Magnetic Flux Through Wire Loop

[bspride]

Homework Statement

A wire loop with radius R is located a distance d from a long straight wire carrying a current I. Find the flux through the loop.

Homework Equations

B = $\mu$I/(4$\pi$)
Magnetic Field above wire
$\phi$=B*da*cos$\vartheta$

The Attempt at a Solution

Somehow you have to combine the two equations to find the flux through the circular loop. Any help would be greatly appreciated as I am stumped.

 

[francesco85]

I don't know whether I have understood rhe problem, but I have worked out the situation in which the wire is on the same plane of the circle (in general I think that the computation is conceptually easy but hard in the practice).

In this case, the magnetic field generated by the wire at distance r is

$B(r)=\frac{2I}{cr}$

in cgs units.


so, if we assume that $d>R>0$, we have that the flux $\Phi$ is


$\Phi=\int_{d-R}^{d+R}dr\frac{2I}{cr}2\sqrt{R^2-(d-r)^2}$

 

[bspride]

Yes you are right that the wire loop is in the same plane as the wire. Care to explain the formula you used for the magnetic field generated by the wire?

 

[francesco85]

Yes: from the 4-th Maxwell's equation (ignoring the electric field: the problem is a magnetostatic problem):

$\nabla \times \vec{B}=\frac{4\pi}{c}\vec{J}$


Integrate over a surface S; transform the integral of the rotor in a line integral through one of the usual theorem (Green's theorem, maybe, but I'm not sure); moreover the integral of J gives the current and you find the relation.

 

[asteriatic]

I would start by breaking down the problem into smaller parts and identifying the key variables and equations involved. In this case, we have a wire loop with a radius R located a distance d from a long straight wire carrying a current I. We also have two relevant equations: one for the magnetic field above the wire and one for the flux through the loop.

To find the flux through the loop, we need to calculate the magnetic field at the location of the loop and then use that value to calculate the flux. Using the equation for the magnetic field above the wire, we can find the magnetic field at the location of the loop by plugging in the values for the current I, distance d, and the constant \mu (the permeability of free space).

Once we have the value for the magnetic field, we can use the equation for flux to calculate the flux through the loop. This equation takes into account the magnetic field, the area of the loop (given by \pi R^2), and the angle between the magnetic field and the area vector (given by \vartheta).

Therefore, to find the flux through the loop, we can combine these two equations by first finding the magnetic field at the location of the loop and then using that value to calculate the flux. This approach should help you solve the problem and find the desired result.