The boundary condition on E isSelect one:a. an×(E1− E2) = 0b. E1 = E2c. an•(E1− E2) = 0d. None of above

Step 1: Break Down the Problem

To determine the correct boundary condition on the electric field $E$, we need to analyze the options provided in relation to Maxwell's equations and the principles of electromagnetism at the boundary between two different media.

Step 2: Relevant Concepts

When considering the boundary conditions of the electric field $E$ across the interface between two media, the relevant concepts are:

1. The electric field must be continuous at the boundary unless there is a surface charge present.
2. The normal component of the displacement field $D$ changes according to the surface charge density.

Step 3: Analysis and Detail

Let's examine the provided options:

1. Option a: $\mathbf{a_n} \times (E_1 - E_2) = 0$ - This statement implies that the difference between the electric fields in the two media is perpendicular to the normal vector, which does not represent a boundary condition.
2. Option b: $E_1 = E_2$ - This condition indicates that the electric field is continuous across the boundary. This could be true if there is no surface charge.
3. Option c: $\mathbf{a_n} \cdot (E_1 - E_2) = 0$ - This states that the component of the electric fields in the direction of the normal vector is continuous across the boundary, which is consistent with the absence of surface charge.
4. Option d: None of the above is incorrect because we have valid conditions in options b and c.

Step 4: Verify and Summarize

• The correct boundary condition for electric fields at the interface between two different media can either be that the fields are equal ($E_1 = E_2$) or that their normal components are equal ($a_n \cdot (E_1 - E_2) = 0$) assuming no surface charge.
• Based on this analysis, both options b and c are acceptable representations of boundary conditions on $E$.

Final Answer

The correct boundary condition on $E$ is: Option c: $\mathbf{a_n} \cdot (E_1 - E_2) = 0$ (assuming no surface charge).