An electricity and magnetism problem by A Former Brilliant Member

Consider a Gaussian cylinder as shown above. Find the electric field at the top / bottom. The area of one of the ends is $A$ . There is no net electric flux on the sides, due to symmetry.

$E \, (2A) = \frac{Q_{enclosed}}{\epsilon_0}$

Find the enclosed charge ( $z_1 = -0.02, z_2 = 0.02)$ .

$\rho = \alpha z^2 \\ dQ = \rho \, dV = (\alpha z^2)(A \, dz) = \alpha A z^2\, dz \\ Q_{enclosed} = \alpha A \int_{z_1}^{z_2} z^2 \, dz = 2 \alpha A \int_{0}^{z_2} z^2 \, dz = 2 \alpha A \frac{z_2^3}{3}$

Plugging back into the original Gauss equation:

$E \, (2A) = \frac{2 \alpha A \frac{z_2^3}{3}}{\epsilon_0} \\ = \frac{\alpha z_2^3}{3 \epsilon_0}$

Putting in numbers:

$E = \frac{(0.005)(0.02)^3}{3(8.854 \times 10^{-12})} \approx \boxed{1506}$