The boundary condition on E is

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Option A: $\hat{a}_n 	imes (\vec{E}_1 - \vec{E}_2) = 0$

Explanation & Concepts:
The tangential component of the electric field intensity ($E$) must be continuous across the boundary between two different dielectric media. Mathematically, this boundary condition is expressed as $\hat{a}_n 	imes (\vec{E}_1 - \vec{E}_2) = 0$, which signifies that the tangential components are equal ($E_{1t} = E_{2t}$).

Working Principle: This principle is derived from Maxwell's equation $\oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi}{dt}$. By applying this to a small rectangular loop straddling the interface between two media as the height of the loop approaches zero, the contribution of the normal components vanishes, leaving the tangential components to be continuous across the boundary to satisfy the conservative nature of the static electric field.

Key Formulas: $\hat{a}_n 	imes (\vec{E}_1 - \vec{E}_2) = 0$ — Represents continuity of tangential E-field; $\vec{E}_{1t} = \vec{E}_{2t}$ — Physical interpretation of continuity

Key Points: The tangential component of electric field intensity is always continuous at a boundary.; The normal component of electric flux density ($D$) is continuous only if there is no surface charge density ($ho_s = 0$).; This relation holds regardless of whether the media are conductors or dielectrics.

Advantages: Simplifies analysis of field behavior at interfaces.; Essential for solving electromagnetic boundary value problems.

Disadvantages: Requires knowledge of interface geometry (unit normal vector).; Does not account for surface currents, which would change the condition for magnetic fields.

Applications: Capacitor design and analysis.; Propagation of electromagnetic waves across interfaces.; Transmission line modeling.

Additional Info: If the boundary has a surface charge density, the normal component boundary condition changes to $D_{1n} - D_{2n} = ho_s$.. Option B represents the dot product condition for the normal component, which is generally not zero unless there is no surface charge density.. Option C is a simplified version but lacks the vector-specific definition required for general boundaries.

Answer: A is correct — The tangential component of the electric field intensity must be continuous at the interface, expressed by the cross product being zero.

Exam Tip: Always remember: E-field tangential is continuous (Cross product = 0), and D-field normal is continuous if there is no surface charge (Dot product = 0 if $ho_s=0$).

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Option B: $\hat{a}_n \cdot (\vec{E}_1 - \vec{E}_2) = 0$

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Option C: $\vec{E}_1 = \vec{E}_2$

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Option D: None of above

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