Induced E-fields

Consider a bar magnet moving with velocity v in x-direction as seen from S. Does it produce electric field?

Let us look at the situation from S'. The magnet is at rest and we know, it produces only a magnetic field. You also know the magnetic field directions at different points due to this magnet. You can now do a field transformation from S' to S and get the fields in S. Indeed there is an electric field.

Any source that produces time varying B-field also produces E-field. Time varying B-field is always accompanied by E-field. Such an electric field is called Induced electric field.


If a conducting loop, coil or circuit is placed in an induced E-field, a current can be driven by the induced E-field. The emf is given by the Flux Rule.

The equations governing Induced E-field are:

∇ X E = -(∂ B /∂ t), ∇. E = 0.

These equations have the same mathematical structure as

∇ X B = μ₀ J and ∇. B = 0.

Hence induced E can be obtained from ∂ B/∂ t in the same way as B can be obtained from J .

Lecture-2 Sept 17, 2010 Motional emf

Today we continued with E-B fields of a point charge q moving with a uniform velocity v. We found that (1) The E-field at a point at an instant is radial from the instantaneous position of the charge. So if you wish to get the field at P at time t, and the charge is at A at this time t, the field at t is along AP (q positive). (2) The magnitude of the E-field is largest in the direction perpendicular to the direction of motion and smallest in the direction of motion.


Then we started Motional emf. Questions: What is emf?, What is driving force and what is maintaining force? What charge distribution creates electric field in a wire connected to a battery, resistances etc? The most juicy part was the calculation of magnetic field and electric field inside a conducting rod moving with a uniform velocity v in a uniform magnetic field. The rod, velocity and the B-field were taken mutually perpendicular.
Why should there be an E-field inside the rod, where I had only applied B-field? This is because the magnetic force pushes free electrons on one side and there is a charge distribution coming up on the surface of the moving conductor.


This charge distribution also moves with the rod and hence produces a B-field other than the applied B-field. Thus the B-field inside the rod (and in the vicinity of the rod) will the different from the applied field B. The task is to get these fields. And to do this we go to the rod frame which is S'. Clearly distinguish applied fields in S, Applied Fields in S', Inside (rod) Fields in S and Inside fields in S'. At this moment we know Applied fields in S, and wish to get inside Fields in S. Using Direct transformation equations, get the Applied Fields in S'. Though the Applied E-field is zero in S, it is not zero in S'. It is in y-direction in the scheme taken in class. The Applied B field has somewhat larger magnitude in S' though the direction is the same.


Now that we know the applied fields in S', we turn our attention to fields inside the rod in S'. As the rod is at rest and there is an E-field in y-direction, charges will redistribute on the surface of the rod and finally the E-field inside the rod will be zero. Remember we are talking from S' frame. The redistributed charges are at rest and so produce no magnetic field. The magnetic field inside the rod is THEREFORE equal to the magnetic field Applied (all talks in S'). Hence we obtain E and B fields inside the rod in S'.


Since we know E and B fields inside the rod in S', we do Back transformation to get E and B fields inside the rod in S. And that completes the task.


What do we find for the fields? Now I am talking from S. The job of S' is done. All our experiments are in S. We took help of S' only to get the fields inside the rod in S. As the surface charge distribution on the rod was not known, we could not have directly calculated fields inside the rod in S.
1) The B-field inside the rod is larger than the applied B-field by the factor 1/(1-v²/c²).
2) There is an E-field inside the rod. You can check, E = - v x B as expected in steady state.


Hope it had been a challenging session. I was benefited by the 30-m questions from various students after the class. Thank you.

Summary of Lecture‐1 of Part II

Summary of Lecture‐1 of 2nd part  
--------------------------------------------
1. E, B are frame dependent. Examples: (a) Charge at rest in S, B=0 in S and not zero in S’ 
(b) E= E ₀ in y‐direction, B= B₀  in z
direction in S. Charge sent at velocity E/B in x‐direction in S, Keeps moving 
with this velocity along x‐direction. In S’, it is at rest showing E=0. 


