Match the right field components at a dielectric interface

Take a thin rectangular loop crossing an interface. In the electrostatic limit its circulation vanishes; shrinking its height leaves equality of the tangential electric fields. Next place a thin pillbox across the same interface and apply the displacement version of Gauss’s law. Its flux jump equals the free surface charge. Choose one normal, pointing from material 1 into material 2, and keep it fixed for both limiting fields.

For homogeneous isotropic linear materials, displacement equals absolute permittivity times electric field. Thus without free surface charge the normal displacement is continuous, while the normal electric field generally changes inversely with permittivity. Relative permittivity is dimensionless and must be multiplied by vacuum permittivity in SI formulas. Potential is continuous across an ordinary interface with finite fields, even when its normal derivative jumps. An ideal dipole sheet would require a separate potential-jump model.

The displacement is defined as vacuum permittivity times electric field plus polarization. Its divergence counts free charge because the divergence of polarization accounts for bound charge with the opposite sign. If there is no free surface charge, a discontinuity of normal electric field can still occur because of bound surface charge. Declaring that every surface charge is zero whenever normal displacement is continuous confuses two different conservation statements.

At an interface the net bound surface charge is the normal component of polarization in material 1 minus that in material 2. Combining it with free surface charge recovers the vacuum-permittivity jump condition for electric field. This is a useful audit: compute the electric-field discontinuity using permittivities, then check that the resulting total charge equals free plus bound contributions. Interface sign errors often become obvious when both descriptions are compared.

A two-layer parallel-plate capacitor has constant normal displacement when the internal interface has no free charge. Each layer’s field is that common displacement divided by its own permittivity. The total imposed voltage is the sum of the field times thickness in each layer. Interface conditions alone do not determine the field magnitude: the external plate voltage or plate charge supplies the missing global constraint. Fringing is neglected only when plate dimensions greatly exceed layer thicknesses.

Use the vector lab to inspect components rather than assume full-vector continuity. Reset and set angle to 0° and magnitude to 2. Reset and set angle to 60° and magnitude to 2; predict the x projection falling from 2 to 1 while the y projection grows. These are independent vector examples, not a material-interface protocol. The lab neither enforces displacement matching nor contains a permittivity control, so material calculations must use the equations here.