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

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

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

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

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

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Dr. Champak B. Das (BITS, Pilani) -e-e +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.

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

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

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

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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 :

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

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

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

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

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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?

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

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

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

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

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

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

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

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

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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 + =

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

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

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

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

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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.

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

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

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

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

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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

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

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

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

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

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

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

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

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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

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

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

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

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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.

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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

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Dr. Champak B. Das (BITS, Pilani) 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

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

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

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

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

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

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

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

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

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Dr. Champak B. Das (BITS, Pilani) 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

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

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

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

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

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

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

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

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

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