Electromagnetism - Magnetic flux between infinite wire and rectangular loop

In summary, the amount of magnetic flux generated by the loop which links the straight wire can be calculated using Ampère's law and the fact that M12 = M21, which implies that Phi12 = Phi21 if the same current flows in both circuits. It is correct to assume that the current flowing in the wire is equal to the current flowing in the loop. To calculate the B-field generated by the rectangular loop, you can use a similar approach as for the wire, but with a different integration path. The integration path will be a rectangle that encloses the loop, with one side being the straight wire and the other three sides being the sides of the loop. The length of the sides of the rectangle can be used for dl, and
  • #1
Asrai
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Homework Statement



The figure shows a rectangular wire loop, around which a current I flows, and and an infinite straight wire. The wire lies in the plane of the loop and is parallel to, and a distance d from, the side AB. (see attachment)

Show that the amount of magnetic flux generated by the loop which links the straight wire is equal to:

(mu(0)*h*I/2*Pi)*ln((1+l/d))

Hint: Use the fact that M12 = M21 implies that Phi12 = Phi21 if the same current I flows in both circuits.

Homework Equations



Ampère's law: int(B.dl) = mu(0)*I

Phi = int(B.dA)

The Attempt at a Solution



I've done some work on this problem and actually solved it, but all based on the following assumption:

The current flowing in the wire is equal to the current flowing in the loop.

Because then you can calculate B that is generated by the wire and the effect it has on the loop using Ampère's law. Phi is then just the solution of the equation above, with the integration limits of d to d+l.

This gives the right solution, but is it correct to assume this? I cannot do it the other way round, i.e. calculate the B-field generated by the rectangular loop; what would dl be in this case? And also dA for the following equation?

Any help or hints would be greatly appreciated!
 

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  • #2


Thank you for your question. Your approach to the problem is on the right track, but I would like to clarify a few things.

Firstly, it is correct to assume that the current flowing in the wire is equal to the current flowing in the loop. This is because of the fact mentioned in the hint - if the same current flows in both circuits, the magnetic flux is equal in both directions.

Secondly, for calculating the B-field generated by the rectangular loop, you can use the same approach as you did for the wire. The key is to use Ampère's law, but with a different integration path. In this case, the integration path will be a rectangle that encloses the loop, with one side being the straight wire and the other three sides being the sides of the loop.

For calculating dl, you can use the length of the sides of the rectangle, and for dA, you can use the area of the rectangle (which is the product of two adjacent sides).

I hope this helps. Let me know if you have any further questions. Good luck with your calculations!
 
  • #3




Your solution is valid and correct. The assumption that the current in the wire is equal to the current in the loop is a reasonable one to make in this scenario. This is because the wire and the loop are connected in series, so the same amount of current must flow through both of them according to Kirchhoff's law.

To calculate the B-field generated by the rectangular loop, you can use a similar approach. You can use Ampère's law to calculate the magnetic field at a distance d from the wire, and then integrate over the loop to find the total flux. In this case, dl would be the length of the side of the loop and dA would be the area of the loop.

Overall, your solution is a valid and effective way to solve this problem. Keep in mind that there may be multiple ways to approach a problem in science, and your solution is just one of them. Great job!
 

1. What is the equation for calculating the magnetic flux between an infinite wire and a rectangular loop?

The equation is given by Φ = μ0I/4π * (ln(2L/d) - ln(L/d)), where Φ is the magnetic flux, μ0 is the permeability of free space, I is the current in the wire, L is the length of the wire, and d is the distance between the wire and the center of the loop.

2. How does the distance between the wire and the loop affect the magnetic flux?

The magnetic flux is inversely proportional to the distance between the wire and the center of the loop. As the distance increases, the magnetic flux decreases.

3. What is the direction of the magnetic flux in this scenario?

The direction of the magnetic flux is perpendicular to both the wire and the loop, and follows the right-hand rule. This means that if you curl your fingers in the direction of the current in the wire, your thumb will point in the direction of the magnetic flux.

4. Can the magnetic flux between the wire and the loop be negative?

Yes, the magnetic flux can be negative. This occurs when the current in the wire and the direction of the loop's normal vector are in opposite directions.

5. How does the length of the wire affect the magnetic flux?

The length of the wire affects the magnetic flux in a logarithmic manner. As the length increases, the magnetic flux increases, but at a decreasing rate.

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