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## Electric Fields in Matter

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**Electric Fields in Matter**• Polarization • Field of a polarized object • Electric displacement • 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**a**+q +q -q -q d 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)**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:**Polar molecules: molecules having permanent dipole moment • when put in a uniform external field:**Alignment of Polar Molecules:**• when put in a non-uniform external field: +q F+ d -q F-**+q**F+ E+ d -q E- F-**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 onp? (ii) What is the force onq?**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**rs**p The Field of a Polarized Object = sum of the fields produced by infinitesimal dipoles**Dividing the whole object into small elements, the dipole**moment in each volume element d’ : Total potential :**Prove it !**Use a product rule :**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 P A dielectric tube Dipole momentof the small piece: = -q +q A Surface charge density:****P A If the cut is not to P : A’ In general:**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**z** P R Field of a uniformly polarized sphere Choose: z-axis || P P is uniform**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**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**+ + + + + +**+ + + + + + + - d - - - - - - - - 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 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: free charge bound charge caused by polarization NOT a result of polarization**Gauss’ Law :**Electric Displacement ( D ) :**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 ► Linear Magnitude of E ► Isotropic Direction of E**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