Lecture 8

Dielectrics

Back in lecture 6, we briefly looked at dielectrics as something intermediate between a conductor and an insulator.

Electrons can be displaced a short distance from their equilibrium position.
No sustained current can flow, but this displacement can still give rise to important phenomena.
At the microscopic scale, you can think of a dielectric as consisting of a large number of dipoles.
Water is a good example - the individual molecules have a dipole moment, which has huge consequences that life literally depends on (see lecture 3).
If there’s no external electric field, there is no reason for these dipoles to be oriented in the same direction.
But if an external electric field is applied, dipoles will experience a torque that will make them align along the field, as we saw in lecture 4

The question is, what is the net effect of all these little dipoles now pointing along the electric field lines?

When seen averaged out over large enough scales deep inside the dielectric, there is no net charge density.
The only place where this cannot hold is at the surface of the dielectric.
If all the negative charges have been moved slightly to the left (say), then at the far left hand edge of the dielectric they have nowhere to go. They must pile up there and make a net negative surface charge density. Similarly there is a net positive surface charge density on the opposite side (the right-hand side in this example).
These net charge densities oppose the electric field that generated them.
The idea is exactly the same as how conductors manage to completely cancel out the electric field inside their interior (see lecture 6). But a dielectric can only build up a small charge on its surfaces, so the effect is far weaker.

In summary: the electric field inside a dielectric is reduced, but not eliminated.

Dielectric constant

The discussion above is about what happens on microscopic scales. But the overall effect is just to reduce the intensity of the electric field. The dielectric constant of a material, $\kappa$, is defined as the factor of this reduction:

\[\kappa \equiv \frac{\vert\vec{E}_{\rm vacuum}\vert}{\vert\vec{E}_{\rm dielectric}\vert}\]

All materials¹ have $\kappa > 1$. Air has $\kappa \approx 1.0006$, porcelain has $6$, while water has $\kappa \approx 80$. Some materials have $\kappa$ that are much larger still; calcium copper titanate, for example, has $\kappa \approx 10^4$. The particular value of $\kappa$ depends on details of the material, which we won’t go into here – but it boils down to how easily dipoles can be aligned by an external electric field.

We saw last lecture that the electric field outside a single conducting sheet with charge density $\sigma$ is $E = \sigma/2\epsilon_0$. If the sheet is surrounded by a dielectric, the electric field is reduced by a factor of $\kappa$, meaning that $E = \sigma/2\kappa\epsilon_0$.
The ‘permittivity’ $\epsilon$ of any material is defined as $\epsilon = \kappa\epsilon_0$, so we can write this as $E = \sigma/2\epsilon$.
The permittivity might sound like how ‘permissive’ a material is to electric fields, but beware: high permittivity materials imply lower electric fields! The naming is historical, but it’s too late to change it now.
More generally, since the electric field in a dielectric is reduced by a factor of $\kappa$, equations for a vacuum can be used in a dielectric by replacing $\epsilon_0$ with $\epsilon$. For example, Gauss’s law becomes
\[\oiint_S d\mathbf{A} \cdot \mathbf{E} = \frac{q}{\kappa \epsilon_0} = \frac{q}{\epsilon}\]
where $q$ is the free charge enclosed by the Gaussian surface $S$. Note that $q$ should not include any charges that are part of the dielectric itself, since the effect of these charges is accounted for in the permittivity $\epsilon$.

Quantifying the surface charge

We can work out what the surface charge density on a dielectric is, by imagining it in contact with a charged conducting plate.

Remember the electric field from a conductor in a vacuum is generated by a charge density $\sigma$ on its surface.
To reduce this field by a factor of $\kappa$, the total charge at the interface between the dielectric and the conductor must be $\sigma/\kappa$.
Since no electrons actually pass between the conductor and the dielectric, the charge on the dielectric surface $\sigma_{\rm dielectric}$ must account for the difference between $\sigma$ (the total charge surface density) and $\sigma/\kappa$ (the total charge surface density), i.e. $\sigma_{\rm dielectric} = \frac{\sigma}{\kappa} - \sigma = \sigma\left(\frac{1}{\kappa} - 1\right)$
As $\kappa$ becomes really large, $\sigma_{\rm dielectric}$ becomes close to $-\sigma$. This makes sense: large $\kappa$ means the dielectric is able to cancel out the electric field from the conductor almost completely, and it does it by building up a compensating charge on its surface.

Capacitance with a dielectric

If one sandwiches a dielectric between the two plates of a capacitor, its capacitance increases. This is because:

the potential difference between the plates is reduced by a factor of $\kappa$, while the charge on the plates remains the same. (Remember that the potential difference is just the electric field times the distance between the plates, $V = Ed$.)
the capacitance is defined as $C = Q/V$, where $Q$ is the charge on the plates. Since $V$ is reduced by a factor of $\kappa$, the capacitance is increased by a factor of $\kappa$.

This has implications for the design of capacitors:

the energy stored in a capacitor is $U = \frac{1}{2}CV^2$. So for a given potential difference $V$, the energy stored in a capacitor with a dielectric is increased by a factor of $\kappa$.
capacitance is also proportional to the area of the plates $A$. So another way to think about why dielectrics are useful is that you can shrink the size of a capacitor, while keeping the same capacitance.
This allows for the design of capacitors which fit in a small footprint (e.g. into a microchip within a phone or smartwatch) but which are also very powerful.

Discussion point: the dielectric constant of water is 80. Think of some challenges that prevent us from using water as a dielectric in a capacitor. (There are at least three.)

Footnotes

A more accurate version of this statement is that all natural materials have $\kappa>1$. “Meta-materials” which are constructed by artificially manipulating materials on small scales, can effectively have values of $\kappa<1$, which leads to some exotic properties – even opening up the possibility of constructing an invisibility cloak. ↩