1. The problem is to understand and apply Gauss's Law in electromagnetism. 2. Gauss's Law states that the total electric flux $\Phi_E$ through a closed surface is equal to the charge enclosed $Q_{enc}$ divided by the permittivity of free space $\epsilon_0$: $$\Phi_E = \oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{enc}}{\epsilon_0}$$ 3. Here, $\mathbf{E}$ is the electric field vector, $d\mathbf{A}$ is the differential area vector on the closed surface, and $\epsilon_0$ is a constant approximately equal to $8.854 \times 10^{-12}$ F/m. 4. Important rules: - The surface must be closed. - The direction of $d\mathbf{A}$ is outward normal to the surface. - The law applies to any closed surface, regardless of shape. 5. To apply Gauss's Law, choose a Gaussian surface that exploits symmetry to simplify the calculation of $\mathbf{E}$. 6. Example: For a point charge $Q$ at the center of a spherical surface of radius $r$, the electric field magnitude is uniform on the surface, so: $$\Phi_E = E \times 4\pi r^2 = \frac{Q}{\epsilon_0}$$ 7. Solving for $E$: $$E = \frac{Q}{4\pi \epsilon_0 r^2}$$ This is Coulomb's law for the electric field of a point charge. This completes the explanation and application of Gauss's Law.