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intervoxel
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Homework Statement
A long metallic sheet of width 'a' and negligible thickness has a current uniformly distributed along its length. Find the magnetic field on the plane of the sheet a distance 'b' from its axis.
The current flows in the x direction; the z direction is perpendicular to the surface.
Homework Equations
The Biot-Savart law:
[itex]
\vec{B}=\frac{\mu_0}{4\pi}\int_S \frac{J_s \times \hat{r-r'}}{|r-r'|^2} dS'
[/itex]
The Attempt at a Solution
Since J_s is constant, it can be put outside the integral.[itex]
\vec{B}=\frac{\mu_0 J_s}{4\pi}\int_S \frac{\hat{x} \times \hat{r-r'}}{|r-r'|^2} dS'.
[/itex]
Considering a finite region of length L we have:
[itex]
\vec{B}=\frac{\mu_0 J_s}{4\pi}\int_{-a/2}^{a/2} \int_{-L/2}^{L/2} \frac{\hat{x} \times \hat{r-r'}}{|r-r'|^2} dx dy
[/itex]
or[itex]
B=\frac{\mu_0 J_s}{4\pi}\int_{-a/2}^{a/2} \int_{-L/2}^{L/2} \frac{1}{|r-r'|^2} dx dy
[/itex]
in the z direction. But [itex]|r-r'|=\sqrt(x^2+(b-y)^2)[/itex]Substituting the denominator, we have[itex]
B=\frac{\mu_0 J_s}{4\pi}\int_{-a/2}^{a/2} \int_{-L/2}^{L/2} \frac{1}{x^2+(b-y)^2} dx dy
[/itex]
Is it correct?
How resolve this integral?