Find Magnetic Flux: Solve Triangle θ & x0 in xy Plane

[auk411]

Homework Statement

There is a triangle in the upper right hand corner of the xy plane. You are given θ, and x₀.

Consider an electric field which does not vary with time or the z direction but which varies in the x and y directions according to E=K(x²y, -xy²) throughout the region of interest (aka the triangle), where K is a constant. Show that the magnitude of the magnetic flux $\phi$_B through the triangle is given by
$\phi$_B = K(3+ tan²θ/{12}) x₀⁴tanθt + c, where c is an arbitrary constant and t represents positive time.

So I know that ∫E. ds = derivative of the magnetic flux.

So the first line integral over the x-axis is zero b/c y = 0.

For the second second line intergral: dx = 0. So -∫x₀y²dy, as y goes from 0 to x₀tanθ. This gives us -x₀/3 (tan³θ).

For the last line integral: ∫x²y dx - x∫y²dy. I get ((tanθ)/3)x₀³ - (x/3)tan³θ. The thought is that for the dx part of this integral we are integrating from 0 to x₀. For the dy part, we are going from 0 to tanθ, since the line is y = tanθ.

So this doesn't seem right. And even if it were, what do I do next. If I integrate to find the magnetic flux, what do I integrate with respect to.

 

[jbsteele]

Homework Equations ∫E. ds = derivative of the magnetic fluxThe Attempt at a Solution So I know that ∫E. ds = derivative of the magnetic flux.So the first line integral over the x-axis is zero b/c y = 0.For the second second line intergral: dx = 0. So -∫x0y2dy, as y goes from 0 to x0tanθ. This gives us -x0/3 (tan3θ). For the last line integral: ∫x2y dx - x∫y2dy. I get ((tanθ)/3)x03 - (x/3)tan3θ. The thought is that for the dx part of this integral we are integrating from 0 to x0. For the dy part, we are going from 0 to tanθ, since the line is y = tanθ.So this doesn't seem right. And even if it were, what do I do next. If I integrate to find the magnetic flux, what do I integrate with respect to.