Force on A Current-carrying Conductor

When a conductor carrying a current is placed in a magnetic field, the conductor experiences a magnetic force.

  • 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.


left hand rule
Diagram by Jfmelero. Link: Available under the Creative Commons Attribution-Share Alike 3.0 Unported license.

Referring to the diagram above, F is Force, B is Magnetic field, I is current.

Factors affecting magnetic force on a current-carrying conductor in a magnetic field:

  • Strength of the magnetic field
  • Current flowing through the wire
  • Length of the wire


magnetic force
$F = B I l \, sin \, \theta$, 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,
  • $\theta$ 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.


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46 thoughts on “Force on A Current-carrying Conductor”

      • The four fingers indicate the direction of flux created which will be perpendicular to the direction of current flow

    • It is based on the direction of current flowing in the conductor of the electromagnet. Use right hand thumb rule. Holding right hand thumb upward and remaining four finger closed, means current coming out of the conductor, and the four finger close in the anticlockwise direction and vice versa.

        • Assalamu Alaikum dear brother!
          If you put cos instead of sin, force will be zero at 90 angle mathematically but during experiment you will observe that force is acting on conductor and it is not zero. Hence considering cos instead of sin does not make sense. Hope you have got some idea, Thanks!

        • From the hand rule, the direction of force is vertical and the field strength is horizontal. Force being a vector quantity, the component of that force in the direction drawn is the sine of the angle inclined at the horizontal lines. Hence sine theta. If the angle is at the vertical line it would have been cos theta.
          Hope you find this helpful.

    • I believe that you are in the wrong section. In A Levels, L represents the length of the conductor and hence, is treated as a scalar. The directions in the equation are handled by the $\sin \theta$.

      At a higher level (University/College), L is NOT the length of the conductor. L represents the element of the current carrying conductor (that is in the magnetic field). Hence, L contains the length of the conductor (scalar part) and the direction of the conductor (vector part). The cross-product of L and B will give rise to the $\sin \theta$.

      Hope this helps.

  1. Can I receive full notes of electromagnetism for a level course through my below email address. The notes are good. Kindly i am asking for references in my studies.

  2. what does this tell us about how changes in current will affect the force using on a wire that is when kept inside a magnetic field

    • ADD-ONS
      Cuz according to my knowledge,when the length of wire increases,the resistance increases too. my question is why the longer the length of conductor in the magnetic field,the greater the force on the conductor? isn’t it should be the shorter the length of conductor in the magnetic field,the greater the force on the conductor?

      • In order to answer your question, we will have to go to the very basics. Here are the few things that you will need before we start:
        1) A moving electron will experience a force in a magnetic field. (See Motion of A Moving Charge In An Uniform Magnetic Field)
        2) A conductor has many free electrons. We can denote $n$ as the linear charge density (i.e. number of free electrons per unit length of conductor)

        From 2), a current flowing through a conductor will essentially mean that free electrons are moving through the conductor. From 1), these moving free electrons will experience a force in a magnetic field. This force is then interpreted as the force on a current-carrying conductor. (But it is essentially a force on the moving free electrons)

        Furthermore, from 2), we know that more free electrons will be available for a longer conductor. This means that the force on the conductor will be larger for a longer conductor.


        Ok, the above explains your first question. For your second question, you are applying your knowledge incorrectly. Let’s take a look at the equation: (assume $\theta = 90^{\circ}$)

        $$F = BIl$$

        From the above equation, the following RELATIONSHIPS can be formed:
        $$\begin{aligned} F &\propto B \\ F &\propto I \\ F &\propto l \end{aligned}$$

        From the third relationship ($F \propto l$) and $F = BIl$, we can say that the force on a current-carrying conductor will increase IF the length of the conductor increases AND the current and magnetic field strength REMAINS THE SAME. (i.e. ALL other conditions REMAINS THE SAME)

        Of course, if the resistance increases due to an increase in the length of the conductor, the current will drop. However, the force may or may not drop!! This will depend on the magnitude of the drop in the current and the magnitude of increase of the length of the conductor. (Just use $F = BIl$)

  3. Sir or ma’am… can u plz write the mathematical equation for finding force on a current carrying conductor placed in a magnetic field or F=BIL sin theta….

  4. i would wish to be helped with the formula on how to calculate the current carrying capacity of a 120mmsq x 3 core 11kv core cable?

  5. Respected Sir/ Madam

    If I consider the conductor in the form of a current carrying loop that experiences a torque and further consider the moment arm, is it possible to replace the sin with a cos?

    Mad Scientist


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