A refraction with no wave in it

A ray of light entering glass bends towards the normal, by a law relating the sines of the two angles. A static electric field line entering a dielectric also bends, and the resemblance is close enough that the effect is usually introduced as “refraction of field lines”.

The resemblance is a good deal shallower than it looks. There is no wave, no speed, no wavelength and no time in the derivation, the law involves tangents rather than sines, and the bending goes the opposite way. What the two share is that both follow from what a boundary does to a field, which is a real similarity and worth having.

The two things a boundary cannot do #

Everything below comes from two statements, each of which is one of Maxwell’s equations applied to a shape that has been shrunk onto the interface.

The component of E along the surface is continuous. Take a small rectangular loop straddling the boundary, with its long sides parallel to the surface, and shrink its height to nothing. The line integral of E round a closed loop is zero in the static case, the short sides contribute nothing once the height vanishes, and what is left is that the tangential fields on the two sides are equal.

The component of D across the surface is continuous. Take a small pillbox straddling the boundary and shrink its thickness. Gauss’s law says the flux of D out of it equals the free charge inside, which is zero if the interface carries none, and what is left is that the normal components of D on the two sides are equal.

The second condition comes from the same place Gauss’s law does. Take a closed surface with no free charge inside it, so that whatever flux enters must leave, and then shrink it onto the boundary until only its two flat faces matter. What survives of the statement is that the normal component of $\mathbf{D}$ is the same on both sides — the enclosed free charge is none, so there is nothing for the two faces to disagree about. No wave has been mentioned and none is needed; this is a bookkeeping argument about a surface with nothing in it.

Both statements are worth reading twice, because the shrinking is doing something subtle. Nothing is being approximated: the loop and the pillbox are being taken to a limit in which the contributions from their sides go to zero faster than the contributions from their faces, so what survives is exact rather than leading-order. That is why boundary conditions in electromagnetism are equalities rather than estimates, and why they hold across an interface of any shape, at any point, however curved.

Both are exact and neither involves the material. What the material supplies is the relation between D and E, which for a linear isotropic dielectric is $\mathbf{D} = \varepsilon\mathbf{E}$, and that is the entire physical input.

Dividing one by the other #

Write the angles from the normal. Tangential continuity says $E_1\sin\theta_1 = E_2\sin\theta_2$. Normal continuity says $\varepsilon_1 E_1\cos\theta_1 = \varepsilon_2 E_2\cos\theta_2$.

Divide the first by the second and the field magnitudes cancel:

$$\frac{\tan\theta_2}{\tan\theta_1} = \frac{\varepsilon_2}{\varepsilon_1}$$

Tangents, not sines. The two laws agree for small angles, where both reduce to a ratio of the angles themselves, and diverge everywhere else. At 68° incidence into a medium four times as permittive the tangent law gives 84°, and there is no incidence at all at which the ratio saturates — the tangent runs to infinity, so grazing on one side is grazing on the other and there is no analogue of total internal reflection.

The right-hand panel of the opening figure is that relation drawn, with the identity line beside it. Every point on the curve is a traced pair of directions built from the two continuity conditions, and the ratio of their tangents comes out at the permittivity ratio to a part in ten thousand billion, which is the check that the construction and the algebra describe the same boundary.

Which way it bends, and why that is the opposite #

A light ray entering a denser medium bends towards the normal. A field line entering a more permittive medium bends away from it, towards the surface.

The reason is visible in the two conditions. The tangential field is carried across unchanged, and the normal field is reduced by the permittivity ratio, because it is D rather than E that is continuous. So the field on the dense side has the same sideways part and a smaller part across, which is a direction closer to the surface.

Physically, the dielectric has polarised: the matter has taken part of the field away, and it has taken it away only from the component that terminates on the interface. Bound charge appears on the surface at a density equal to the discontinuity in the normal E, and that bound charge is what the reduction consists of.

The direction has a consequence worth carrying. A field line meeting the surface at grazing incidence stays nearly parallel to it whatever the materials, because a tangent of infinity divided by four is still infinity. So the ratio matters least where the field is most nearly along the surface, and most where it is nearly across.

The conductor as a limit #

Push the permittivity ratio up and the lines on the dense side come out ever closer to the surface; equivalently, on the thin side they come out ever closer to the normal.

In the limit of infinite permittivity, every line on the thin side arrives exactly perpendicular. That is the familiar statement that the field meets a conductor’s surface at a right angle, and it is this boundary condition taken to its extreme rather than a separate rule.

The correspondence is exact in the static case and is a useful way to think about intermediate cases. A high-permittivity ceramic behaves like an imperfect conductor as far as field-line geometry goes: lines are drawn into it and turned towards its surface, and the field just outside is nearly normal. That is why a high-$\varepsilon$ substrate concentrates the field of a microstrip line and why the effective permittivity of such a line is neither the substrate’s nor the air’s.

Running the same limit the other way gives a cavity in a dielectric, where the lines on the inside are turned away from the surface, and that is the geometry behind the depolarising factors that decide how a small inclusion responds to an applied field.

