Show that the total magnetic force on the loop is zero

[blueyellow]

Homework Statement

a closed wire loop with current I is in a uniform magnetic field B

show that the total magnetic force on the loop is zero

Homework Equations

The Attempt at a Solution

everywhere that I've looked just states it without explaining why

 

[issacnewton]

think of force acting on a small element of the wire which is carrying current I.
Its given by

$$\vec{dF}= I\; \vec{dl}\times \vec{B}$$

so for the loop the total force acting on it would be

$$\vec{F} = I\; \oint\left(\vec{dl}\times \vec{B}\right)$$

so what can you say about this integral around the loop , where dl is length
element ?

 

[blueyellow]

the loop integral is zero because it's a conservative field?
but why is a magnetic field conservative? why would it be path independent?

 

[issacnewton]

well, the loop is kept in a constant magnetic field B. and its carrying current I.
since B is constant, we can take it out of integral.

$$\vec{F} = I\; \left( \oint\vec{dl}\right) \times \vec{B}$$

now, what is integral ? isn't it the net displacement from some point to itself in the loop
(we are adding infinitesimally small vector elements around the loop) so what is it ?