Electromagnetism II (spring term 2020)

Lecture 6

Electric field in dielectric materials

E Goudzovski  [email protected] http://epweb2.ph.bham.ac.uk/user/goudzovski/Y2-EM2

Previous lecture

 External electric fields induce polarization of dielectric materials, which is characterized by the electric susceptibility:

 Volume distribution of polarization P can be replaced by an equivalent distribution of surface and volume charges:

and

 Molecular theory of gaseous dielectrics leads to

The two terms are due to the induced polarization and the permanent dipole moment p of the molecules. 1 This lecture

Lecture 6: Electric fields in dielectric materials

 The electric displacement field D  Relative permittivity  Gauss’s law for the D field  Boundary conditions for the E and D fields  Field lines at boundaries  Energy density of the electric field

2 The electric displacement field

Dielectrics in non-uniform fields: (lecture 5) induced volume polarization charges with density

In general, dielectrics can also carry free charges with density

Total charge density:

Gauss’s law:

Therefore and

Define the electric displacement field

3 Gauss’s law for the D field

Gauss’s law for the electric displacement:

In the integral form (see lectures 1,2):

It is a generalized version of Maxwell’s equation (M1):

Using the definition of electric susceptibility, ,

where is the relative permittivity (aka the dielectric constant) of the medium. Gauss’s law for uniform : E-field decreases by a factor of >1 relative to the field produced by the same distribution of free charges in free space. 4 Physical meaning Assumptions behind the idea of a constant for a dielectric:

 Linearity: P is proportional to E. This breaks down at high electric fields.

 Isotropy: polarization is always collinear to E. Some crystals are anisotropic, particularly under mechanical stress (the piezoelectric effect).

 Homogeneity: no dependence on the position within material.

Most insulators satisfy these conditions in practice (and are known as “LIH dielectrics”)

Physical meaning:

 Electric field E [V/m]: force acting on unit charge.  Polarization P [C/m2]: induced dipole moment per unit volume.  Electric displacement D [C/m2]: introduced to facilitate field calculations in the presence of dielectrics (polarization charges do not enter Gauss’s law for D). 5 Boundary conditions for E and D

S1 l1 Boundary of two dielectrics 1

with relative permittivities 1 and 2 2 S2 l2 Gauss’s law for a very thin disk (if no free surface charge):

therefore

Considering , obtain and

More generally, the discontinuity of Dn equals the surface charge density:

A very thin rectangular loop: infinitely small area, . Conservation of energy

similarly leads to 6 Field lines at boundaries

Dn is continuous:

θ1 Et is continuous:

θ2 Therefore

(similar to Snell’s law in optics  not by accident!)

 Electrostatic field in cavities in high- dielectrics is partially shielded.

 Cf. complete shielding in

a hollow conductor 2>1 (Faraday’s cage, lecture 4) 7 Dielectric slab in a capacitor (1)

+ P  

Area S

E E/ b a

 The displacement field D is uniform (Dn is conserved): .

 Therefore E = D/(0) is reduced by a factor of  in the dielectric. Field energy density (W~E) is reduced: high- dielectrics are attracted into regions of stronger electric field. 8 Dielectric slab in a capacitor (2) Potential difference:

For b=0 (no dielectric slab),

For b=a (slab occupying all space), : reduces by a factor .

Capacitance in the two cases:

: increases by a factor . 9 Electric field energy (1) d Work required to move the charge from plane A to plane B in a dielectric: E

C (S: area; : surface charge density; V: volume) B A +  Considering that the D-field is zero outside the capacitor, the boundary condition gives , therefore

Therefore The increment in energy density [J/m3] is

More generally (also for anisotropic dielectrics),

10 Electric field energy (2)

In the assumption (true for LIH dielectrics), the energy density is

of which is due to polarization of the dielectric.

More generally (also for anisotropic dielectrics),

In electrostatics, this can be interpreted as the potential energy of electric charges (assuming action at a distance).

In electrodynamics, energy is localized in space and is carried by the electric field. 11 Summary

 The electric displacement vector is defined as

 Gauss’s law for the electric displacement field:

or equivalently

 For linear, isotropic and homogeneous (e.g. most) dielectrics,

 Boundary conditions: and , leading to

 Energy density of the electric field: 12