2. Equations for E,B transformation stated without proof. 


3. Example: Line charge λ at rest in S. E Field in S written using Coulomb’s law, B=0. Fields in S’ 
obtained from transformation equations. Comparing with Coulomb’s, Biot‐Savart law shows that  linear charge density is  
λ/√(1‐v²/c²).


Idea of length contraction introduced to get increased λ. 
To get such a contraction, x’ =x‐vt modified by the factor 
1/√(1‐v²/c²). y’=y, z’ = z stated. 


4.Example: A point charge moving in S with velocity v. In S’ the charge is static and fields written  using Coulomb’s law, B’=0. Using inverse transformation equations, E‐field obtained in S.  There are questions about length contractions, and modification in coordinate transformation equations. These will be dealt in somewhat more detail, hopefully, in PHY102 under special theory of  relativity. But once you take the E-B transformation equations granted, they follow naturally. 
   

Lecture Schedule for PHY 103 N Part 2. ( Prof. H.C. Verma)

Lecture Plan of PHY103 2010-11 ( Part II)


Lect No    Date           Topic
---------------------------------------------------------------------------------

1    15 Sept 10    Relativistic Transformation of E-B fields      

2    17 Sept 10    Motional  emf, Flux rule, Lenz law      

3    20 Sept 10    Flux rule in Generalized  case, Faraday's law,  
                            Calculation of induced Electric field.

4    22 Sept 10    Inductance, L-R circuit      

5    24 Sept 10    Magnetic  energy density, Poynting theorem      

6    27 Sept 10    Displacement current, B-field from time varying  E-field      

7    29 Sept 10    Maxwell's Equation in material medium, Boundary conditions      

8    1 Oct 10 (GS)    EM waves in free space, Integrating Optics with Electromagnetism      

9    4 Oct 10            Interference of EM Waves, Light, spatial and temporal coherence      

10    6 Oct 10            Fraunhofer Diffraction of Light at a single slit, N-slits      

11    18 Oct 10            EM waves in material medium, conducting and dielectric medium      

12    20 Oct 10            Reflection and refraction of EM waves      

13    25 Oct 10            Low intensity YDSE, Photoelectric effect, Compton Scattering      

14    27 Oct 10            Davisson Germer Expt,  YDSE with electrons, Heisenberg uncertain

15    29 Oct 10           Wave function,  Probability density, Relation with position and
                                momentum distributions, Delta function wave fn and plane waves      

16    1 Nov 10           Operators for observables,  Schrodinger Equation, Definite Energy states      
 
17    3 Nov 10           Deep Square well ,  Finite Square well, hetero junctions      

18    8 Nov 10           Barrier penetration, Nuclear Fusion and Coulomb Barrier      

19   10 Nov 10           Hydrogen atom wave functions      

20    12 Nov 10          Revision and Discussion      
           
Books:
1.    Electrodynamics by D J Griffiths
2.    Optics by P K Srivastava
3.    Quantum Physics by H C Verma

Magnetostatics in Material Media II

Linear Magnetic Materials
----------------------------------
Just like that for Linear Dielectrics we also have Linear Magnetic Materials for which the Magnetization M is proportional to the applied field. The applied field is taken to be the H field because that is the field which is measured in the Lab. This is unlike the case for electrostatics where the E-field is the applied field . So we have M=χ_mH where χ_m is called the Magnetic susceptibility. Since B=μ₀( H + M )=μH where μ is the permeability of the material we have μ=μ₀(1 + χ_m). As in Dielectrics it is also possible to define a relative permeability μ_r=μ/μ₀=(1 + χ_m).


Since in the linear magnetic materials both the H field and the Magnetization M are proportional to the B- field, it may appear that in this case as ∇.B=0 this implies ∇.H=0. This is wrong, for the same reason as in Dielectrics where Curl E=0 does not imply Curl D=0. This is because at the boundary the proportionality changes and a careful examination of the corresponding integral forms of the law show that both M and H have non zero divergences except in the case when the entire space is filed with a single magnetic material in which case there are no boundaries. For linear materials since the bound current J_b=∇ X M and M =χ_m H 
we have J_b=χ_m J_free. 


