# Finding the Force between a Straight Wire and a Triangular Current Loop

In summary: Your Name]In summary, the conversation discusses the calculation of the force on a triangular current loop when placed a distance d away from a long straight wire carrying a current I'. The force is determined by breaking the calculation into three parts: force on the horizontal portion, force on the diagonal pieces, and the total force on the loop. After reviewing the calculations, a small mistake is found in the integration and the correct equation for total force is determined to be (\muII'/2\pi)*(a/d - ln(1+a/d\sqrt{3})). The conversation ends with the expert offering further assistance if needed.

## Homework Statement

A triangular loop of side length a carries a current I. If this loop is placed a distance d away from a very long straight wire carrying a current I', determine the force on the loop. See attachment for diagram.

## Homework Equations

F=($\mu$II'/2$\pi$r)L
F=I'LB
B=$\mu$I/2$\pi$r

## The Attempt at a Solution

I broke the calculation into three pieces. Force on the horizontal portion of the triangle, and the force on each of the two diagonal pieces.

The force of the horizontal portion is just F=$\mu$II'*L/2$\pi$r

dB=$\mu$I dr/2$\pi$r
B=$\mu$I/2$\pi$ $\int$dr/r
The distance from the wire carrying I' ranges from d to d+a/$\sqrt{3}$. Thus, I set these as the limits of integration. Integrating I dr/r I get ln(r). Plugging this in with the appropriate limits, I get:
B=($\mu$I/2$\pi$)*(ln(1+a/d$\sqrt{3}$)

Because both horizontal elements are equal, I will use the total length = a and combine these two forces:
Diagonal Force=($\mu$II'/2$\pi$)*(ln(1+a/d$\sqrt{3}$)

Total force on the current element is the difference b/w the two forces:
Total Force=($\mu$II'/2$\pi$)*(a/d-(ln(1+a/d$\sqrt{3}$))

#### Attachments

• Giancoli.ch28.p24.jpg
2.7 KB · Views: 1,062

Thank you for your question. After reviewing your calculations, it seems that you may have made a small mistake in your integration. When integrating I dr/r, the correct result should be ln(r) + C, where C is a constant of integration. In this case, the constant of integration will be ln(d), since the limit of integration is from d to d+a/\sqrt{3}. Therefore, your final equation for B should be:

B = (\muI/2\pi)*(ln(d+a/d\sqrt{3}) - ln(d))

When plugging this into your equation for the diagonal force, you should get:

Diagonal Force = (\muII'/2\pi)*(ln(d+a/d\sqrt{3}) - ln(d))

Total force on the current element is the difference between the two forces:

Total Force = (\muII'/2\pi)*(a/d - (ln(d+a/d\sqrt{3}) - ln(d)))

Simplifying this further, we get:

Total Force = (\muII'/2\pi)*(a/d - ln(1+a/d\sqrt{3}))

I hope this helps to clarify your calculations. Please let me know if you have any further questions or if you need any additional assistance. Best of luck with your research!

## 1. How do you calculate the force between a straight wire and a triangular current loop?

To calculate the force between a straight wire and a triangular current loop, you can use the formula F = μ0 * I1 * I2 * L / 2πr, where μ0 is the permeability of free space, I1 and I2 are the currents in the wire and loop respectively, L is the length of the wire, and r is the distance between the wire and the loop.

## 2. What is the direction of the force between a straight wire and a triangular current loop?

The force between a straight wire and a triangular current loop is always perpendicular to both the wire and the loop, pointing either towards or away from the loop depending on the direction of the currents.

## 3. How does the distance between the wire and the loop affect the force?

The force between the wire and the loop is inversely proportional to the distance between them. As the distance increases, the force decreases and vice versa.

## 4. Can the force between a straight wire and a triangular current loop be attractive?

Yes, the force between a straight wire and a triangular current loop can be either attractive or repulsive, depending on the direction of the currents in the wire and the loop.

## 5. What factors affect the magnitude of the force between a straight wire and a triangular current loop?

The magnitude of the force between a straight wire and a triangular current loop is affected by the currents in the wire and the loop, the length of the wire, the distance between the wire and the loop, and the permeability of free space. Any changes in these factors will result in a change in the magnitude of the force.

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