At the boundary between two dielectric media, the following conditions hold:
The electric field in medium 1 is: \[ \vec{E_1} = 3\hat{x} + 2\hat{y} + 4\hat{z}. \] - Tangential components (\( \vec{E_{1t}} \)): These are the components parallel to the interface (\( z = 0 \)): \[ \vec{E_{1t}} = 3\hat{x} + 2\hat{y}. \] - Normal component (\( \vec{E_{1n}} \)): This is the component perpendicular to the interface (\( z \)-direction): \[ \vec{E_{1n}} = 4\hat{z}. \]
Using the continuity conditions:
The displacement vector \( \vec{D} \) is related to \( \vec{E} \) by: \[ \vec{D} = \epsilon_0 \epsilon_r \vec{E}. \] For medium 2 (\( \epsilon_r = 3 \)): \[ \vec{D_2} = \epsilon_0 \cdot 3 \cdot (3\hat{x} + 2\hat{y} + \frac{16}{3}\hat{z}). \] Distribute \( \epsilon_0 \cdot 3 \): \[ \vec{D_2} = \epsilon_0 (9\hat{x} + 6\hat{y} + 16\hat{z}). \]
The displacement vector in medium 2 is: \[ \vec{D_2} = (9\hat{x} + 6\hat{y} + 16\hat{z})\epsilon_0. \]