A certain spherically symmetric charge configuration in free space produces an electric field given in spherical coordinates by $$ \mathbf{E}(r)=\left\\{\begin{array}{ll} \left(\rho_{0} r^{2}\right) /\left(100 \epsilon_{0}\right) \mathbf{a}_{r} \mathrm{~V} / \mathrm{m} & (r \leq 10) \\ \left(100 \rho_{0}\right) /\left(\epsilon_{0} r^{2}\right) \mathrm{a}_{r} \mathrm{~V} / \mathrm{m} & (r \geq 10) \end{array}\right. $$ where \(\rho_{0}\) is a constant. (a) Find the charge density as a function of position. (b) Find the absolute potential as a function of position in the two regions, \(r \leq 10\) and \(r \geq 10 .(c)\) Check your result of part \(b\) by using the gradient. (d) Find the stored energy in the charge by an integral of the form of Eq. (43). (e) Find the stored energy in the field by an integral of the form of Eq. (45).

#tag_title# (Find absolute potential from given electric field) #tag_content# To find the absolute potential function, we will integrate the electric field over the radial distance for both regions. First, let's recall the definition of the potential difference: $$ \Delta V = -\int_\mathbf{A}^\mathbf{B} \mathbf{E} \cdot \mathrm{d} \boldsymbol{l} $$ For \(r \leq 10\), we have: $$ V(r) - V(0) = -\int_{0}^{r} \frac{\rho_0 r^2}{100\epsilon_0} dr $$ Similarly, for \(r \geq 10\), we have: $$ V(r) - V(10) = -\int_{10}^{r} \frac{\rho_0 (20-r)}{10\epsilon_0} dr $$ Now we can find the potential function for both regions by integrating and applying appropriate boundary conditions. #(c) Checking the Results by Taking the Gradient# #tag_title# (Check electric field consistency with potential) #tag_content# To check if our results in part (b) are correct, we can take the gradient of the obtained potential functions for both regions and compare with the given electric fields. For \(r \leq 10\), we have: $$ \nabla V_{r \leq 10}(r) = -\mathbf{E}_{r \leq 10}(r) $$ Similarly, for \(r \geq 10\), we have: $$ \nabla V_{r \geq 10}(r) = -\mathbf{E}_{r \geq 10}(r) $$ If the results match, then our potential functions are indeed correct. #(d) Finding the Energy Stored in the Charge# #tag_title# (Find stored energy in the charge) #tag_content# Using equation (43), we can find the energy stored in the charge and field by integrating the charge density and potential over the entire volume: $$ W_{charge} = \frac{1}{2} \int_{all\; space} \rho(\mathbf{r}) V(\mathbf{r}) d^3\mathbf{r} $$ #(e) Finding the Energy Stored in the Electric Field# #tag_title# (Find stored energy in the electric field) #tag_content# Using equation (45), we can find the energy stored in the electric field by integrating the square of the electric field magnitude over the entire volume: $$ W_{field} = \frac{1}{2} \int_{all\; space} u(\mathbf{r}) d^3\mathbf{r} $$ where \(u(\mathbf{r}) = \frac{1}{2} \epsilon_0 E^2 (\mathbf{r})\) is the energy density associated with the electric field. By following these steps, we can calculate each of the required quantities and solve the problem.