At a dielectric interface with no surface charge, the tangential components of E satisfy which condition?

At a dielectric interface with no surface charge, the tangential components of E satisfy which condition?

• A. E1_t = - E2_t
• B. E1_t = E2_t
• C. E1_t = 0
• D. E2_t = 0

Answer: (B)

The tangential component of the electric field is continuous across a boundary when there’s no surface charge. Imagine a tiny rectangular loop that straddles the interface with sides parallel to the surface. Faraday’s law says the line integral of E around that loop equals minus the time rate of change of the magnetic flux through it. As the loop shrinks, the area goes to zero, so the right-hand side vanishes, forcing the difference of the tangential E components on the two sides to vanish. In other words, E on each side has the same tangential value: E1_t = E2_t. The absence of surface charge matters for the normal component (through the boundary condition on D_n, since D_n jumps by the surface charge: D1_n − D2_n = σ_s). With σ_s = 0, the normal components satisfy ε1 E1_n = ε2 E2_n, but that does not affect the tangential continuity.

The tangential component of the electric field is continuous across a boundary when there’s no surface charge. Imagine a tiny rectangular loop that straddles the interface with sides parallel to the surface. Faraday’s law says the line integral of E around that loop equals minus the time rate of change of the magnetic flux through it. As the loop shrinks, the area goes to zero, so the right-hand side vanishes, forcing the difference of the tangential E components on the two sides to vanish. In other words, E on each side has the same tangential value: E1_t = E2_t.

The absence of surface charge matters for the normal component (through the boundary condition on D_n, since D_n jumps by the surface charge: D1_n − D2_n = σ_s). With σ_s = 0, the normal components satisfy ε1 E1_n = ε2 E2_n, but that does not affect the tangential continuity.