EM-10 · Electrostatics field

Dielectrics and Polarization

Placing a dielectric in an electric field polarizes its molecules and weakens the net electric field inside. As you raise the relative permittivity ε_r, you see the dipoles separate more and the field weaken by a factor of ε_r.

Raise ε_r and the field inside weakens

A dielectric slab sits in a uniform external electric field. The cyan arrows are the external field, and the small dipoles inside the slab, together with the red and blue bound charges on its faces, show the polarization. Raising the relative permittivity ε_r with the slider separates the dipoles more and deepens the charge on the faces, and the gold arrow, the net field inside the slab, gets shorter.

Relative permittivity ε_rε_r = 3.0

Drag to rotate; the slider changes the relative permittivity ε_r.

The net field inside the dielectric

E = E₀/ε_r · the net field weakens by a factor ε_r

P = ε₀(ε_r − 1)E · molecular dipoles line up

Net field (vs external) · 33%

Middle

What a dielectric is

A dielectric is an insulator with almost no free electrons, so charge cannot flow freely as it does in a conductor. Yet when a field is applied, the positive and negative charges in each molecule are pushed very slightly in opposite directions and form a dipole. The charge does not flow away; it only separates a little in place, and this is called polarization.

Polarization, dipoles align

Apply an external field E₀ and countless molecular dipoles line up along it. Molecules like water that already carry a dipole rotate into line, while molecules without one have their charges pulled apart by the field to form a new dipole. The degree of alignment, expressed as the dipole moment per unit volume, is the polarization P. Inside the bulk, neighboring + and − touch and cancel, but on the two end faces uncanceled bound charge appears: − spread over one face and + over the other.

ObserveE = E₀/ε_r

The net field inside a dielectric is weaker by a factor of ε_r.

The net field weakens

The bound charge on the faces creates, inside the dielectric, a field opposite to the external one. Pointing from the + face to the − face, this field shaves off part of E₀, so the net field inside the dielectric is weaker than outside. The factor of weakening is the relative permittivity ε_r, and E = E₀/ε_r. This formula holds when the field enters perpendicular to the faces of the slab, as in the figure. For air ε_r is nearly 1, while for water it is about 80, so water weakens the field a great deal.

ChooseP = ε₀(?)E

The polarization is ε₀ times (ε_r − 1) times the field.

D and permittivity

To handle the share of polarization separately, the electric displacement D is defined. D = ε₀E + P. In a linear dielectric P is proportional to E, so it tidies up to D = εE, with permittivity ε = ε_r ε₀. The advantage of D is that Gauss's law can be written with free charge alone, without counting bound charge separately. ∮D·dA = Q_free, so even with a dielectric present you count only free charge. The displacement itself is not independent of bound charge, though, so finding it directly still requires symmetry.

Fill inε = ?

The permittivity is the relative permittivity times the permittivity of free space.

On your ownD = ? = εE

The displacement is the vacuum term plus the polarization, that is, εE.

Back to the first screen

On the first screen, raising ε_r separated the dipoles in the dielectric more and deepened the bound charge on the faces. The opposing field from that bound charge shaved down the external field, and the gold arrow for the net field grew shorter and shorter. The factor of weakening is exactly ε_r. E = E₀/ε_r. A dielectric calms the electric field through polarization alone, without letting charge flow.

Put a dielectric in an electric field and its molecules are polarized, their positive and negative charges separating slightly. The bound charge on the faces creates a field opposite to the external one, so the net field is weakened by a factor of ε_r. E = E₀/ε_r. The polarization is P = ε₀(ε_r − 1)E, the permittivity ε = ε_r ε₀, and the displacement D = ε₀E + P = εE.

What comes next

Dielectrics go hand in hand with capacitors. Filling the space between two plates with a dielectric lets them hold more charge at the same voltage. For the same charge, the field and the voltage drop by a factor of ε_r, so the capacitance grows by a factor of ε_r. The next unit (EM-11) treats capacitance C = εA/d and counts this effect precisely.