Force Due to Magnetic Field on a Charge Carrying Wire

[wk1989]

Hi, I have a bit of a problem understanding one of the solutions for my assignment.

Homework Statement

Normally we use the equation
F = iL X B
to find the force of a magnetic field on a wire with a current.

One of the questions asks to find the force on a section of wire between x=3 and x= 1, the wire's current is in the the negative x direction. The magnetic field is something like B=3i+8.0x^2

They used integration to solve the problem, integrating the change of force from 1 to 3. I'm wondering why this is used? Is it because the magnetic field is not uniform (it's in terms of x)? If the magnetic field is uniform, could we just have done F = iL x B with 2 being the L?

Thanks in advance!

 

[dynamicsolo]

Is the magnetic field supposed to be B = 3i + 8.0x^2j?

If that's the case, then, yes, the field is not uniform over the section x = 3 to x = 1, so an integration is necessary over the length of the wire.

 

[Domnu]

You can only use $$\vec{F} = I (\vec{L} \times \vec{B})$$ when the magnetic field is constant. Since here, your magnetic field is different over a particular area, you have to use integration.