing through a length of the conductor in time t is , the current is given by,

I = frac{q}{t}I=

q = It

The magnetic force acting on the conductor in time is obtained by substituting in the expression of magnetic force,

overrightarrow F=Itoverrightarrow vtimes overrightarrow B


Clearly, the direction of current is towards the direction of velocity of the flow of charges. Since the velocity is uniform,

overrightarrow v= frac{{overrightarrow L }}{t}

overrightarrow L= toverrightarrow v

Here, is the length of the conductor.The magnetic force acting on a length of the conductor carrying uniform current is,

overrightarrow F=Ioverrightarrow Ltimes overrightarrow B


If the direction of current makes an angle with the direction of magnetic field, the magnitude of magnetic force is given by,

F = ILBsin thetaF=ILBsin

The force acts in a direction that is perpendicular to both, the current and the magnetic field. The direction of the magnetic force can then, be understood using,


The thumb, the index finger and the middle finger of the left hand are held in a way, that they are mutually perpendicular to each other. A depiction of this is shown in the figure, above. Based on this positioning, we can interpret the directions of these variables. If the index finger points in the direction of the magnetic field and the middle finger points towards the direction of the current, the thumb then, shows the direction of the magnetic force.

Applications of magnetic force on a current – carrying conductor
The magnetic force acting on a current-carrying conductor can be applied in several ways.

Hall effect: When a magnetic field is applied perpendicularly to the current in a conductor, a voltage difference is created. This voltage difference is transverse to the current. This resulting voltage difference is called the Hall voltage. In the presence of a magnetic field, the current carriers bend due to the magnetic force. The deviated charges accumulate in one side of the conductor. This pattern of charge accumulation causes an accumulation of opposite charges in the two sides of the conductor. So, resulting from this, a voltage difference arises. The direction of the voltage depends on the nature of the current carriers. The experiment of Hall helped to infer that negatively charged particles carry current in conductors.


Using the magnitude and direction of the magnetic force on such a conductor, the magnetic field strength can be determined, in this case.

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