Question #77838

An exotic electric field in a region of space is given by $\vec{E} = k r^2 \sin^2\theta \hat{r}$, where $k$ is a constant, $r$ is the radial distance, and $\theta$ is the polar angle in spherical coordinates. A uniform magnetic field $\vec{B} = B_0 \hat{k}$ is also present in the region. A student attempts to determine the constant $k$ by measuring the total outward electric flux $\Phi$ through a closed, slightly deformed spherical surface $S$. The surface $S$ is defined by the equation $r(\theta) = R_0 (1 + \beta \cos^2\theta)$, where $\beta \ll 1$ is a known dimensionless deformation parameter.

To simplify calculations, the student INCORRECTLY assumes that the surface $S$ can be treated as a perfect sphere with an effective uniform radius equal to the solid-angle average radius $R_{avg}$ of the deformed surface. They use this $R_{avg}$ to calculate a value $k_{calc}$ from the measured flux $\Phi$.

Assuming the flux $\Phi$ is measured perfectly, what is the relative error $\frac{k_{calc} - k_{true}}{k_{true}}$ in the student's calculated value to first order in $\beta$?