Electrostatic Fields in Matter

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7 Permittivity and SusceptibilityThe permittivity (and dielectric constant) are related to the susceptibility.For simplicity we consider the case of a constant field, as foundbetween parallel plates. We can then omit the vector nature of thefield.From Gauss’s lawUsing (9) and (4):E = σ f − Pɛ 0E = σ f − σ bɛ 0(13)= σ f − ɛ 0 χ e Eɛ 0giving E = σ fɛ 011 + χ e(14)The field is reduced by the factor χ e + 1. The capacitance isincreased by the same factor. Keeping in mind the definition of thedielectric constant, we have:K = ɛ r = χ e + 1 (15)The permittivity of a dielectric is : ɛ = ɛ r ɛ 0Equation (14) could be written in a variety of ways:E =σ fɛ 0 (1 + χ e ) = σ f= σ fɛ r ɛ 0 ɛ(16)Equation (16) illustrates an important point. In dielectrics it isusually possible to use the normal equations for calulating electricfields due to free charges by replacing ɛ 0 by ɛ.8
This could all get confusing!Remember• ɛ will be a very small number (since ɛ 0 = 8.84 × 10 −12 ).• Relative permittivity and dielectric constant are the same thing,with values usually between 1 and 100 (although 50,000 is possible).• χ e is one less than ɛ r . (So what is its value for free space?).8 Electric DisplacementThe differential form of Gauss’s law is:∇ · ⃗E = ρ ɛ 0(17)We now think of the charge as two parts: ρ = ρ f + ρ b .This means that ∇ · ⃗E = ρ f + ρ bɛ 0= ρ f − ∇ · ⃗Pɛ 0and this gives us ∇ · (ɛ 0⃗ E + ⃗ P ) = ρf (18)We now define a new quantity, the electric displacement vector⃗D, defined as⃗D = ɛ 0⃗ E + ⃗ P (19)(This also means that ⃗ D = ɛ ⃗ E) (20)We then have ∇ · ⃗D = ρ f (21)We can think of equation (21) as Gauss’s law in a dielectric.9
Page 3 and 4: 3 Polarisation MechanismsThere are
Page 5 and 6: This leaves a layer of negative cha
Page 7: The amount of charge in the element
Page 11 and 12: SolutionSince in this simple geomet
Page 13 and 14: 9 Boundary Conditions.What happens
Page 15 and 16: Take the length of the plates to be
Page 17 and 18: 12 SummaryThe state of a dielectric

Electrostatic Fields in Matter

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