The most important and widely used magnetic materials are however Ferromagnetic materials. These are non linear magnetic materials containing permanent magnetic dipoles associated with unpaired electrons in odd electron atoms just as in Paramagnetism. However the difference in this case is that there is a very strong
interaction between neighbouring dipoles due to quantum mechanical reasons. For this reason neighbouring dipoles tend to point the same way even in the absence of an external magnetic field. In small regions the dipole orientation is almost 100 % due to this reason. These are called Ferromagnetic domains and the entire region may be described by a single total dipole moment vector called the magnetic domain vector .


There are a large number of such domains in a Ferromagnetic material with randomly oriented domain vectors subject to random thermal vibrations. When a a Ferromagnetic material is subjected to an external magnetic field the domain vectors tend to align
together and this causes domains to merge and grow and for string fields the entire material may be described by a single domain and results in a very strong Magnetization. When the external field is switched off some of the domain vectors stay aligned and gives rise to permanent magnetization. This is called Hysteresis .


Normally random thermal motion determined by the Temperature of the material compete with domain alignments. However at a certain critical temperature called the Curie Temperature the alignment of domain vectors are favored over random thermal motions. For IRON this is T= 770 deg Centigrade above which it is Paramagnetic wit no domain formations and below the Curie Temperature it is Ferromagnetic. The transition between the Paramagnetic phase and the Ferromagnetic phase is thermodynamically alike to a liquid-solid phase transition in materials, like water-ice transition. The properties of Magnetic materials are decided by quantum mechanics and is a subject of frontline research in Condensed mater physics.


This ends our discussion on the Electric and Magnetic effects of static charges and steady currents. The second part of the course will deal with dynamic situations where charge densities and currents are functions of time and we will see
that this would unify the Electric and Magnetic aspects into a single framework of Electrodynamics involving time dependent electric and magnetic fields and described by a set of 4 equations
called the Maxwell's equation. It was Maxwell that unified the 
apparently different phenomena into a single framework. You will also see how Electrodynamics is intimately connected to
the Special Theory of Relativity.


Hysteresis link: 
http://www.tpub.com/content/chemical-biological/TM-1-1500-335-23/css/TM-1-1500-335-23_208.htm




http://en.wikipedia.org/wiki/Ferromagnetism 

Week 7 Lex 3: Magnetostatics in a Material Media I

Having completed Magnetostatics in free space ( vaccuum) we now turn to study the Magnetostatics in Magnetic Materials. This is the magnetostatic analog of Electrostatics in Dielectric Material.


Recall that the source of Magnetism are currents. In magnetic materials the magnetism is due to atomic currents. These currents essentially arise due to two causes. (i) The orbital motion of the electron around the nucleus (ii) The intrinsic ( quantum mechanical) spin of the electron. These small atomic current loops are equivalent to tiny atomic dipoles from a macroscopic ( large scale ) point of view. Normally their effects cancel out due to (i) random orientation (ii) thermal motion of the atoms in a material. However in an applied external magnetic field B these magnetic dipoles tend to either align parallel or anti-parallel to the applied field giving rise to weak magnetic effects called (i) paramagnetism (ii) diamagnetism respectively. For permanent magnetic materials there is a FROZEN-IN or PERMANENT magnetization. Apart from this we also have Ferromagnetic Materials about which we will discuss later.




