Chapter 4 - Electric Fields in Matter
Reference "Introduction to Electrodynamics" by David Griffiths.
This chapter looks at electric fields within matter that aren't necessarily conductors - i.e.most matter in the universe.
Polarization
Griffith's 4.1
If we place a neutral atom in an electric field, the net force on it is zero (as it is uncharged) - however, the positively-charged nucleus is pushed along into the field, while the negatively charged electron cloud is pulled down into the field.
The neutral atom is now polarized, with some tiny dipole moment : is the atomic polarizability and varies from atom to atom.
If the electric field is strong enough, the cloud will be ripped away from the nucleus and the atom will become ionized. Coulomb's law determines both.
Molecules become more complicated, since they're a few individual atoms coupled together - CO for instance has different polarizabilities parallel or orthogonal to its axis (), so for molecules, a polarization tensor might be more useful.
Force on dipoles
In the presence of an electric field, a dipole will rotate to point in the direction of the -field with a torque given by
At any other point on the dipole than the origin, .
The net force on a dipole in a uniform field is zero as seen in the first image. However, in a nonuniform electric field, the force on a dipole is
The potential energy of a dipole in an electric field is
The Field of a Polarized Object
Griffiths 4.2
In the presence of an external electric field, the individual atoms of a given material will become polarized, leading to a lot of little baby dipoles, all pointing in the direction of the applied field.
Since all of its constituent atoms are polarized, the material itself is said to be polarized with some polarization density .
Potential
Since the potential for an individual dipole is The electric potential for the entire object with polarization is With surface and volume bound charge densities of We can rewrite the potential of our whole polarized object in terms of these densities:
The net potential is just the sum of the potentials produced by the bound surface and volume charges.
, where is the angle between and . in spherical.
To calculate the electric field caused by polarization, use
Physical interpretation of bound charge
Say we have a long string of dipoles, as we might in a polarized dielectric in some field . The head of one dipole will cancel the tail of its neighbor, except at the ends, so that:
No one electron makes the whole trip - but the magnitude of the in some field will be matched by , such that it's almost conductor-like - in a uniform field, we see only accumulations of bound charge on the surface, such that
If, however, the polarization is non-uniform, then we'll see accumulations of bound charge within the object, as well as on the surface,
The Electric Displacement
Griffiths 4.3
Most materials aren't perfect conductors or perfect insulators. Conductors have free flowing electrons , while insulators have exclusively bound charges - a normal dielectric will have both free and bound charges, such that Let the displacement field take into account the fields of the polarization field and electric field, such that Gauss's law for is
We only have control over free charge - bound charge is a consequence of the properties of the material itself. Free charges themselves generate a field.
Though Gauss's law has a parallel for , Coulomb's law does not.
The curl of is
Boundary conditions
For a given dielectric, the boundary conditions are:
Note that since the curl of is zero ),
Linear dielectrics
Griffiths 4.4
For linear substances, the polarization is linearly proportional to the total electric field:
is the electric susceptibility of the medium (dimensionless), with being the permittivity of the material (with in vacuum, the permittivity of free space). A unit-less version is the dielectric constant, .
The displacement field in linear media is thus
Notes on examples
Example 4.5: conducting sphere of radius and charge surrounded by linear dielectric of radius and permittivity , find potential at the center.
- The conducting sphere of charge is our enclosed free charge, so
- Inside the sphere,
- Integral to find potential is (with reference to infinity):
Example 4.6: parallel plate capacitor filled with dielectric material of dielectric constant , what effect does this have on capacitance?
- Dielectric will reduce , and hence , by factor of
- Since , capacitance increased by :
Boundaries of linear dielectrics
In a homogenous isotropic (uniform) linear dielectric, : The boundary conditions are (for ): In terms of :
The potential above and below the boundary is continuous.
Energy in dielectric systems
The work to fill (charge) a capacitor is If a capacitor is filled with dielectric, the capacitance increases by ; thus the work does as well. The total work done (derivation in Griffiths 4.4.3) to fill the capacitor from 0 to its final configuration is
Work is still "energy to assemble a system".
Forces on dielectrics
Dielectrics are attracted into fields just as conductors are, since bound charge tends to accumulate near the free charge of opposite sign.
In a capacitor with a partially-inserted dielectric, this might look like this:
Fringing fields are "notoriously difficult to calculate" (agreed!) - see Griffiths 4.4.4 for a method to deal with them, the result of which is
where is the width of the plates (for a square plate, ).