Maxwell's Equations in Matter

Question 1

When can Maxwell’s equations be used?

Answer 1

• 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

Question 2

What are the names of the four Maxwell’s Equations (microscopic)?

Answer 2

• Gauss’s Law for Electric Fields
• Gauss’s Law for Magnetic Fields
• Faraday’s Law
• Ampere-Maxwell Law

Question 3

Which of the four Maxwell’s Equations are simplified for use in matter?

Answer 3

• 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

Question 4

Dielectric

Definition

Answer 4

-a non-conducting material is a dielectric

Question 5

Charge Distribution in Atoms and Molecules

Answer 5

• 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

Question 6

Electric Dipole Moment

Answer 6

-the electric dipole moment, |p, for two charges -q and +q a vector |L apart is given by

|p = q|L

Question 7

Induced Electric Dipole Moment

Answer 7

-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

Question 8

Distortion Polarisation

Answer 8

• 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

Question 9

Orientation Polarisation

Answer 9

-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

Question 10

Dielectric Between the Plates of a Capacitor

Description

Answer 10

• 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

Question 11

Bound Charge

Answer 11

-charge is bound if it is not free to move and therefore cannot conduct a current

Question 12

Dielectric Constant

Answer 12

-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

Question 13

Dielectric in a Capacitor

V

Answer 13

-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

Question 14

Dielectric in a Capacitor

C

Answer 14

-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

Question 15

Permittivity of Dielectric

Answer 15

ε = κ εo = εr εo

where κ = εr

Question 16

Electric Polarisation / Dielectric Polarisation Density

Answer 16

-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

Question 17

Electric Susceptibility

Answer 17

χe = εr - 1

Question 18

Applied Magnetic Field Inducing Magnetisation

Summary

Answer 18

-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

Question 19

Magnetic Fields in Matter

Answer 19

• 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

Question 20

Bound Currents

Definition

Answer 20

-bound currents are not free to take charge macroscopically from one place to another

Question 21

Which three main categories to magnetic materials fall into?

Answer 21

• paramagnetic
• ferromagnetic
• diamagnetic

Question 22

Paramagnetic

Definition and Examples

Answer 22

• 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

Question 23

Ferromagnetic

Definition and Examples

Answer 23

• 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

Question 24

Diamagnetic

Definition and Examples

Answer 24

• 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

Question 25

Magnetisation

Answer 25

-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

Question 26

Magnetisation in Paramagnetic and Diamagnetic Materials

Answer 26

-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

Question 27

Magnetic Susceptibility

Answer 27

μo |M = χm |Bapp

-where χm is the magnetic susceptibility (dimensionless)

Question 28

Permeability of Material

Answer 28

μ = μr μo

μr = relative permeability
μo = permeability of free space

Question 29

Relative Permeability in Different Materials

Answer 29

• 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

Question 30

Polarisation in Matter

Answer 30

-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

Question 31

Electric Displacement Field

D

Answer 31

|D = εo |E + |P

Question 32

Gauss’s Law for Electric Fields Simplified for Use in Matter

Integral Form

Answer 32

∫ |D . ^n dA = Qf

-the displacement flux through a closed surface is equal to the total free charge enclosed by that surface

Question 33

Gauss’s Law for Electric Fields Simplified for Use in Matter

Differential Form

Answer 33

|∇ . |D = ρf

Question 34

Current Density in Matter

Answer 34

|J = |Jf + |Jb + |Jp

-total current density is made up of contributions from free charge, bound charge and polarisation

Question 35

Magnetic Field Strength

H

Answer 35

|H = |B/μo - |M

Question 36

Ampere-Maxwell Law Simplified for Use in Matter

Differential Form

Answer 36

|∇ x |H = |Jf + ∂|D/∂t

Question 37

Ampere-Maxwell Law Simplified for Use in Matter

Integral Form

Answer 37

∮|H . d|L = If + d/dt ∫ |D.^n dA

Question 38

Relationship Between |H and |B

Answer 38

|B = μ|H

-true when |M is proportional to |B

Question 39

Relationship Between |D and |E

Answer 39

|D = ε|E

-true when |P is proportional to |E

Question 40

Divergence of Polarisation in a Dielectric

Answer 40

|∇ . |P = - ρb

Question 41

Curl of Magnetisation

Answer 41

|∇ x |M = |Jb

Question 42

Electric Field in a Dielectric

Answer 42

|Ed = - |P/εo