Reading the picture as bound charge #

There is a second account of the same bending that uses no boundary conditions at all, and having both is worth the effort because each answers a question the other cannot.

It is worth reading the picture in terms of charge rather than of fields, because the two descriptions are the same arithmetic and one of them is easier to hold. Polarise a body and bound charge appears on its surface; that charge makes a field of its own inside the body, opposed to the one that produced it, and the reduction of the interior field is entirely that charge’s doing. On a plane interface the same bound charge is what bends a field line across it. Nothing extra has been introduced — the permittivity was a way of not mentioning the bound charge, and here it is again.

Put a dielectric in a field and its molecules polarise. In the bulk the resulting charges cancel against their neighbours; at a surface they do not, and what is left is a bound surface charge of density $\mathbf{P}\cdot\hat{\mathbf{n}}$ — the component of the polarisation across the boundary.

That charge produces a field of its own, directed so as to oppose the applied one, and only in the direction normal to the surface. Add it to the applied field and the normal component is reduced while the tangential component is untouched, which is the tangent law arrived at from the other end.

The two accounts are not alternatives. The boundary-condition route gets the field without ever having to find the charge, which is why it is the practical one. The bound-charge route says where the force on the dielectric acts — on the surface, where the charge is — which the boundary conditions do not, and it says what happens if the polarisation is not uniform, which produces a bulk bound charge as well. A calculation of a real device usually needs the first to get the field and the second to get the mechanics.

Where the energy goes and the force acts #

A field line diagram is a picture of directions and says nothing about forces. The forces are worth a section, because they are where this geometry has practical consequences.

The energy and the force follow from the same conditions without any new principle. Slide a dielectric slab part-way into a capacitor and the field at the edge of the slab is not uniform: the lines fringe, and it is in that fringing region that the force on the slab acts. Which way it acts is decided by exactly the boundary conditions above, and the answer — that the slab is pulled in, whether the plates are held at fixed voltage or at fixed charge — comes out of the geometry rather than out of a separate rule about energy.

A dielectric slab partly inserted between charged plates is pulled in. The uniform field in the middle exerts no net force on it; the entire force comes from the fringing field at the edge, where the lines are neither along nor across the surface and the bound charge feels a component of field that is not perpendicular to it.

That is a case where the field-line picture is genuinely instructive, because the correct answer is easy to get wrong. Treating the slab as though the force acted where the field is strongest gives no force at all; recognising that the force acts where the field is non-uniform gives the right sign and, with the boundary conditions, the right size.

The same reasoning explains why a dielectric is drawn towards a region of stronger field generally, why a stream of water bends towards a charged rod, and why an uncharged object is attracted to a charged one at all.

The same conditions, moving #

None of this needed the field to be static — only that it be slowly varying enough for the electrostatic equations to hold. Letting it oscillate turns the same two conditions into optical refraction, and it is worth seeing how the tangent becomes a sine.

For a wave, the two continuity conditions still hold at every instant, but there is a third requirement: the phase must match along the interface, so the tangential component of the wavevector is the same on both sides. That single extra condition is Snell’s law, and it involves sines because a wavevector’s tangential component is $k\sin\theta$.

So the sine law is about phases and the tangent law is about fields. They coexist: a light wave crossing an interface obeys both, with the sine law fixing the propagation direction and the field conditions fixing the amplitudes — which is where the Fresnel coefficients come from.

The static case has no phase to match, so only the field conditions survive, and the tangent law is what is left. That is the cleanest way to see that the resemblance between the two refractions is real but structural rather than a shared mechanism.

Where it is used #

The bending of static field lines is not a curiosity; it is a constraint that field solvers, insulation designers and electrostatics engineers work with constantly.

The place this is used is insulation design, and the reason is the kink. Field lines refract at the boundary and equipotentials must stay perpendicular to them, so both families bend where two materials meet — and the field concentrates on whichever side has the lower permittivity. That is why a void inside a solid insulator is dangerous out of all proportion to its size: the air in it has the lowest permittivity anywhere in the assembly, the field there is the highest, and the weakest material is carrying the largest stress.

Insulation design. A high-voltage cable is a stack of dielectrics, and the field is largest where the permittivity is lowest — because the normal D is continuous and the normal E is D over $\varepsilon$. A small void in a solid insulator therefore carries several times the field of the material around it, which is why voids cause partial discharge and why the manufacture of high-voltage insulation is largely a campaign against bubbles.

Field grading. The same relation run the other way is a design tool: putting a graded-permittivity layer where the field would otherwise concentrate smooths it out, and stress-grading tubing on a cable termination is exactly that.

Electrostatic separation. Sorting plastics or minerals in an electrostatic separator relies on particles of different permittivity experiencing different forces in the same non-uniform field, and the size of the difference is set by the same ratio. So is the operation of an electret microphone’s bias, an inkjet’s droplet deflection, and the electrostatic clamping used to hold a silicon wafer flat in a vacuum tool.

