Question #42843

In a region of space, two independent static electric potentials are established, given by: $V_1(x,y,z) = E_0 x \exp\left(-\frac{z^2}{R^2}\right)$ $V_2(x,y,z) = E_0 y \exp\left(-\frac{z^2}{R^2}\right)$ where $E_0$ is a positive constant with units of $\text{V/m}$, and $R$ is a positive constant with units of meters. The corresponding electric fields are $\vec{E}_1 = -\nabla V_1$ and $\vec{E}_2 = -\nabla V_2$. A new vector field is defined as $\vec{G} = \vec{E}_1 \times \vec{E}_2$. What is the magnitude of the flux of $\vec{G}$ through the open hemispherical surface defined by $x^2 + y^2 + z^2 = R^2$ with $z \ge 0$?