Flux Through a Circle

Electricity and Magnetism Level 3

Consider a circle of unit radius centered at the origin of the $xy$ -plane. There exists an infinitely long current-carrying wire along the straight line $x = 2$ . The magnitude of the current flowing through this wire is:

$I = \frac{2\pi}{\mu_o}$

Here $\mu_o$ is the permeability of free space. Compute the magnitude of the magnetic flux through the circle.

Note: I am not sure whether this exercise is original. If someone provides a source, I will mention it in this problem statement.

The answer is 1.68357443.

1 solution

Aaghaz Mahajan
Jul 29, 2019

The magnetic field at a distance $\displaystyle 1+x$ from the wire is $\displaystyle \frac{1}{1+x}$ .

The area of a small rectangular strip of length $\displaystyle dx$ on the ring is $\displaystyle 2\sqrt{x\left(2-x\right)}dx$ where $x$ is the distance on the X-Axis from point $\left(1,0\right)$

So, the magnetic flux comes out to be

$\int_0^2\frac{2\sqrt{x\left(2-x\right)}}{x+1}dx$

Which evaluates to $\displaystyle \left(4-2\sqrt{3}\right)\pi$

@Karan Chatrath I think we could do at least two followups to this as well.

1) This problem, except with the wire at $z = -\alpha$ instead of $z = 0$
2) Non-standard orientations for both the wire and the circle

Steven Chase - 1 year, 10 months ago

@Steven Chase The follow ups to this will be a little delayed. Turns out I'm traveling for the next couple of weeks. Just fyi.

Karan Chatrath - 1 year, 10 months ago

Ok, thanks for the info. Meanwhile, I will make my general solver in anticipation of them

Steven Chase - 1 year, 10 months ago

@Aaghaz Mahajan Thank you for the solution.

@Steven Chase Thanks for the suggestions. I will post follow ups as soon as I can

Karan Chatrath - 1 year, 10 months ago

Flux Through a Circle

The answer is 1.68357443.

1 solution

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