How to find the net force from a non uniform magnetic field?

Question

egenriether · Answer

you will likely need to integrate the field equation.  Are you referring to the force on a charged particle?

anonymous · Answer

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Consider a current ,I flowing through a wire. Let a small charge dq, passes through a small element dl in a small time,dt.

Magnetic field ,B is in a arbitrary direction. Then force on dq is given by

$\large{\overrightarrow{dF} = dq \ (\overrightarrow{v}\times\overrightarrow{B})}$ 

Putting dq=Idt  

$\large{\overrightarrow{dF} = I \ (\overrightarrow{v}dt\times\overrightarrow{B})}$ 

 $\large{\overrightarrow{v}dt=\overrightarrow{dl}}$

$\large{\overrightarrow{dF} = I \ (\overrightarrow{dl}\times\overrightarrow{B})}$

We can get total force by integrating over the entire line dl

$\large{\overrightarrow{F} = \int\limits_{L} I \ (\overrightarrow{dl}\times\overrightarrow{B})}$

This the most general expression of force on a current element where B may vary  from point to point on the wire.

If B is constant and perpendicular to the length of wire and also the wire is straight ,the force on every element dl will be in same direction. Then the formula takes the familiar form of force on a current carrying straight wire in a uniform magnetic field

$\large{F=BIL}$

anonymous · Answer

The original question is incomplete.  The entity that the force is to act on is not specified.

It could be a moving charge, a current element or a magnetic dipole for example.