Question #77831

A non-uniform electric field is given by $\vec{E} = E_0 (1 + \alpha z) \hat{k}$, where $E_0$ is an unknown constant and $\alpha = 2.0 \text{ m}^{-1}$ is a known exact constant. A student determines $E_0$ by measuring the electric flux $\Phi$ through an open surface $S$ bounded by the curve $x^2+y^2=R^2$ in the $xy$-plane. The true surface $S$ is a paraboloid defined by $z = R - \frac{x^2+y^2}{R}$ for $z \ge 0$. The student measures the base radius as $R = 1.0 \text{ m}$ with a relative error $e_R = \frac{\Delta R}{R}$, and the flux $\Phi$ with a relative error $e_\Phi$. However, the student incorrectly assumes the surface $S$ is a standard hemisphere of radius $R$ ($z = \sqrt{R^2 - x^2 - y^2}$). Let $e_{true}$ be the true maximum relative error in $E_0$ (derived from the correct surface geometry), and $e_{student}$ be the maximum relative error calculated by the student using their flawed geometric assumption. What is the magnitude of the difference $|e_{student} - e_{true}|$? (Assume the normal to the surface has a positive z-component).