Maxwell's Equations in Matter Flashcards
When can Maxwell’s equations be used?
- Maxwell’s equations are always valid in classical physics
- they can be applied within matter eventhough some of them use constants like εo or μo of free space
- the four Mawell’s equations are sometimes referred to as Maxwell’s equations in a vacuum (although they can be applied in any situation)
- Maxwell’s equations in matter (macroscopic) are simplified versions of these four
What are the names of the four Maxwell’s Equations (microscopic)?
- Gauss’s Law for Electric Fields
- Gauss’s Law for Magnetic Fields
- Faraday’s Law
- Ampere-Maxwell Law
Which of the four Maxwell’s Equations are simplified for use in matter?
- the enclosed charges and currents in the four Maxwell’s equations include ALL the charges and currents in an object
- within matter this makes the equations very difficult to use (although they are still valid)
- Gauss’s law for magnetic fields and Faraday’s law do not need matter friendly versions as they don’t include charges or currents
- Gauss’s law for electric fields and the Ampere-Maxwell law do require simplification
Dielectric
Definition
-a non-conducting material is a dielectric
Charge Distribution in Atoms and Molecules
- atoms and molecules are electrically neutral
- some atoms and molecules have a charge distribution that is not spherically symmetrical
- this means that the centre of the negative charge is not necessarily at the same point as the centre of the positive charge
- these can be thought of as a positive point charge at the centre of the positive charge and a negative point charge at the centre of negative charge
- therefore they have a permanent electric dipole moment
Electric Dipole Moment
-the electric dipole moment, |p, for two charges -q and +q a vector |L apart is given by
|p = q|L
Induced Electric Dipole Moment
-even atoms and molecules that are not polar (i.e. do not have a permanent electric dipole moment) can still have an induced electric dipole moment when placed in an electric field
Distortion Polarisation
- when a dielectric is placed in an electric field the centres of positive charge and negative charge within the molecules/atoms are distorted so that they are no longer at the same point
- therefore an electric dipole moment is induced
Orientation Polarisation
-if a dielectric material is placed in an electric field then the permanent and/or induced dipoles will align to the field and the material itself will be polarised
Dielectric Between the Plates of a Capacitor
Description
- consider a dielectric material between two metal plates of a capacitor, the electric field between the two plates will polarise the dielectric material
- the dielectric material ends up with a bound charge at its surface
- in the middle the system is entirely neutral, at one side there will be a net positive charge and on the other a net negative charge
- the bound charge produces an electric field within the dielectric that opposes the electric field due to the free charge on the plate
Bound Charge
-charge is bound if it is not free to move and therefore cannot conduct a current
Dielectric Constant
-the dielectric constant, κ, is also referred to as the relative permittivity, εr
-it is defined by the electric field strength at a point in a vacuum Eo compared to the electric field E at that same point when a dielectric material is present at that point
E = Eo/κ
κ=εr
κ≥1
-in a vacuum, κ=1
Dielectric in a Capacitor
V
-consider a charged but disconnected capacitor with plates separated by a vacuum a distance d apart
-there is a potential difference Vo between the plates, each plate has charge Q and area A
-now place a dielectric between the plates:
E = |-dV/dx| = V/d
V = Ed = Eo/κ * d = Eo*d/κ = Vo/κ
Dielectric in a Capacitor
C
-consider a charged but disconnected capacitor with plates separated by a vacuum a distance d apart
-there is a potential difference Vo between the plates, each plate has charge Q and area A
-now place a dielectric between the plates:
C = Q/V = Q/(Vo/κ) = κ*(Q/Vo) = κ Co
C = κ Co = κ εoA/d = εA/d
where ε = κ εo = εr εo
Permittivity of Dielectric
ε = κ εo = εr εo
where κ = εr
Electric Polarisation / Dielectric Polarisation Density
-the electric field E inside a dielectric is related to the applied electric field Eo by:
E = Eo/εr where εr = ε/εo
-bound charges are polarised in an applied field and this sets up an electric field that opposes the applied field
-inside the material there is a dipole moment per unit volume, |P
|P = εo χe |E
-where χe is the electric susceptibility, χe = εr - 1
Electric Susceptibility
χe = εr - 1
Applied Magnetic Field Inducing Magnetisation
Summary
-an applied magnetic field can induce a magnetisation in a material and create an induced magnetic field tht tends to add to the applied field
Magnetic Fields in Matter
- inside matter there are molecules and/or atoms, these comprise nuclei with orbiting electrons:
- -some nuclei have intrinsic magnetic dipole moments
- -orbiting electrons give rise to magnetic dipole moments
- -an electron has an intrinsic magnetic dipole moment
- -can induce currents and so induce magnetic fields in the electron cloud / orbiting electrons
- these are all bound currents
Bound Currents
Definition
-bound currents are not free to take charge macroscopically from one place to another
Which three main categories to magnetic materials fall into?