The torque on a magnetic dipole in an uniform field B is given as N=m X B whereas in a non uniform field the dipole experiences a force ∇ ( m.B) where the dipole moment m=I A where A is the vector area of the current loop. ( For flat current loops this is the usual area vector). This torque tends to orient the dipoles along the applied field direction resulting in a Magnetization. Due to the Pauli exclusion principle paired electrons in atoms with up and down spins cancel each others torques. So Paramagnetism is most readily observed in atoms with unpaired electrons. As paired electrons do not contribute to the magnetic effects Paramegnatism is a weak magnetic phenomena. Diamagnetism on the other hand arises due to the orbital motion of the electrons. The electron due to the magnetic forces speeds up/slows down in an external magnetic fields B. This causes a change in the dipole moment associated with the orbital motion anti-parallel to the field. ( Check Griffiths). The weak diamagnetic effects arise from this incremental dipole moment.


Hence at a macroscopic ( large scale) level a magnetic material has magnetic polariaztion and can be described by a magnetic dipole moment density which is called Magnetization M which is the magnetic dipole moment/unit volume. Just as in dielectrics it is now possible to describe the macrsocopic field due to a magnetized material by a distribution of surface and volume bound current densities. However the equations for the magntetic field being different the bound currents are now given as a surface bound current density K_b=M X η and a volume bound current desnity J_b=∇ X M ( Check Grifiths for the proof which is just like that in Dielectrics but now involving the Stokes Theorem.)


The bound current densities arise just as in dielectrics from now the cancellation between adjacent atomic current loops leving only the boundary surface contributions for uniform Magnetization M when every loop carries the same current. For non-uniform M the currents are different for different loops and only a partial cancellation occurs giving rise to also a volume bound current density inside the material apart from a surface bound current density.


The differential form of Ampere's Law can now be written as 
∇ X B= μ₀ J_tot where J_tot= J_free + J_b is the total current density consisting of the free current density if any in the material and the bound current density due to the Magnetization M. ( We are neglecting the surface bound current density for the same reason as in Dielcetrics. For real magnetic materials the Magnetization goes to ZERO rapidly within a small surface thickness and surface bound currents do not develop in real life.) But for our problems we will consider such IDEALIZED surface bound currents. ( DIPA/IPSA)


It is now easy to define the magnetic analog of the electric displacement D as the H field which satisfies ∇ X H= J_free. So the source for the H field is the free current density only and  
H= (B/μ₀ - M). We can easily define the corresponding integral form of the Amperes law for H as ∫_C H.dl=I^enc_free. When symmetry allows us we can calculate H from knowing the free currents ( see Eg 6.2 Griffths ).


Similar warning as in Dielcetrics for the displacement vector D is also in force with H. H is not quite like B as ∇.B=0 but ∇.H=-∇.M. Only if M is constant or uniform is B like H. In particular do not assume that H is ZERO because there are no free currents. The H field satisfies similar boundary conditions with both normal and tangential components discontinuous at a surface current density. This is obvious from a look at the equations for H namely ∇.H=-∇.M and  ∇ X H=-K_free for a surface current density.

Week 7 Lex 2: Magnetostatics -II ( Magnetic Vector Potential)

Just as in Electrostatics where the formulation of a scalar potential V ( scalar field)  made the problem of determination of the field simpler because E=-∇ V in Magnetostatics also a Potential formulation is possible. The electrostatic potential formulation for V was a consequence of the fact that ∇ X E =0, and this is ensured as Curl ( Grad V)=0 identically for any scalar field.  For magnetostatics ∇. B=0 implies that the magnetic field B=∇ X A where A is a vector field and called the Magnetic Vector Potential since Div( Curl A)=0 identically for any vector field ( proved in the vector calculus part of the course).


So ∇ X B=∇ X ( ∇ X A)=∇(∇. A) -∇² A=μ₀ J. Notice however that there is an inherent non-uniqueness of the Vector Potential A as A + ∇χ where χ is a Scalar field gives the same magnetic field B=∇ X A + ∇ X (∇ χ) and the second term is identically ZERO because Curl ( Grad χ)=0 for any scalar field χ. This transformation of the vector potential by the gradient of scalar field is called a Gauge Transformation. ( gauge transformations
are at the heart of modern fundamental physics and electromagnetism is an example of the simplest type of gauge theories which describe the 3 forces weak, strong and the electromagnetic


So there are an infinite number of vector potential A differing from each other by a gauge transformation, all of which gives the same magnetic field upon taking the Curl of A due to the reason mentioned above. We can use this non-uniqueness of A to fix the divergence of the vector potential as ∇.A=0 9 once this is fixed the vector potential is unique). This is called a Gauge condition or a Gauge choice. There can be many such gauge choices but we will only consider the condition Div A=0. For this gauge choice then the first term of the RHS of ∇ X B vanishes and we have the equation ∇²A=-μ₀ J.


