Electric Fields in Matter v Polarization v Field

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

Conductors Matter Insulators/Dielectrics 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; … centre of the negatively charged cloud is pushed in the opposite direction of E; • If E is large enough ► the atom gets pulled apart completely => the atom gets IONIZED

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

Induced Dipole Moment: (pointing along E) Atomic Polarizability

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). -q +q a +q -q d E At equilibrium, ( produced by the negative charge

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

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: Polar molecules: molecules having permanent dipole moment Ø when put in a uniform external field:

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

+q F- d -q E- F+ 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: Induced dipoles (for non-polar constituents) Aligned dipoles (for polar constituents) A lot of tiny dipoles pointing along the direction of the field

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

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

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: Surface Bound Charge Volume 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; but represent perfectly genuine accumulations of charge !

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

If the cut is not to P : A In general: A ’ 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 z P P is uniform R

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)

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

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. 2. 18 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:

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

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

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

Electric Constitutive Relations and / or Represent the behavior of materials

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

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: