Force and torque on a wire carrying a current

[theKeeblerElf]

Homework Statement

A wire along the x-axis carries current I in the negative x-direction through the magnetic field
$$\vec{}B$$ = B₀*(x/L) $$\hat{}k$$ 0$$\leq$$x$$\leq$$L
= 0 elsewhere

Part a was to draw a graph of B versus x over the interval -L$$\leq$$x$$\leq$$L, which I did.

b. Find an expression for the net force F_{\rm net} on the wire. Express your answer in terms of the variables I, L, and B₀

c. Find an expression for the net torque on the wire about the point x = 0.
Express your answer in terms of the variables I, L, and B₀

Homework Equations

F_net=I*L*B*sin$$\alpha$$
$$\tau$$_net=I*L²*B*sin$$\alpha$$

The Attempt at a Solution

I thought that because I is carried in the -$$\hat{}i$$ direction and B points in the $$\hat{}k$$ direction that $$\alpha$$=90 degrees, meaning I would be multiplying by 1. However, for both parts, when I submit I*L*B₀ and I*L²*B₀, respectively, the website says "Your answer either contains an incorrect numerical multiplier or is missing one."

For part b I've tried submitting -I*L*B₀ because actually calculating I$$\hat{}L$$ X $$\hat{}B$$ yields a negative answer, but that didn't work either.

I've also tried to use x in my answer, and the website says the correct answer doesn't depend on it.

Any help would be much appreciated!

 

[gabbagabbahey]

F_net=I*L*B*sin$$\alpha$$
$$\tau$$_net=I*L²*B*sin$$\alpha$$

This is only true when the magnetic field is constant over the length of the wire. More generally, you will have to divide the wire into pieces $d\textbf{l}$ so small that the magnetic field is effectively constant/uniform over the length of the piece, calculate the force on each piece:

$$d\textbf{F}=Id\textbf{l}\times\textbf{B}$$

and then add up (integrate) all these little forces:

$$\textbf{F}=\int Id\textbf{l}\times\textbf{B}$$

You must do something similar for the torque.