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.