- paramagnetic
- ferromagnetic
- diamagnetic
Paramagnetic
Definition and Examples
- there is partial alignment of the microscopic magnetic moments by an applied
- this takes place in the direction of the field
- the increase in the magnetic field inside the material is relatively small
- e.g. most chemical elements
Ferromagnetic
Definition and Examples
- strong interactions between the microscopic magnetic moments result in a high degree of alignment even in a weak external applied field
- this gives rise to a strong increase in the magnetic field in the material
- even when there is no field the microscopic magnetic moments can stay aligned giving a permanent magnet
- e.g. iron, iron alloys, nickel, cobalt, compounds of rare earth metals a few minerals
Diamagnetic
Definition and Examples
- there are induced dipole magnetic moments that oppose the applied field
- this is typically a small effect and usually masked by the effects of the permanent dipoles that give rise to paramagnetic and ferromagnetic effects
- e.g. water, wood, organic compounds, petroleum, some plastics, copper, mercury, gold and frogs
Magnetisation
-a material has a magnetisation |M defined by:
|B = |Bapp + μo|M
-where |B is the field inside the material and Bapp is the supplied field
Magnetisation in Paramagnetic and Diamagnetic Materials
-for paramagnetic and diamagnetic materials, |M is proportional to the applied field strength:
μo |M = χm |Bapp
-where χm is the magnetic susceptibility
-from this we can write:
|B = |Bapp + μo|M = |Bapp + χm |Bapp
|B = (1 + χm) |Bapp = μr |Bapp
-where μr is the relative permeability
Magnetic Susceptibility
μo |M = χm |Bapp
-where χm is the magnetic susceptibility (dimensionless)
Permeability of Material
μ = μr μo
μr = relative permeability μo = permeability of free space
Relative Permeability in Different Materials
- for most materials μr is very close to unity, within +/- 0.005%
- > paramagnets μr > 1
- > diamagnets μr < 1
- for ferromagnets, μr is not a constant an it can range from ~10^3 to ~10^5
- permanent magnets do not have a μr since they don’t need an applied field
- superconductors are perfect diamagnets, μr=0, you cannot induce a magnetic field in a superconductor
Polarisation in Matter
-consider a unit volume of the dielectric material
-the width of the material is L so the cross sectional area must be 1/L for the volume to me 1m³
-let the surface charge density on each side have size σb, the surface charge density of bound charge
-the magnitude of the charge q on each side is then given by:
q = σb*A = σb/L
-the dipole moment per unit volume is then:
|p = q |L = σb/L * |L where |L = L ^i so,
|p = σb/L * L ^i = σb ^i
-the dipole moment per unit volume is |p/V, and in this case V=1m³, so:
|P = σb ^i
Electric Displacement Field
D
|D = εo |E + |P
Gauss’s Law for Electric Fields Simplified for Use in Matter
Integral Form
∫ |D . ^n dA = Qf
-the displacement flux through a closed surface is equal to the total free charge enclosed by that surface
Gauss’s Law for Electric Fields Simplified for Use in Matter
Differential Form
|∇ . |D = ρf
Current Density in Matter
|J = |Jf + |Jb + |Jp
-total current density is made up of contributions from free charge, bound charge and polarisation
Magnetic Field Strength
H
|H = |B/μo - |M
Ampere-Maxwell Law Simplified for Use in Matter
Differential Form
|∇ x |H = |Jf + ∂|D/∂t
Ampere-Maxwell Law Simplified for Use in Matter
Integral Form
∮|H . d|L = If + d/dt ∫ |D.^n dA
Relationship Between |H and |B
|B = μ|H
-true when |M is proportional to |B
Relationship Between |D and |E
|D = ε|E
-true when |P is proportional to |E
Divergence of Polarisation in a Dielectric
|∇ . |P = - ρb
Curl of Magnetisation
|∇ x |M = |Jb
Electric Field in a Dielectric
|Ed = - |P/εo
