In empty space, electromagnetism is clean: a charge sits here, a current flows there, and Maxwell's equations tell you the fields. Drop the same charge into a block of glass, water, or iron, though, and the tidy picture explodes into chaos. Every one of the 10^{23} atoms in the material feels the field, stretches or twists in response, and becomes a tiny source in its own right. To find the true field you would have to track the pull of every stretched atom on every other — an impossible bookkeeping problem.

This page is about the beautiful trick physicists use to escape that mess. Instead of tracking each atom, we lump the material's collective response into two new fields, called \mathbf{D} and \mathbf{H}, that are cleverly built so they depend only on the charges and currents we control — the "free" ones we push in with a battery — and never on the swarm of atomic charges the material conjures up by itself. Master that one idea and a slab of dielectric or a lump of iron becomes almost as easy to handle as empty space.

Polarisation: matter grows an electric grain

Put a neutral atom in an electric field \mathbf{E}. The positive nucleus is nudged one way and the electron cloud the other, so the atom stretches into a little dipole pointing along the field. A whole material full of these aligned dipoles is said to be polarised. We measure how strongly with the polarisation \mathbf{P} — the dipole moment per unit volume.

Here is the key consequence. Inside the bulk, neighbouring dipoles sit head-to-tail: the + end of one cancels the - end of its neighbour, so the interior stays neutral. But at the surfaces there is nothing to cancel against — one face is left with a sheet of exposed positive ends, the opposite face a sheet of negative ends. These are the bound charges: real charge, but shackled to their atoms, not free to flow away. Counting them up gives two exact rules:

The minus sign in \rho_b = -\nabla\cdot\mathbf{P} is worth a moment: where \mathbf{P} diverges (fans outward), positive ends are marching away and leaving negative ends behind — hence a build-up of negative bound charge.

The electric displacement \mathbf{D}

Gauss's law in matter has to answer to all charge, free and bound alike:

The bound term is a nuisance — we don't know \mathbf{P} until we know \mathbf{E}, which is what we're trying to find. So sweep it to the left and give the combination a name. Define the electric displacement

and the ugly divergence rearranges into something gorgeous:

That is the payoff in a single line. The divergence of \mathbf{D} sees only the free charge — the charge you actually deposited with a battery or a rubbed rod. All the messy bound charge has been absorbed into the definition of \mathbf{D} and vanishes from the source. Its integral form, \oint \mathbf{D}\cdot d\mathbf{A} = Q_{\text{free,enc}}, lets you find \mathbf{D} from a symmetric free-charge arrangement without ever thinking about the material.

Linear dielectrics: the easy, common case

For most everyday insulators the induced polarisation is simply proportional to the field that caused it — double the field, double the stretch. Such a linear dielectric obeys

where the dimensionless electric susceptibility \chi_e measures how easily the material polarises. Feed this back into the definition of \mathbf{D}:

So in a linear medium \mathbf{D} and \mathbf{E} are just proportional, through the permittivity \varepsilon = \varepsilon_0(1+\chi_e). Dividing out \varepsilon_0 gives the number quoted on datasheets, the relative permittivity or dielectric constant:

Air is barely polarisable (\varepsilon_r \approx 1.0006); glass sits around \varepsilon_r \approx 5; water is a whopping \varepsilon_r \approx 80. The graph below shows how, in a linear medium, \mathbf{D} climbs with \mathbf{E} along a straight line whose slope is the permittivity — steeper material, more displacement for the same field.

Picture it: a slab in a field

The whole story of \mathbf{P}, bound charge, and the reduced interior field fits in one diagram. Reveal it step by step: an external field points right, we slide a dielectric slab in, its atoms line up as dipoles, the uncancelled ends pile up as bound surface charge on the two faces — and that bound charge sets up a back-field that partly cancels the applied field, so the field inside the slab is weaker than outside.

The magnetic story: magnetisation and \mathbf{H}

Magnetism plays the same game with a magnetic accent. Atoms carry tiny current loops (from orbiting and spinning electrons), each a magnetic dipole. Apply a field \mathbf{B} and these loops tend to align; the material becomes magnetised, described by the magnetisation \mathbf{M} — the magnetic dipole moment per unit volume.

Just as aligned electric dipoles leave uncancelled bound charge, aligned current loops leave uncancelled circulating current. Inside the bulk, adjacent loops run antiparallel where they touch and cancel; only at the surface (and wherever \mathbf{M} varies) does a net bound current survive:

Ampère's law must include every current, free and bound:

Same trick: move the bound term to the left and christen the combination the auxiliary field \mathbf{H}:

And the curl of \mathbf{H} sees only free current — the current you drive through your wires. Its line-integral form, \oint \mathbf{H}\cdot d\mathbf{l} = I_{\text{free,enc}}, is how you find \mathbf{H} inside a solenoid or toroid full of iron without touching the atomic currents. For a linear magnetic material,

with magnetic susceptibility \chi_m and permeability \mu = \mu_0(1+\chi_m). Note the deliberate parallel — and one sneaky asymmetry: for the electric field we wrote \mathbf{P}=\varepsilon_0\chi_e\mathbf{E} (proportional to \mathbf{E}), but for magnetism we tie \mathbf{M} to \mathbf{H}, not \mathbf{B}. That is exactly why \mathbf{H} earns its keep as the practical magnetic variable in matter.

Boundary conditions: what happens at a surface

Fields don't just live inside one material — they cross from one to another, and at that interface they must obey matching rules. These fall straight out of the four field laws by shrinking a Gaussian pillbox or an Amperian loop across the boundary. Split each field into the component perpendicular (\perp) to the surface and the component parallel (\parallel) to it:

Notice the elegant pairing: the auxiliary fields (D^{\perp}, H^{\parallel}) are the ones that jump — and they jump only by free sources — while the true fields (E^{\parallel}, B^{\perp}) glide across smoothly. This is why \mathbf{D} and \mathbf{H} make problems with interfaces tractable.

Worked example: a capacitor filled with dielectric

This is the payoff every student should carry home. Take a parallel-plate capacitor, plates carrying free charge \pm Q, area A, free surface charge \sigma_{\text{free}}=Q/A. Now fill the gap with a linear dielectric of constant \varepsilon_r.

Step 1 — find \mathbf{D} from free charge alone. A Gaussian pillbox across one plate gives, exactly as in vacuum,

The dielectric never entered — that is the whole point of \mathbf{D}.

Step 2 — get the real field \mathbf{E}. In the linear medium D=\varepsilon E=\varepsilon_0\varepsilon_r E, so

where E_0=\sigma_{\text{free}}/\varepsilon_0 is the field the same free charge would make in vacuum. The dielectric weakens the field by a factor \varepsilon_r — precisely the reduction the slab diagram above was showing, now with a number on it.

Step 3 — the voltage and the capacitance. The plate voltage is V=Ed=E_0 d/\varepsilon_r=V_0/\varepsilon_r: the dielectric lowers the voltage for the same charge. Since C=Q/V,

Filling a capacitor with a dielectric multiplies its capacitance by \varepsilon_r. Slip glass (\varepsilon_r\approx 5) between the plates and you store five times the charge at the same voltage — which is exactly why real capacitors are stuffed with dielectric, and why the numbers on the side of one are so much bigger than an air gap could ever give.