Question

An electric field is described by the vector e=yj -xi. what is the electric flux through a rectangular box in the x-z plane that is bounded by x between 0 and 1, and z between 0 and 1? (You need to identify dA to do this problem.)

Answer 1

The flux of
`\vec E = -x\,\vec\imath + y\,\vec\jmath` is given by the surface integral

`\displaystyle \iint_S \vec E \cdot d\vec\sigma`

where
`S` is the given square region, which we can parameterize by

`\vec s(x, z) = x\,\vec\imath + z\,\vec k`

with
`0\le x\le 1` and
`0\le z\le 1`. The area element is

`d\vec\sigma = \vec n \, dx\,dz`

where
`\vec n` is the normal vector to
`S`. Depending on the orientation of
`S`, this vector could be

`\vec n = (\partial\vec s)/(\partial x) * (\partial\vec s)/(\partial z) = -\vec\jmath`

or
`-\vec n = \vec \jmath`; either way, the integral reduces to

`\displaystyle \iint_S \vec E \cdot d\,\vec\sigma = \int_0^1 \int_0^1 (-x\,\vec\imath + z\,\vec k) \cdot (\pm\vec\jmath) \, dx\,dz = \boxed{0}`