Dielectrics: the Slab In, the Battery On or Off

Insulators in a field

A metal has free electrons that flow until the field inside is zero. An insulator, or dielectric, has none. Every electron stays bound to its own molecule. But a field can still move the charges of each molecule a little, and that is enough to change the field.

Some molecules, such as oxygen, have their positive and negative charges centred at one point. A field pulls the two apart slightly and makes each molecule a small dipole along the field. Other molecules, such as water, are dipoles already, pointing every which way. In the chapter Electric Potential, Energy and the Dipole you saw the torque on a dipole in a field. It turns these dipoles towards the field. Either way the insulator becomes full of small dipoles lined up with the field. This is polarisation.

Play Fig 5.1 and follow one row of molecules. Once they line up, the plus end of each sits next to the minus end of the next. Inside the slab these cancel, so the inside stays neutral. At the left face a layer of minus ends has no partner; at the right face a layer of plus ends has none. These layers are the induced charge of the dielectric, \(-\sigma_i\) facing the positive plate and \(+\sigma_i\) facing the negative plate.

The comb of the opening does the same to the paper. The paper polarises, with the charge of the opposite sign on the side near the comb. The comb's field is stronger nearer the comb, so the near layer is pulled harder than the far layer is pushed. The net pull lifts the paper.

The field falls by K

The induced layers are like a second, weaker pair of plates facing the other way. Between the plates, their field points against the field of the free charges. The net field in the slab is \(E\) \(= (\sigma - \sigma_i)/\varepsilon_0\), smaller than \(E_0 = \sigma/\varepsilon_0\). The ratio \(E_0/E\) is a property of the material, its dielectric constant \(K\).

Since \((\sigma - \sigma_i)/\varepsilon_0\) \(= \sigma/K\varepsilon_0\), the induced density is \(\sigma_i = \sigma(1 - 1/K)\). For air \(K\) is about \(1.0006\), so air is almost the same as vacuum. Most plastics and paper have \(K\) of \(2\) to \(4\). Water, whose molecules are strong dipoles, has \(K\) about \(80\). A metal is the extreme case: its free charges cancel the field completely, as if \(K\) were infinite.

With the charge fixed, the field falls by \(K\), so the voltage \(Ed\) falls by \(K\). The same charge at a \(K\) times smaller voltage means a capacitance \(K\) times larger. The plates of Lesson 1, \(88.5\,\text{pF}\) in air, become \(4 \times 88.5\) \(= 354\,\text{pF}\) filled with \(K = 4\). In every formula for the capacitance, \(\varepsilon_0\) becomes \(K\varepsilon_0\). So a spherical capacitor of \(100\,\text{pF}\), filled with oil of \(K = 4\), has \(400\,\text{pF}\).

The slab in, battery removed

As in Lesson 2, first ask what stays fixed. Charge a \(20\,\text{nF}\) air capacitor to \(200\,\text{V}\), so it holds \(Q = 4\,\mu\text{C}\) and \(U = 4 \times 10^{-4}\,\text{J}\). Remove the battery, then slide in a slab of \(K = 4\) that fills the gap. The charge has nowhere to go, so \(Q\) stays \(4\,\mu\text{C}\).

The capacitance rises to \(80\,\text{nF}\). The voltage falls to \(Q/C\) \(= 50\,\text{V}\), and the field in the gap to a quarter of its value. The energy \(Q^2/2C\) falls to a quarter too: \((4 \times 10^{-6})^2/(2 \times 8 \times 10^{-8})\) \(= 10^{-4}\,\text{J}\). Three quarters of the store, \(3 \times 10^{-4}\,\text{J}\), has gone.

It went into the slab's motion. Near the edge of the plates the fringing field acts on the induced charges of the slab and pulls it in. If the slab is let go, it shoots in and oscillates. To push it in slowly, a hand must hold it back. The hand then does negative work, \(W_\text{agent} = \Delta U\) \(= -3 \times 10^{-4}\,\text{J}\).

The slab in, battery connected

Now keep the \(200\,\text{V}\) battery connected while the slab goes in. The battery holds \(V\), so the field \(V/d\) cannot change. With \(C\) now \(80\,\text{nF}\), the charge rises to \(CV\) \(= 16\,\mu\text{C}\), four times as much. The battery pushes the extra charge onto the plates. There the extra free charge just makes up for the induced layers, which cancel three quarters of it.

In both columns the agent's work is negative, so in both the slab is pulled in. With the battery removed, the pull comes out of the capacitor's own store. With it connected, the battery pays for the rise in the store and for the pull. Lesson 6 shows this pull directly, as the slab slides in.

Breakdown, and why capacitors are filled

Every insulator has a limit. In a strong enough field it tears electrons from its molecules, a current flows, and a spark jumps. This is breakdown, and the largest field a material can bear is its dielectric strength. For dry air it is about \(3 \times 10^6\,\text{V/m}\). Lightning is the breakdown of air on a grand scale.

This is why real capacitors are filled. A dielectric gives a larger capacitance in the same box, and it bears a larger field than air. So the plates can be brought very close without a spark. The fan capacitor of Lesson 1 uses both tricks: a plastic film a few micrometres thick and a dielectric constant above one.

A capacitor with an air gap and one filled with dielectric, in series, show where breakdown begins. Take two \(20\,\text{nF}\) air capacitors, fill one with \(K = 4\), and put them in series across \(100\,\text{V}\). The filled one has \(80\,\text{nF}\), so the pair has \(16\,\text{nF}\) and each holds \(1.6\,\mu\text{C}\). The air one takes \(1.6\,\mu\text{C}/20\,\text{nF}\) \(= 80\,\text{V}\); the filled one only \(20\,\text{V}\).