Notice that this equation ( IN CARTESIAN CO-ORDINATES ONLY ) are actually 3 Poissons equations for the three cartesian components of the vector potential A current density J. These equations are exactly like the Poisson equation in electrostatics for the scalar potential V except that the source term on the RHS is a cuurent density J instead of a charge density ρ. So it is obvious that these equations will have the same solution as that for the electrostatic scalar potential provided we identify ( replace) ρ/ε₀=μ₀J_x,y,z whichever component is appropriate for the RHS. 

Note that this holds only for a Caretesian coordinate system because in a curvilnear co-ordinate system it is easy to see that the original equation do not break up into 3 Poissons equations. So even if you use curvilinear co-ordinates for solving teh integrals for the vector potential A, the current density J must first be expressed in Cartesian coordinates ( Check the DIPA problems and Grifiths examples.)


So if we know the solution to a corresponding electrostatic problem we can simply read off the solution of the magnetostatic problem with the replacement as mentioned above. The trick is this is to identify the correct electrostatic problem. Check out exaples in Griffiths and the DIPA problems.


Magnetostatic Boundary Conditions : The magnetic field B follows certain boundary conditions just like electrostatic fields. These follow just as in the electrostatic case from the integral form of the equations ∇.B=0 and the Amperes Law ∇ X B=mu₀ J . Just as in electrostatics the integral form of these equations evaluated over a infinitesimal Gaussian pillbox and an Amperian loop straddling the boundary shows that the B field has a discontinuity at a surface current just as the electric field has a discontinuity at a surface charge. However in the case of B the tangential component is discontinuous which is perpendicular to the direction of the surface current. The normal component of B is continuous. As the vector potential is chosen to have ∇.A=0 and the magntic flux through an Amperian loop of vanishing width is zero, the vector potential A is continuous across the boundary for both its normal and tangential components. However the normal derivative of the vector potential inherits the discontinuity of B ( note that B comes from a derivative of A which is Curl A).

Magnetic Multipole Expansion: As the Vector Potential A is given by an integral of the current density ( line , surface or volume) similar to the scalar potential V ( which is given by an intergral of the charge density) it is obvious that the Vector potential A will also have a multipole expansion just like the scalar potential by expanding the denominator in a power series. As in the electrostatic case the multipole expansion of the Vector potential is also a series in powers of 1/r over simple current configurations. However as there is no free magnetic poles ( no magnetic monopoles in clasical electromagnetism) the magnetic multipole expansion starts from the pure magnetic dipole term. From the expansion it is possible to determine the vector potential due to a pure magnetic dipole in terms of the magnetic dipole moment m of a pure dipole where m=I a and a is the area of the corresponding current loop. The pure dipole is when the area of the loop goes to zero and the current to infinity keeping m=I a fixed. It is now possible to find the magnetic vector potential due to a small loop at the origin and has a similar form to that of the electric field of a pure electric dipole. Check Griffiths for the field line picture of a pure and a physical magnetic dipole.

Week 7 Lex 1: Magnetostatics in free space-I

Having studied the force on a static charge due to a static chrage distribution
elsewhere (source) which was the fundamental problem of Electrostatics we now turn to studying the forces due to charges moving steadily on each other. Steadily moving charges constitute steady currents and Magnetostatics studies the forces affecting such currents. Two such current carrying conductors exert a force/unit length on each other whose direction is dependent on the direction of the steady current I. These forces are not electrostatic forces as current carrying conductors are overall neutral.


