1. Maxwell’s equations continued
Lecture 2:
Maxwell’s equations continued
Electric fields in matter: Griffiths Chapter 4
Polarisation
A neutral atom is polarised by an electric field:
i.e. the electrons are shifted slightly one way and the nucleus the other. The atom has an induced dipole moment.
α (atomic polarisability) depends on the atom i.e. the element
We now concentrate on dielectrics i.e. non-conductors or insulators. We introduce P, the dipole moment per unit volume, or polarisation.
The effect of polarisation is to produce accumulations of bound charge: 10

2. So, the total charge density is now:
Gauss’s law becomes:
i.e.
We define the electric displacement:
So Gauss’s law becomes:
Question: Why no infinite loop? i.e. polarisation causes bound charge which causes electric field which causes polarisation which causes bound charge (at least 2 points to consider)
point 1 - matter is not infinitely polarisable
point 2 - polarisation tends to oppose the electric field 11

3. electric susceptibility (of the material) relative permittivity or
In many systems the polarisation is proportional to the electric field, provided that the electric field is not too strong.
where:
permittivity
(of the material)
electric susceptibility
(of the material)
relative permittivity or
dielectric constant
(of the material) 12

4. Capacitors and dielectrics: (example 4.6 Griffiths)
We fill a capacitor with a dielectric.
What happens?
So E decreases and thus V by a factor of 1/εr . So the capacitance is increased by a factor of the dielectric constant. 13

5. + - Why? Magnetostatics: Griffiths Chapter 5 (and 7)
+ -
Two current carrying wires attract or repel each other, depending on the direction of the current.
Why? 14

6. μ0 is the permeability of free space = 4π×10-7 N/A2
Reason 1: The Biot-Savart Law: Steady currents produce constant magnetic fields
μ0 is the permeability of free space = 4π×10-7 N/A2
dl is an element of length along the wire
Right-hand rule applies 15

7. So the total force due to electric and magnetic fields is:
Reason 2: The Lorentz force law: A magnetic field exerts a force on a moving charge
So the total force due to electric and magnetic fields is:
Right-hand rule applies
Particle accelerators use electric and magnetic fields to accelerate, bend and focus beams of charged particles.
Visit the tandem accelerator at iThemba 16

8. volume current density (current per unit area-perpendicular-to-flow)
Ampère’s law
volume current density
(current per unit area-perpendicular-to-flow)
permeability of free space = 4π×10-7 N/A2
which in integral form is:
Electric current generates a magnetic field
Ampère’s law follows from the Biot-Savart law: 17

9. Maxwell’s correction to Ampère’s law
Here we move from magnetostatics to magnetodynamics
We apply the divergence to Ampère’s law:
zero only for steady currents
always zero
Note: If we do the same with Faraday’s law everything is OK (see notes page 22)
take the del of the bottom equation
So we use the conservation of local charge (the continuity equation)
and Gauss’ law to get: 18

10. which in integral form is:
One of Maxwell’s equations - has no name
which in integral form is:
no magnetic monopoles!
compare with Gauss’s law - electric and magnetic charges are not equivalent
Integrate over any surface here and the sum of magnetic field lines will be zero.
Point to remember: magnetic forces do no work 19

11. Magnetic fields in matter: Griffiths Chapter 6
A magnetic field influences electron spins and orbits - diamagnetism, paramagnetism...
is the resultant magnetic dipole moment per unit volume.
The effect of magnetisation is to establish bound currents:
So, the total current is now:
Ampère’s law (before Maxwell) becomes:
i.e.
We define :
So Ampère’s law becomes:
and after Maxwell:
Magnetic fields in matter: Griffiths Chapter 6
check yourself 20

12. magnetic susceptibility (of the material) relative permeability
Similarly to polarisation, for magnetisation we have:
where:
and
permeability
(of the material)
magnetic susceptibility
(of the material)
relative permeability
(of the material)
Magnetic susceptibility is what is measured when one studies magnetic properties of materials. 21

13. Faraday’s law A B
Move a coil of wire in a magnetic field and a current flows A B
Move the magnetic field and a current flows in the coil A B
Vary the magnetic field and a current flows in the coil
A changing magnetic field induces an electric field
change the magnetic field 22

14. Mathematically Faraday’s law is:
which in integral form is:
In practical terms an electromotive force (emf) in volts is generated every time that the magnetic flux through the loop changes
This is the principle of most power generation:
(1) water is heated to form steam
(2) the steam moves a coil in a magnetic field or moves a magnet to generate electricity in the coil. 23

15. Maxwell’s equations in matter
So, finally we have:
Maxwell’s equations in matter
Gauss’s law (or Coulomb’s law)
no name (or absence of magnetic poles)
Faraday’s law
Ampère’s law with Maxwell’s correction
Remark on Maxwell’s equations: the principle of superposition applies 24

16. Examples to know: (from Griffiths 3rd edition)
examples: 4.2, 4.3, 4.4, 4.5, 4.6 (done in class)
5.1, 5.2, 5.3, 5.4, 5.5, 5.6, 5.7, 5.9, 5.10
6.1, 6.3
7.1, 7.2, 7.5, 7.6, 7.7
Know the derivation of Ampère’s law (pgs )
have a to study section (as in lecture 3) 25