--- type:slide --- ### Exploration - Boundary Conditions for Conductors --- #### Free space-Conductor Interface Consider the interface between free space and perfect electric conductor (PEC) shown in Fig. 2. To determine the boundary conditions for this specific example, let's start with the most general form: $$ \vec{E}_{1t} = \vec{E}_{2t} $$ $$ \hat{n} \cdot (\vec{D}_1 - \vec{D}_2) = D_{1n} - D_{2n} = \epsilon_1 E_{1n} - \epsilon_2 E_{2n} = \rho_s $$ --- Since material 2 is PEC ($\sigma_2 = \infty$), the electric field must vanish––Why? Recall the vector form of Ohm's law ($\vec{J} = \sigma \vec{E}$). If we want to solve for $\vec{E}$ and rewrite this equation as $\vec{E} = \vec{J}/\sigma$, the electric field must go to zero as $\sigma$ approaches $\infty$. Thus, $\vec{E}_2 = 0$ at our free space-conductor interface. Applying that to our original boundary conditions yields the following: $$ \vec{E}_{1t} = \vec{E}_{2t} = 0 $$ $$ \epsilon_0 E_{1n} - 0 = \rho_s $$ --- We can conclude that the tangential components of the electric field at the interface are zero, and that the only component we expect to be non-zero is the normal component in material 1 ($\vec{E}_{1n}$) if there is a surface charge density ($\rho_s$) at the interface. --- #### Conductor-Conductor Interface Consider the interface between two conductors with different conductivities ($\sigma_1 \ne \sigma_2$) shown in Fig. 3. Let's once again start with the most general form of our electric field boundary conditions: $$ \vec{E}_{1t} = \vec{E}_{2t} $$ $$ \hat{n} \cdot (\vec{D}_1 - \vec{D}_2) = D_{1n} - D_{2n} = \epsilon_1 E_{1n} - \epsilon_2 E_{2n} = \rho_s $$ --- To do this, we can invoke the vector form of Ohm's law, which relates the conduction current density ($\vec{J}$) to the electric field as: $$ \vec{J} = \sigma \vec{E} $$ Substituting this relationship in our original boundary conditions yields: $$ \frac{\vec{J}_{1t}}{\sigma_1} = \frac{\vec{J}_{2t}}{\sigma_2} $$ $$ \epsilon_0 \bigg( \frac{J_{1n}}{\sigma_1} - \frac{J_{2n}}{\sigma_2} \bigg) = \rho_s $$ ---