We can obtain the equation for the Force on a current-carrying conductor in a magnetic field using the formula of the **force on a moving charge in a magnetic field**.

## Force on a current-carrying conductor in a magnetic field – formula

We have seen in one of our posts, that F_{on a moving charge} = q (v X B). [ read this post here: formula of the force on moving charge in a magnetic field ]

q (v X B) formula can be modified in the following way to get the formula for the Force on a current-carrying conductor in a magnetic field.

### derivation of the formula

The following steps will help us to derive the equation for the Force on a current-carrying conductor in a magnetic field.

**F** = q (**v** X **B**) = (I t) [(**L**/t) x **B** ] …. (1)

Here, we have replaced charge q with the product of current I in the conductor and the time duration t. We have also replaced velocity **v** with L/t where **L** is the length of the current-carrying conductor (hence, the displacement of charge) and t is again the time duration being considered.

From (1) we get:

**F** = I **L** x **B** (where IL can be taken as the current element)

F = I

LxBForce on a current carrying conductor in a magnetic field(formula derivation)

F = I L B Sin θ

where θ is the angle between the current element and the magnetic field.

### The maximum value of the Force on a current-carrying conductor in a magnetic field

When θ = 90 degrees, then the force on a current-carrying conductor in a magnetic field becomes maximum and it is expressed as F_{max} = I L B

### The minimum value of the Force on a current-carrying conductor in a magnetic field

When θ = 0 degrees, then the force on a current-carrying conductor in a magnetic field becomes minimum and it is expressed as F_{min} = 0