# Determining the force on a loop cause by an infinite line

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1. Jul 3, 2016

### mobay

1. The problem statement, all variables and given/known data
An infinitely long line of current $I_1=6[A]$ is following along the positive z-axis in the direction of +$\hat{a_z}$. Another current is following a triangular loop counter clockwise from the points A(0,2,2), B(0,6,2) and C(0,6,6).

2. Relevant equations
To start I applied Ampere's Law since an amperian path could be applied to current $I_1$ which results in
$$\vec{H}=\frac{I_1}{2\pi\rho} \hat{a_{\phi}}$$

Applying the force on two conductors formula $$F=\int{I_2 \vec{dl_2}} \times{\vec{B_1}}$$

3. The attempt at a solution
Where I am stuck is at segment AB, I have determined that the following for this segment $$dl=dy\,\hat{a_y}$$

The problem that I am having is determining the direction of the magnetic field vector, is it it directed in the positive $\hat{a_z}$ because of how the problem is described or is it another direction all together? The teacher provided a solution but did not outline why they did certain steps.

2. Jul 3, 2016

### TSny

Welcome to PF!
You know $\vec{H}=\frac{I_1}{2\pi\rho} \hat{a_{\phi}}$. Thus, $\hat{a_{\phi}}$ gives you the direction.

Are you familiar with the magnetic field pattern of a long straight current? (Right hand rule)

3. Jul 3, 2016

### mobay

Do you mean that since the current is moving in the positive $\hat{a_x}$ that I apply the right hand rule? If so then the result is that the force is directed in the negative $\hat{a_x}$ direction. Therefore, $$\vec{B}=\frac{-{\mu}_0\,I_1}{2\pi\,y} \hat{a_x}$$

At this point I integrate the and solve for segments AB and BC which results in $$\vec{F_{1,AB}}= 6{\mu}_0\,ln(3) \,\hat{a_z}$$ $$\vec{F_{1,BC}}= -4\mu \,\hat{a_y}$$

Up to this point I understand and did correctly according to the given solution, however, the final integral confounds me...After the cross product is applied and simplified my result is this $$6{\mu}_0[\int\frac{-dz}{y}\,\hat{a_y} + \int\frac{dy}{y}\,\hat{a_z} ]$$

I believe the second integration should result in $6{\mu}_0\,ln(1/3)\,\hat{a_z}$ but I dont know about the about the first integral since there is the is "y" term in the denominator. Do I make that y=6 because that is where the integration started at or y=2?

4. Jul 3, 2016

### TSny

Typo? The current is in the direction of $\hat{a_z}$. The field is in the negative x direction.

OK

I can't check the above, since $I_2$ was not specified.

For the first integral you'll need to relate $y$ and $z$ along the path CA.

5. Jul 3, 2016

### mobay

Ok I will see what I can do, sorry I forgot but $I_2 = 2 [A] and I_1=6\pi [A]$. I am still trying to get a handle on latex

6. Jul 3, 2016

### TSny

OK. Your results so far look good.