1. Choose a Gaussian surface. 2. Calculate the electric flux through the surface. 3. Determine the enclosed charge. 4. Apply Gauss's Law to solve for the electric field.

1. Choose a spherical Gaussian surface. 2. Calculate flux: $\Phi = E(4\pi r^2)$. 3. Find enclosed charge. 4. Apply Gauss's Law: $E = \frac{Q_{enc}}{4\pi \epsilon_0 r^2}$.

1. Define the area vector. 2. Determine the electric field vector. 3. Calculate the dot product: $\Phi = \vec{E} \cdot \vec{A} = EA \cos(\theta)$.

1. Define the volume element $dV$. 2. Express charge density as function of position $\rho(r)$. 3. Integrate: $q_{enc} = \int \rho(r) dV$.

1. Define the path of integration. 2. Calculate the line integral: $V(r) = - \int_{\infty}^{r} \vec{E} \cdot d\vec{r}$.

1. Count electric field lines entering. 2. Count electric field lines exiting. 3. If entering = exiting, net flux = 0.

There is a non-zero electric flux through the surface.

The flux is zero.

The electric flux increases proportionally.

The net electric flux through the surface is zero.

Integration is required to find the enclosed charge: $q_{enc} = \int \rho(r) dV$.

The electric field inside the conductor becomes zero.

1: Electric Field Lines, 2: Area Vector, 3: Angle $\theta$

1: Lines Entering (Negative Flux), 2: Lines Exiting (Positive Flux)

1: Electric Flux ($\Phi_E$), 2: Surface Integral ($\oint \vec{E} \cdot d\vec{A}$), 3: Enclosed Charge ($Q_{enc}$), 4: Permittivity of Free Space ($\epsilon_0$)

1: Gaussian Surface, 2: Radius (r), 3: Charge (Q)

1: Spherical Charge Distribution, 2: Gaussian Surface, 3: Electric Field (E)