Electric Fields in Matter  Polarization  Electric displacement  Field of a polarized object  Linear dielectrics

Matter Insulators/Dielectrics Conductors All charges are attached to specific atoms/molecules and can only have a restricted motion WITHIN the atom/molecule.

When a neutral atom is placed in an external electric field (E): … positively charged core ( nucleus) is pushed along E; If E is large enough ► the atom gets pulled apart completely => the atom gets IONIZED … centre of the negatively charged cloud is pushed in the opposite direction of E;

For less extreme fields ► an equilibrium is established => the atom gets POLARIZED ……. the attraction between the nucleus and the electrons AND ……. the repulsion between them caused by E

Induced Dipole Moment: Atomic Polarizability (pointing along E)

To calculate  : (in a simplified model) The model: an atom consists of a point charge (+q) surrounded by a uniformly charged spherical cloud of charge (-q). At equilibrium, ( produced by the negative charge cloud) -q E d +q a -q

At distance d from centre, (where v is the volume of the atom)

Prob. 4.4: A point charge q is situated a large distance r from a neutral atom of polarizability . Find the force of attraction between them. Force on q :

Alignment of Polar Molecules:  when put in a uniform external field: Polar molecules: molecules having permanent dipole moment

Alignment of Polar Molecules:  when put in a non-uniform external field: d F+F+ F-F- -q +q

F-F- d F+F+ -q +q E+E+ E-E-

For perfect dipole of infinitesimal length, the torque about the centre : the torque about any other point:

Prob. 4.9: A dipole p is a distance r from a point charge q, and oriented so that p makes an angle  with the vector r from q to p. (i) What is the force on p? (ii) What is the force on q?

Polarization: When a dielectric material is put in an external field: A lot of tiny dipoles pointing along the direction of the field Induced dipoles (for non-polar constituents) Aligned dipoles (for polar constituents)

A measure of this effect is POLARIZATION defined as: P  dipole moment per unit volume Material becomes POLARIZED

The Field of a Polarized Object = sum of the fields produced by infinitesimal dipoles p rsrs

Dividing the whole object into small elements, the dipole moment in each volume element d  ’ : Total potential :

Prove it ! Use a product rule :

Using Divergence theorem;

Defining: Volume Bound Charge Surface Bound Charge

surface charge density  b volume charge density  b

Field/Potential of a polarized object Field/Potential produced by a surface bound charge  b Field/Potential produced by a volume bound charge  b + =

Physical Interpretation of Bound Charges …… are not only mathematical entities devised for calculation; perfectly genuine accumulations of charge ! but represent

-q +q d A Surface Bound Charge A dielectric tube Dipole moment of the small piece: = Surface charge density: P

A P  A’ In general: If the cut is not  to P :

_ _ _ _ _ _ _ __ Volume Bound Charge A non-uniform polarization accumulation of bound charge within the volume diverging P pile-up of negative charge

Net accumulated charge with a volume Opposite to the amount of charge pushed out of the volume through the surface =

Field of a uniformly polarized sphere Choose: z-axis || P P is uniform z PR 

Potential of a uniformly polarized sphere: (Prob. 4.12) Potential of a polarized sphere at a field point ( r ): P is uniform P is constant in each volume element

Electric field of a uniformly charged sphere

At a point inside the sphere ( r < R )

Inside the sphere the field is uniform

At a point outside the sphere ( r > R )

(potential due to a dipole at the origin) Total dipole moment of the sphere:

Uniformly polarized sphere – A physical analysis Without polarization: Two spheres of opposite charge, superimposed and canceling each other With polarization: The centers get separated, with the positive sphere moving slightly upward and the negative sphere slightly downward

At the top a cap of LEFTOVER positive charge and at the bottom a cap of negative charge Bound Surface Charge  b d

Recall: Pr Two spheres, each of radius R, overlap partially. ++ -- _ + d _ +

Electric field in the region of overlap between the two spheres d For an outside point:

Prob. 4.10: A sphere of radius R carries a polarization where k is a constant and r is the vector from the center. (i) Calculate the bound charges  b and  b. (ii) Find the field inside and outside the sphere.

The Electric Displacement Polarization Accumulation of Bound charges Total field = Field due to bound charges + field due to free charges

Gauss’ Law in the presence of dielectrics Within the dielectric the total charge density: bound charge free charge caused by polarization NOT a result of polarization

Gauss’ Law : Electric Displacement ( D ) :

Gauss’ Law

D & E :

Boundary Conditions: On normal components: On tangential components:

For some material (if E is not TOO strong) Electric susceptibility of the medium Linear Dielectrics Recall: Cause of polarization is an Electric field Total field due to (bound + free) charges

Location ► Homogeneous Magnitude of E ► Linear Direction of E ► Isotropic In a dielectric material, if  e is independent of :

In linear (& isotropic) dielectrics; Permittivity of the material The dimensionless quantity: Relative permittivity or Dielectric constant of the material

and / or Electric Constitutive Relations Represent the behavior of materials

But in a homogeneous linear dielectric : Generally, even in linear dielectrics :

When the medium is filled with a homogeneous linear dielectric, the field is reduced by a factor of 1/  r.

Capacitor filled with insulating material of dielectric constant  r :

Energy in Dielectric Systems Recall: The energy stored in any electrostatic system: The energy stored in a linear dielectric system: