Question

Find the current I flowing through a square with corners at (0,0,0), (2,0,0), (2,0,2), (0,0,2). The current density is: bold italic J equals bold y with bold hat on top open parentheses y squared plus 5 close parentheses space space space space space open parentheses straight A divided by straight m squared close parentheses

Answer 1

Parameterize the square (call it S) by

$\mathbf s(u,v)=2u\,\mathbf x+2v\,\mathbf z$

with both $u\in[0,1]$ and $v\in[0,1]$.

Take the normal vector pointing in the positive y direction to be

$\dfrac{\partial\mathbf s}{\partial v}\times\dfrac{\partial\mathbf s}{\partial u}=4\,\mathbf y$

Then the current is

$\displaystyle\iint_S(y^2+5)\,\mathbf y\cdot4\,\mathbf y\,\mathrm dA=20\int_0^1\int_0^1\mathrm dA=\boxed{20\,\mathrm A}$

where $y^2+5$ reduces to just 5 because $y=0$ for all points in S.