Welcome to PYL100 course

Lecture-4 on 09/01/2015 By: Rajendra S. Dhaka (rsdhaka@physics.iitd.ac.in)  

PYL100: Electromagnetic Waves and

Quantum Mechanics

1  Ch.4: Electric Fields in Matter

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Effect of uniform polarization: Bound charges Ø  Dielectric material becomes polarized in the E-field

² We measure this effect as the polarization P….. ² P Dipole moment per unit volume

Cartoon representation of dipoles aligning in an E-field

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Effect of uniform polarization: Bound charges ² Now we have two E-fields: Ø 1) External applied electric field….

Dielectric Material Induce many dipole moments Ø  2) ``Internal field``

due to the induced Polarization

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The Field of a Polarized Object:

Ø  Concept of bound charges: Ø  What is the field produced by the induced dipoles?

Ø  So we do??.. As we know the field/potential (far field limit) for a single dipole..

Ø  Subdivide the dielectric into infinitesimal dipoles & integrate

Ø  It is easier to work with potential….

Field Lines & Potential surfaces for a single dipole

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What is the field produced by the polarized material?

Ø  Subdivide the dielectric into infinitesimal dipoles & integrate

Integrate by parts using the product rule

Using the Divergence theorem:

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What is the field produced by the polarized material?

Ø  Resembles potentials of surface & volume charges.

with &

² The potential of a polarized dielectric material is the same as is produced by:

1.  Surface charge density

2.  Volume charge density ²  This is true for the E-field too since the gradient is a linear operator

bound  charges  

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Physical interpretation of bound charges:

Ø  What is the physical meaning of bound charges?

Ø  Consider a long string of dipoles:

²  Along the line head of one effectively cancels the tail of it’s neighbor.

²  Only two charges at the ends are left.

²  The net charges at the ends are called bound charge.

²  Results from polarization and has E..

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How do we calculate the amount of bound charge? bound charge density

Ø  Consider a cylinder of dielectric parallel to P

Ø  If the slab is cut at angle to the cylinder, the charge is still the same, but A becomes larger

     

Bound charge accumulated on right face

A = cross-sectional area of the tube d = the length of the chunk

Surface charge density

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Surface bound charge density

Ø  Consider a cylinder of dielectric parallel to P

     Since these charges reside on

the surface and are bound to the dipoles they are called the bound surface charge.

Physically, the effect of polarization is, to “paint” bound charge over the surface of the material.

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Effect of non-uniform polarization?

²  strength of the individual dipoles will vary ²  bound charge will also accumulate inside the

dielectric material, as well as at the surface

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Effect of non-uniform polarization? ²  Diverging P results in a pileup of negative

charges within volume.

²  If the material is uniformly polarized then the volume charge density is equal to zero…

²  Since the net charge on the polarized material must be zero, the sum of the volume charges & surface charges must be equal to zero.

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Effect of non-uniform polarization?

²  The net bound charge in a given volume is equal and opposite to the amount that has been pushed out through the surface.

This is true for any volume bound charge.

(Gauss’s law)

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Summary: Bound Charges ²  Effect of polarization = Bound charges ²  Uniform polarization == Surface bound charges

²  Non-uniform polarization == Volume bound charges (also surface charges)

²  Bound surface charge density: ²  Bound volume charge density: ²  Field of a polarized object = produced by a surface

charge of density plus a volume charge of density ²  First find the bound charges, and then calculate the

fields they produce.

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Another way of analyzing the uniformly polarized sphere

v  Two spheres of charge: a +ve sphere and a –ve sphere

v  Without polarization, the two spheres are superimposed and cancel completely.

v  But when the material is uniformly polarized,

the +ve charges move slightly upward and

the –ve charges move slightly downward.

v  The two spheres no longer overlap Perfectly.

v  At the top, there is a “cap” of leftover +ve charge and at the bottom a cap of -ve charge.

v  This “leftover” charge is precisely the bound surface charge

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Field inside a dielectric: Microscopic field & Macroscopic filed

²  Microscopic field.. very complicated, will not discuss.

²  Macroscopic field is defined as the average field over regions large enough to contain millions of atoms.

²  The macroscopic field then consists of two parts: Ø  Average field over the sphere due to all charges outside Ø  Average field due to all charges inside

the   integral   runs   over   the  en>re  volume  of  the  dielectric.  

total  dipole  moment  

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Important points: Ø  Effect of polarization is to produce bound

charges. Ø  Field due to polarization of the medium is the

field produced by these bound charges Ø  Considering all together, we can have field

due to bound charges and field due to (say) free charges (charges not as a result of polarization, ex: ions in the dielectric material).

Ø  Within the dielectric, the total charge density :

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The Electric Displacement: Ø  Gauss’s law in the presence of Dielectrics: For closed surface:   !

E ⋅d!aS"∫ =

1ε0Qenc

= total charge enclosed within the surface.

= permittivity of free space

encQ

0εLet’s apply the divergence theorem:

We can also write: encV

Q dρ τ= ∫ = Volume charge density

ρ

(!∇⋅!E

V∫ )dτ = ( ρ

ε0)dτ

V∫This gives

Finally, we get

!∇⋅!E = 1

ε0ρ

This is the Gauss’ law in differential form

!E ⋅d!a

S!∫ = (

!∇⋅!E)dτ

V∫

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The Electric Displacement: Ø  Gauss’s law in the presence of Dielectrics:

Ø  E is the total field inside the dielectric, not just that portion generated by polarization.

,where

Ø  The new field D is called the ‘Electric Displacement’.

Gauss’ law

Total charge density (bound +free)

This is the Gauss’ law for Dielectrics in differential form

(integral  form)  

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The Electric Displacement & Free Charges: Ø  In situations in which Gauss’ Law helps, one

can use this new relation to calculate D, and then to determine E from D, from the free charges alone.

Ø  The use of D turns out to be most helpful where the polarization is not built in, but instead is induced by an external applied electric field.

**Home Work**

Examples: 4.1, 4.2 and 4.3 of Griffiths…