At point $P$, the magnitude of the electric force on the test particle is $F_E$ and the magnitude of the gravitational force on the test particle is $F_g$.

Derive an expression for the magnitude $E_P$ of the electric field at point $P$ (where $r=0.030\ \text{m} < R$) in terms of $\rho$, $r$, $\varepsilon_1$, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information. Your final expression should be valid for any point inside the sphere (any $r<R$).

Indicate whether $F_{E,\,new}$ is greater than, less than, or equal to $F_E$ by writing one of the following. Later, the dielectric material inside the sphere is replaced with a different dielectric with permittivity $\varepsilon_2=4.0\varepsilon_0$, while the charge density remains uniform at $\rho=+6.0\times10^{-6}\ \text{C/m}^3$ and the radius remains $R=0.050\ \text{m}$. The test particle is again placed at rest at the same point $P$ (still at $r=0.030\ \text{m}$). Let $F_{E,\,new}$ be the new magnitude of the electric force on the test particle at $P$.