Capacitance calculation. A layered capacitor’s capacitance depends on whether the layers are stacked across the field or along it, and the two arrangements give the series and parallel combinations respectively. That difference is the boundary conditions again, with the geometry deciding which component is continuous where.

The same law with a current in it #

The construction used two facts: a line integral round a vanishing loop is zero, and the flux out of a vanishing pillbox is the free charge inside. Replace the second with the statement that charge is conserved — in a steady state, as much current leaves the pillbox as enters it — and the whole argument runs again with a different quantity.

Tangential E is continuous as before. Normal $\mathbf{J}$ is now the continuous one, and Ohm’s law in a conductor is $\mathbf{J} = \sigma\mathbf{E}$. Divide as before and

$$\frac{\tan\theta_2}{\tan\theta_1} = \frac{\sigma_2}{\sigma_1},$$

which is the identical law with conductivities in place of permittivities. Current refracts at the boundary between two conducting media, and it refracts by the conductivity ratio. That is what a geophysical resistivity survey exploits: current injected at the surface is bent at each layer boundary, and the apparent resistance measured between electrodes at increasing separations inverts to a depth profile.

The interesting part is what happens when the two laws disagree. A real material has both a permittivity and a conductivity, and the electrostatic law wants the normal E to jump by $\varepsilon_1/\varepsilon_2$ while the steady-current law wants it to jump by $\sigma_1/\sigma_2$. Both cannot hold unless $\sigma_1/\varepsilon_1 = \sigma_2/\varepsilon_2$, and there is no reason two materials should satisfy that.

They do not have to. What happens instead is that free charge accumulates on the interface until the two demands are reconciled, and the time it takes is the charge relaxation time $\varepsilon/\sigma$ of the more resistive material. That is a very long time for a good insulator: a polymer at $10^{-16}$ siemens per metre relaxes in something like a day.

The consequence is that a layered insulator has one field distribution the instant it is energised — the capacitive one, set by permittivities — and a completely different one after the charge has settled, set by resistivities. High-voltage DC cable insulation is designed for the second, and its resistivity is strongly temperature-dependent, so a loaded cable’s field can invert between the conductor and the sheath as it warms. The same interfacial charge, driven at a frequency, is what gives layered materials an enormous apparent permittivity below a few hertz.

Which field the lines are drawing #

There is an ambiguity in every picture of refracting field lines, and it is worth naming because the two possible readings differ in a way that is visible.

E-lines and D-lines are not the same lines. D has no divergence where there is no free charge, so a D-line that enters the boundary leaves it: the lines bend and none of them begins or ends. E has a divergence wherever there is bound charge, and the interface is exactly where the bound charge is — so E-lines are created or destroyed on it, in the ratio of the two permittivities.

So “the number of lines is conserved across the boundary” is true of one field and false of the other, and a diagram that does not say which it is drawing has left its most quantitative feature undefined. The kinked direction is common to both; the density is not.

The distinction shows up whenever a layered capacitor is worked out. Stack the layers across the field and D is the same in each, so it is the D-lines that pass through both and the E in each layer is D over its own permittivity — which is where the series formula comes from, and why the thinnest, least permittive layer carries the largest field. Stack them along the field and E is the same in each instead, D differs, and the capacitances add. Neither result needs a calculation once it is settled which quantity is being carried across.

What the pictures cannot show #

Both media are linear, isotropic and homogeneous. A real dielectric may be anisotropic, in which case D and E are not parallel and the whole construction has to be redone with a tensor; ferroelectrics are nonlinear, and the ratio depends on the field.

The interface carries no free charge. If it does — a charged surface, a leaky boundary with a current through it — the normal D jumps by that charge and the tangent law is modified. Conducting media in the steady state obey a version with conductivities in place of permittivities, which is a different law with the same shape.

Nothing is drawn at the edge of the boundary. Real interfaces are finite, and near their edges the field does something the plane-boundary conditions do not describe; that region is where the interesting forces are.

Frequency is absent from the static case and is the whole story in the optical one. A material’s permittivity is a strong function of how fast it is asked, and the constant that appears here is the low-frequency one — water’s 80 rather than the 1.77 that its refractive index implies. Using the wrong one is the commonest error in applying this construction.

And the lines are a representation. A field line is a choice about how to draw a vector field, and what is physical here is the pair of continuity conditions rather than the picture. The kink in a drawn line is real in the sense that the direction of E really does change; it is not real in the sense that anything travels along the line.

The ladder from here #

Later rungs on this anchor: the depolarising factor, which is this boundary condition solved for an ellipsoidal inclusion and gives how a small particle responds to an applied field; the Clausius–Mossotti relation, which builds a bulk permittivity out of molecular polarisabilities and the field a molecule actually sees; image charges in a dielectric half-space, which is the same problem solved by a trick; and the analogous conditions for magnetic media, where the ratio involves permeabilities and the roles of the two fields are exchanged.

The neighbouring ladders are the field the matter takes away, which is where the permittivity comes from, the bend at the boundary, which is the optical refraction this one resembles and is not, and the inside of a conductor, which is this boundary condition with the ratio taken to infinity.