Saturday, January 2, 2010

2.3 Electric field in materials

Previous sections:
1. Introduction
2. Electrostatic
2.1 Coulomb's Law
2.2 Gauss's Law of electrostatic

2.3 Electric field in materials

So far we have considered the electric field in free space. Obviously, if we now consider the electric field in materials (insulators or conductors), things will be quite different.

In the previous section, we shown that ∇.E=ρ/ε. Let's call this ρ as free charge since it represents the electric charge that is 'free' to move around (note: only free charge contributes to electrical current). Let us now assume that when we apply an electric field to the material, the material will modify the electric field either by strengthening it or weakening it, depending what kind of material it is. Let us represent this effect by introducing an additional charge, called the bound charge, ρ_bound. (These charges are not free and are 'bound' to the material and they cannot freely move about to generate electrical currents.)

One of the ways to imagine bound charge is to consider applying an electric field to a perfect conductor. Assume that the material consists of many electrical dipoles, i.e. opposite charges separated at short distance. (Such dipoles may exist, for example, in molecules with ionic bonding. The electrical field will displace the positive and negative ions slightly to create a dipole.) Now, further assume that these dipoles are not free to move about. They are fixed or bounded to the material. They can only rotate about their axis. What the applied electric field would do to these dipoles is to align them along the electrical field lines with the head of one dipole lining up behind the tail of the other dipole (refer to the diagram below). The result of this 'bound charges' aligning is that the internal electric field cancel each other out (due to the positive and negative charge lying close to each other). This result is important, so remember it: Electrical field inside a perfect conductor is zero. What happens if it is not a perfect conductor? Then, the dipoles cannot align perfectly and the electric fields will attenuate/diminish but it will not be completely cancelled out. You may also ask, what allows us to make such assumption about the electrical dipoles? Nothing, except that experimental results seem to suggest that such assumption is reasonable. As with previous, science is always about suggesting a good assumption to explain the experimental results.

Therefore, in the presence of any material (insulator or conductor), we modify the equation to become ∇.E=(ρ+ρ_bound)/ε to include the bound charges in a material. Having different kinds of charges (bound and free) in the equation is very confusing, this is why in Maxwell's equation all charges always refer to free charges. In order to be consistent with this, we rearrange to 'get rid' of the bound charge in the equation, i.e.

∇.εE - ρ_bound = ρ ;

∇. (εE + P) = ρ ; where ∇.P= - ρ_bound

∇. D = ρ ; where D = εE + P

The polarisation vector P, is the electric dipole moment density. By considering that an electric field causes dipoles to re-arrange in materials, one can calculate that the actual effect an electric field has on a material is to generate a 'surface charge' and a 'volume charge' which is related to P. You can refer to many textbooks on how this is calculated, or you can take my approach, which is just to assume P is a value (like kilograms is for weight) to indicate how much the electric field is affected by the material.

Since the electric dipole is induced by E, we may suspect that P is related to E too, and this is indeed the case. However, the relation may not be a linear one. In most cases we can assume it is linear, i.e. P=kE. But we write P=εχE, where ε=permeability of free space (as usual) and chi, χ=electric susceptibility.

Then D = εE + P = εE + εχE = ε(1+χ)E = ε . ε_r . E ;
where ε_r = (1+χ) is the relative permeability

If ε_r is independent on position, i.e. the same throughout the material, then the material is said to be homogenous. If it is homogeneous, then in general, D = ε . ε_r . E is in matrix form where D and E is a 3 x 1 matrix and (ε . ε_r) is a 3 x 3 matrix. If only the diagonal elements of (ε . ε_r) is non-zero, i.e. the relative permeability is only dependent on the principal (x, y and z) axes, the material is called biaxial, or isotropic:
Dx = ε . ε_r11 . Ex
Dy = ε . ε_r22 . Ey
Dz = ε . ε_r33 . Ez
(where the number indicates the position in the 3 x 3 matrix)

Furthermore, if ε_r11=ε_r22, then the material is said to be uniaxial.

In summary, in the presence of (any) material, the electric field will be different than from the free space and this difference is accounted for by using ε_r, the relative permeability. The equation that relates relative permeability to the electric field, E and displacement field, D is

D = ε . ε_r . E.

But this equation can be confusing sometimes. We can essentially move the relative permeability to other side of the equation and it will now look like this: E = D / (ε . ε_r ). So does the relative permeability serve to modify D or E to account for the presence of a material? If, for example, a dielectric material is placed between two conductor plates (like a capacitor), a constant (electric or displacement?) field will be generated across the dielectric material. What happens at the interface between free space and this dielectric material? Does the displacement field or the electric field change due to the presence of this material? Or both? Although we can use sheer mathematics to find out the answer, it is much more meaningful if we instead rely on our intuition to understand why and which should be the answer. Obviously if both D and E change, and by the same amount, then there is no difference between the two quantity, so this is not allowed. The equation tells us that D and E is related by the permeability, but it does not tell us which is the constant and which is being 'affected' in this case. But if we pay attention to the words I have used so far, I have always said "the materials affect the electric field" and NOT the displacement field. This is a very reasonable statement, since inside a material, especially a conducting one, the charges are BOUNDED, NOT FREE. Displacement field's relation to the charge as indicated by Maxwell's equation ONLY refer to free charges. The free charges, accumulated at the surface of the conductor plates, are constant and therefore D should be constant. The electrical field, on the contrary, is related to the free charge AND the bound charge inside the material. And therefore it is the electric field that will be affected in the presence of a dielectric. (note: there are no free charges INSIDE a conductor but free charges can reside at the SURFACE of a conductor)

In short, the difference between D and E is that "D is the field due to the free charge only" and "E is the field due to both the free and bound charge", the effect of the bound charge is included indirectly through the relative permeability. The quantity D is introduced so that we can make Maxwell's equation look much neater, i.e. always only referring to free charge only. But the usefulness of this displacement field will be more obvious when we disscuss the dynamics of electromagnetism.

Although we have introduced many terms like the polarisation vector, the electric susceptibility, and the relative permeability, it is the relative permeability that is most commonly used to describe the effect of materials on electric field. However, we must always bear in mind that relative permeability is obtained through a series of assumptions. There will be time when we cannot use the relative permeability but instead must use the 'original' equation that contains the polarisation vector or susceptibility, especially for the case of describing in depth behaviour of materials. As a special case, consider a perfect conductor. What is the relative permeability of a perfect conductor? Because D = ε . ε_r . E and E is zero inside a perfect conductor, D will always be zero inside a perfect conductor but this is not true! If we instead use D = εE + P, then when E is zero inside a perfect conductor, D = P. The polarisation vector represents the contribution from the bound charge and thus D is non-zero even in a perfect conductor. Remember, relative permeability is useful because it is a GOOD APPROXIMATION to the 'overall' behaviour of electrostatic systems but it does not work all the time.

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