Across a dielectric boundary with no surface current, what is true about the tangential component of the electric field E_t?

Across a dielectric boundary with no surface current, what is true about the tangential component of the electric field E_t?

• A. E_t is continuous across the boundary (E1_t = E2_t)
• B. E_t is zero on both sides
• C. E_t changes sign across the boundary
• D. E_t is proportional to the surface current density

Answer: (A)

The key idea is how the electric field behaves at a boundary in electrostatics. In this static situation, curl E equals zero, so E is the gradient of a scalar potential and cannot have a jump in its tangential value when you cross a boundary. If the tangential component of E were different on the two sides, you could form a small loop that spans the boundary and along which the line integral of E would be nonzero, implying a nonzero curl—contradicting curl E = 0. Therefore the tangential component must match on both sides: E_t is continuous across the boundary. The note about no surface current doesn’t change this: it’s the absence of a time-varying magnetic field (which would introduce Faraday’s law effects) that enforces the continuity of E_t in electrostatics. Other options would imply a discontinuity or dependence on current, which aren’t required by the boundary conditions in this static case.

The key idea is how the electric field behaves at a boundary in electrostatics. In this static situation, curl E equals zero, so E is the gradient of a scalar potential and cannot have a jump in its tangential value when you cross a boundary. If the tangential component of E were different on the two sides, you could form a small loop that spans the boundary and along which the line integral of E would be nonzero, implying a nonzero curl—contradicting curl E = 0. Therefore the tangential component must match on both sides: E_t is continuous across the boundary.

The note about no surface current doesn’t change this: it’s the absence of a time-varying magnetic field (which would introduce Faraday’s law effects) that enforces the continuity of E_t in electrostatics. Other options would imply a discontinuity or dependence on current, which aren’t required by the boundary conditions in this static case.