# Electric Fields in Matter

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Electric Fields in Matter
Polarization Field of a polarized object Electric displacement Linear dielectrics Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani)
Conductors Matter Insulators/Dielectrics All charges are attached to specific atoms/molecules and can only have a restricted motion WITHIN the atom/molecule. Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani)
A simplified model of a neutral atom electron cloud nucleus The positively charged nucleus is surrounded by a spherical electron cloud with equal and opposite charge. Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani)
When the atom is placed in an external electric field (E) E The electron cloud gets displaced in a direction (w.r.t. the nucleus) opposite to that of the applied electric field. Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani)
If E is large enough ► the atom gets pulled apart completely => the atom gets IONIZED For less extreme fields ► an equilibrium is established => the atom gets POLARIZED Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani)
-e +e ► The net effect is that each atom becomes a small charge dipole which affects the total electric field both inside and outside the material. Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani)
Induced Dipole Moment: (pointing along E) Atomic Polarizability Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani)
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 E d +q +q a -q At equilibrium, ( produced by the negative charge cloud) Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani)
At distance d from centre, (where v is the volume of the atom) Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani)
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 : Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani)
Alignment of Polar Molecules: Polar molecules: molecules having permanent dipole moment when put in a uniform external field: Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani)
Alignment of Polar Molecules: when put in a non-uniform external field: +q F+ d F- -q Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani)
F+ -q +q E+ E- F- Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani)
For perfect dipole of infinitesimal length, the torque about the centre : the torque about any other point: Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani)
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? Dr. Champak B. Das (BITS, Pilani)

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 Dr. Champak B. Das (BITS, Pilani)

P  dipole moment per unit volume
Material becomes POLARIZED A measure of this effect is POLARIZATION defined as: P  dipole moment per unit volume Dr. Champak B. Das (BITS, Pilani)

The Field of a Polarized Object
= sum of the fields produced by infinitesimal dipoles rs p r r Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani)
rs p r r Total potential : Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani)
Prove it ! Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani)
Using Divergence theorem; Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani)
Defining: Surface Bound Charge Volume Bound Charge Dr. Champak B. Das (BITS, Pilani)

Potential due to a surface charge density b
& a volume charge density b Dr. Champak B. Das (BITS, Pilani)

= + Field/Potential of a polarized object
Field/Potential produced by a surface bound charge b + Field/Potential produced by a volume bound charge b Dr. Champak B. Das (BITS, Pilani)

Physical Interpretation of Bound Charges
…… are not only mathematical entities devised for calculation; but represent perfectly genuine accumulations of charge ! Dr. Champak B. Das (BITS, Pilani)

BOUND (POLARIZATION) CHARGE DENSITIES
Consequence of an external applied field ►Accumulation of b and b Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani)
E  ( n : number of atoms per unit volume ) Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani)
E  Net transfer of charge across A : Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani)
Net charge transfer per unit area : P is measure of the charge crossing unit area held normal to P when the dielectric gets polarized. Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani)
When P is uniform : P  M N Q   Q E  … net charge entering the volume is ZERO Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani)
Volume bound charge P A Net transfer of charge across A : Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani)
Surface bound charge P  E  N M G Net accumulated charge between M & N : Dr. Champak B. Das (BITS, Pilani)

Field of a uniformly polarized sphere
Choose: z-axis || P z P R P is uniform Dr. Champak B. Das (BITS, Pilani)

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 Dr. Champak B. Das (BITS, Pilani)

Electric field of a uniformly charged sphere Esphere
Dr. Champak B. Das (BITS, Pilani)

At a point inside the sphere ( r < R )
Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani)
Field lines inside the sphere : P ( Inside the sphere the field is uniform ) Dr. Champak B. Das (BITS, Pilani)

At a point outside the sphere ( r > R )
Dr. Champak B. Das (BITS, Pilani)

Total dipole moment of the sphere:
(potential due to a dipole at the origin) Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani)
Field lines outside the sphere : P Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani)
Field lines of a uniformly polarized sphere : Dr. Champak B. Das (BITS, Pilani)

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 Dr. Champak B. Das (BITS, Pilani)

Bound Surface Charge b
At the top a cap of LEFTOVER positive charge and at the bottom a cap of negative charge + + + - d Bound Surface Charge b Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani)
Recall: Pr. 2.18 Two spheres , each of radius R, overlap partially. + - _ + d _ + Dr. Champak B. Das (BITS, Pilani)

Electric field in the region of overlap between the two spheres
+ + + - d For an outside point: Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani)
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. Dr. Champak B. Das (BITS, Pilani)

The Electric Displacement
Polarization Accumulation of Bound charges Total field = Field due to bound charges + field due to free charges Dr. Champak B. Das (BITS, Pilani)

Gauss’ Law in the presence of dielectrics
Within the dielectric the total charge density: free charge bound charge caused by polarization NOT a result of polarization Dr. Champak B. Das (BITS, Pilani)

Gauss’ Law Defining Electric Displacement ( D ) :
( Differential form ) ( Integral form ) Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani)
D & E : … “looks similar” apart from the factor of 0 ( ! ) …….but : Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani)
D & E :  Field = - Gradient of a Scalar Potential  No Potential for Displacement Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani)
Boundary Conditions: On normal components: On tangential components: Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani)
Prob. 4.15: A thick spherical shell is made of dielectric material with a “frozen-in” polarization where k is a constant and r is the distance from the center. There is no free charge. a b Find E in three regions by two methods: Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani)
Prob. 4.15: (contd.) (a) Locate all the bound charges and use Gauss’ law. For r < a : For r > b: For a < r < b: Answer: a b Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani)
Prob. 4.15: (contd.) (b) Find D and then get E from it. Answer: a b Dr. Champak B. Das (BITS, Pilani)

The Equations of Electrostatics Inside Dielectrics
or with Dr. Champak B. Das (BITS, Pilani)

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 Dr. Champak B. Das (BITS, Pilani)

Permittivity of the material
In such dielectrics; Permittivity of the material The dimensionless quantity: Relative permittivity or Dielectric constant of the material Dr. Champak B. Das (BITS, Pilani)

Electric Constitutive Relations
and / or Represent the behavior of materials Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani)
In a dielectric material, if e is independent of : Location ► Homogeneous ► Linear Magnitude of E ► Isotropic Direction of E Most liquids and gases are homogeneous, isotropic and linear dielectrics at least at low electric fields. Dr. Champak B. Das (BITS, Pilani)

Generally, even in linear(& isotropic) dielectrics :
But in a homogeneous linear dielectric : Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani)
Free charges  D , as: In LD : When the medium is filled with a homogeneous linear dielectric, the field is reduced by a factor of 1/r . Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani)
Capacitor filled with insulating material of dielectric constant r : Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani)
So far……… …source charge distribution at rest ELECTROSTATICS 1st/4 Maxwell’s Equations Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani)
Coming Up….. …source charge distribution at motion MAGNETOSTATICS ELECTROMAGNETISM A New Instructor Dr. Champak B. Das (BITS, Pilani)

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