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**Electric Fields in Matter**

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

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**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)

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**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)

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**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)

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**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)

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**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)

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

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

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

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**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)

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

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**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)

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**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)

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

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

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

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

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**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)

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**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)

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**The Field of a Polarized Object**

= sum of the fields produced by infinitesimal dipoles rs p r r Dr. Champak B. Das (BITS, Pilani)

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

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

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

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

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

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

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

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

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**Potential due to a surface charge density b**

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

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**= + 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)

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**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)

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**BOUND (POLARIZATION) CHARGE DENSITIES**

Consequence of an external applied field ►Accumulation of b and b Dr. Champak B. Das (BITS, Pilani)

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

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

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

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

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

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**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)

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

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

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**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)

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**Field of a uniformly polarized sphere**

Choose: z-axis || P z P R P is uniform Dr. Champak B. Das (BITS, Pilani)

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**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)

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**Electric field of a uniformly charged sphere Esphere**

Dr. Champak B. Das (BITS, Pilani)

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**At a point inside the sphere ( r < R )**

Dr. Champak B. Das (BITS, Pilani)

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**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)

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**At a point outside the sphere ( r > R )**

Dr. Champak B. Das (BITS, Pilani)

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**Total dipole moment of the sphere:**

(potential due to a dipole at the origin) Dr. Champak B. Das (BITS, Pilani)

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

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

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

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

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**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)

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**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)

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**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)

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**Electric field in the region of overlap between the two spheres**

+ + + - d For an outside point: Dr. Champak B. Das (BITS, Pilani)

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

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**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)

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**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)

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**Gauss’ Law Defining Electric Displacement ( D ) :**

( Differential form ) ( Integral form ) Dr. Champak B. Das (BITS, Pilani)

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

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

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

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

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

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

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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 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)

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**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)

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**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)

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**The Equations of Electrostatics Inside Dielectrics**

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

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**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)

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**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)

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**Electric Constitutive Relations**

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

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**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)

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**Generally, even in linear(& isotropic) dielectrics :**

But in a homogeneous linear dielectric : Dr. Champak B. Das (BITS, Pilani)

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**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)

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

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

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**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)

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**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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