Question #42933

A solid non-conducting sphere of radius $R$ (in meters) has a uniform volume charge density $\rho$ (in $\text{C/m}^3$). The sphere is centered at the origin $(0,0,0)$ of a Cartesian coordinate system. A spherical cavity of radius $R/2$ is created inside the sphere, with its center located at $(R/2, 0, 0)$. The entire setup is placed in an external electric field given by $\vec{E}_{ext} = -\frac{\rho}{\varepsilon_0} \left( x \hat{i} - \frac{y}{2} \hat{j} - \frac{z}{2} \hat{k} \right) \text{ V/m}$, where $x, y,$ and $z$ are the coordinates in meters. Find the coordinates $(x, y, z)$ of the neutral point (the point where the net electric field is zero) inside the cavity. Assume the permittivity of free space is $\varepsilon_0$.