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

`(\partial\mathbf s)/(\partial v)*(\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.