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

$\vec n = \dfrac{\partial\vec s}{\partial x} \times \dfrac{\partial\vec s}{\partial z} = -\vec\jmath$

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

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