Just as in electrostatics, the fundamental problem of magnetostatics is to find the force on a test current placed at the field point due to an arbitrary current distribution elsewhere ( source currents). The starting point of magnetostatics is the Lorentz force law which is a fundamental law the force on a charge q moving with a velocity v in a magnetic field B is given as F=q v X B . For an electric field also present the force is as we know to be F=qE and this must be added to the previous expression. Moreover as this force is always perpendicular to the velocity v, magnetic forces do not do any work. It is now simple to integrate this force over the length of the conductor to find out the total force on a conductor carrying a steady current I to be F=I∫ dl X B


This is the case for what we call as a line current. Of course just like in electrostatics we can move surface charges in steady motion constitute a steady surface current density K which is the current flowing/ unit length perpendicular to the direction of flow. The magnetic force in this case is given as F=∫ ( K X B).da where the integral is now taken over a surface S. We can similarly define a volume current density J which is the current/unit area oriented perpendicular to the direction of flow. The magnetic force is now given as F=∫ ( J X B).dτ where the integral now must be taken over a volume τ.


As ∫ J.da= over a CLOSED surface S is the total current through the surface this can be related to the total change of charge density with time inside the volume enclosed by the surface S because total charge is always conserved. Using the divergence theorem we arrive at the continuity equations which states that ∇ .J is equal to the -ve of the rate of change of charge density ρ with time t. For steady currents, the charge density is constant in time as steady flow of charges should not have charge piling up anywhere. So for steady currents Div J=0


The analog of the Coulombs law for Magnetostatics is the Biot Savarts Law for magnetic fields due to steady currents. Please note that the fundamental law of Magnetism is the Lorenz force law and Biot Savart Law is a consequence of the Lorentz Force Law. Just as in coulombs law the magnetic field B due to a steady current is given by the integral of the cross product of the current vector
I=I dl'cap with the unit separation vector from the source point to the field point divided by the square of the separation vector.
The generalization to surface currents K and volume currents J is as usual.
Please check Eg 5.1 to 5.6 in Griffiths


Its easy to see that the magnetic field of an infinite staright line current circulates around the conductor according to the right hand rule in an azimuthal ( plus/minus  φ cap ) direction. The Curl of B can easily be calculated by seeing that the circulation of B over a loop C is ∫ B.dl=μ₀ I_enc where I is the current ENCLOSED by the loop C. This called Amperes law and is the analog of the Gauss Law for Electrostatics. Now using Stokes theorem its easy to see that 
∇ X B =μ₀ J where J is the volume current density. For surface currents J is to be replaced by K.


The divergence of B is ∇. B=0 . This is unlike electrostatics where Div E is equal to the charge density/epsilon zero. There are no free magnetic charges and the magnetic field lines are always CLOSED. This can be easily proved for a volume current density J using similar techniques to those in electrostatics. ( Refer to Griffiths for the derivation where like before they show that a surface contribution from the boundary is ZERO.)


Please read Section 5.34 Griffiths for a comparison of Electrostatics and Magnetostatics. Also look through the Eg. 5.7-5.10 Griffiths.

Week 6 Lex 1: Dielcetrics II

In this lecture we continue with our discussions about Electrostatics in a Dielectric medium. Having obtained the field at a point outside a dielectric material due to the surface and volume charges developing in the dielectric material due to Polarisation,
we now turn to study the internal microscopic field in the dielectric. For the field outside we used the potential for a pure dipole, however polarisation which is due to charge displacements involve physical dipoles. Outside the material since the field point is far away this does not matter as the dipole potential dominates ( atom is neutral so it has zero net charge and hence no monopole moment. The discreteness of the molecular dipoles is also negligible far away and a continuous Polarisation vector field is justified. This is however not the case inside.


The microscopic field inside the dielectric material is very complex and impossible to calculate. But we can focus on macroscopic or average behavior which is the one observable in experiments. The macroscopic or the large scale field is obtained as an average of the microscopic field over a region is large enough to neglect discreteness of molecular dipoles but small enough so that we do not lose all the field variations. This is roughly of the order of 1000 molecular lengths.


For the macroscopic field inside the dielectric, we consider a spherical surface
of radius R around that point. The field has two contributions  
E=E_out + E_in
(i) E_out from all the charges (molecular dipoles) outside S the average field is the field they would produce at the center. This is given by just the dipole potential for the dipole moment density outside integrated over the region OUTSIDE.
(ii) E_in from all the charges ( molecular dipoles) inside S the average field is the field due to a dipole of dipole moment p which is the total dipole moment within the sphere assuming the sphere S to be small enough so that the polarisation vector P is roughly constant.


Now the term left out of the integral in (i) is due to the field at the center of a uniformly polarised sphere. This is calculated in eg 4.2 in Griffiths and it is exactly the same as that of the average field inside S. So this term puts back in the region that was left out of the integral for V due to outside charges.

So the final relation is a potential V due to all dipoles inside the full volume of the dielectric material and hence the integral for V any now be written as over the full volume of the dielectric material. This analysis justifies the use of the V due to dipoles for the macroscopic field both inside and outside the dielectric material.
This is not matter how complicated the internal ( microscopic) field is, for macroscopic fields we can replace it with a smooth distribution of pure dipoles. The argument is independent of the spherical shape S used over here and holds for any general shape.


Gauss Law in Dielectrics: We now turn to the form of the Gauss law in a Dielectrics. As Gauss Law refers to the total enclosed charge only, the differential form for Gauss Law will now be
∇.E=ρ_free + ρ_bound
where ρ is the charge density due to the free charges and the bound charges respectively. Neglecting the surface bound charges, the volume bound chrage density
ρ_b=-∇.P where P is the Polarisation vector. Rearranging terms we get
∇.[ε₀ E +P]=ρ_free
or ∇.D=ρ_free where D=ε₀ E +P is called the Electric Displacement Vector .


The Displacement vector only refers to the free field charge density. We can now write down the integral form of the Gauss Law in Dielectrics by integrating both over some volume τ bounded by a closed surface S and then using the Divergence Theorem to convert the LHS to a surface integral over the closed surface S. So the closed surface integral
∫_S D. da =Q_free^enc
where Q is the total free charge enclosed within the closed surface S. The surface bound charges are ignored in the edge thickness approximation where the polarisation falls rapidly to zero at the edge of the dielectric and hence no surface charge density.


Although the displacement vector D sees only the free charges it is important to notice that it cannot be used as a substitute for the electric field E.
They are very different from each other,
(i) No Coulombs Law for D unlike that of E
(ii) Curl D is not zero unlike Curl E=0. ∇ x D=∇ ; X P.
So no potential formulation for D as its not a conservative field in general. The integral form is ∫ _C D.dl=∫ _C P.dl
by using Stokes theorem on the differential form.


The electrostatic boundary conditions on the Displacement Vector D can now be easily derived exactly in the same fashion as derived for the Electric Field E by using the corresponding integral forms ∫D.da=Q_free^enc and ∫ _C D.dl=∫ _C P.dl. This is worked out in Griffiths and is left as a reading exercise for the students. Please check Ex 4.8 in Griffiths


A certain class of Dielectric materials offers a simplification of these relations.
In these materials P, D are both proportional to the total Electric field E ( applied field + Polarisation field). So for these materials
D =ε E an P =ε₀χ_e E. Its easy to see that
ε=ε₀(1 + χ_e )
The ratio ε/ε₀=(1 + χ_e )=ε_r is called Relative Permittivity or Dielectric Constant.


However please note that even for Linear Dielectrics the Curl D is NOT ZERO unless the entire space is filled with a single homogeneous dielectric. The reason for the decrease of the Electric Field of a point charge in a dielectric medium as compared to a point charge in free space is now clear. The polarisation charges actually screen the point charge so its actual effective magnitude within the dielectric decreases and hence the corresponding electric field is reduced.