class 28: faraday’s law and motion emf
TRANSCRIPT
Class 28: Faraday’s Law and Motion EMF
Maxwell’s 1st equation: (Gauss’s Law foe electric field)
in0 q AdE
Maxwell’s 2nd equation: (Gauss’s Law for magnetic field)
0 AdB
Maxwell’s 3rd equation: (Ampere’s Law ‐ incomplete)
in0I sdB
Maxwell’s equations describe only the fields, it does not include the effect of the field on charges or currents:
)BsId Fd(or Bvq F
Eq F
BB
E
Not included in Maxwell’s equations
Summary of Maxwell’s Equations so far
Magnetic Field from Electrons in an Atom I
A charge particle possessing angular momentum will give rise to magnetic moment. Magnetic moment is a vector in the same direction (for positive charge) as the angular momentum.
Area A
I
L
IA
2
r ve r r 2
ve A I
r vm L
2
L
2me
Magnetic moment is a result of the circular motion of the electrons, which can be described by the angular momentum.
For electron
Quantization of angular momentum, spin, and magnetic moment
Spin of electrons can contribute to magnetic moment too.
Sme
s
For electron
Quantization of spin and angular momentum:
L and S are quantized in units of ħ. ħ = 1.05 10-34 Js.
As a result, magnetic momentum is also quantized, in units of Bohr magnetron
1-24-b JT 109.27
2me
Paramagetism and Diagmagnetism
Paramagnetism:If an atom has a net magnetic moment, it will display parammagnetic property – the net moment will align with external field and create a slightly stronger total field. This alignment process has to compete with the thermal effect, which tends to randomize the alignment.
Diagmagnetism:If an atom has no net magnetic moment (e.g. inert gas like He, Ar, Ne etc.), it will display diamagnetic property – an external field will induce a magnetic moment in opposite to the field direction and produce a slightly weaker field. This is a result of the Faraday’s Law, which we will study in the next two chapters.
In general, the diamagnetic effect is much weaker than the paramagnetic effect.
Ferromagnetism
The individual magnetic moments in some paramagnetic materials will align together to form domains when the temperature is below a critical temperature called Curie Temperature.
When an external field is applied, the domains with moment in the field direction will grow in size. The original configuration may not be recovered after the field is removed an a permanent magnet is formed.
Structure of Equations
E B
Interaction between charges Interaction between moving charges/ currents
Coulomb’s Law Biot‐Savart Law
Gauss’s Law Gauss’s Law (Conceptual)
Gauss’s Law Ampere’s Law (Calculation)
Parallel capacitor gives uniform E field
Solenoid gives uniform B field
r̂rdq
41 Ed 2
0
rr̂sd
4I
Bd 20
in0 q AdE
0 AdB
in0 q AdE
in0 I sdB
nI B 0
0
E
E
B
Imaginary loop in an electric and magnetic field
We will do two types of integrals for the closed loop:
1.Magnetic flux
Note that B0 (Maxwell’s 2nd equation) because this is not a 3 dimensional closed surface.
2.Electromotive force (emf, )
loop = 0 for electrostatic case. Note that loop = 0 does not mean E =0.
AdB B
sdE loop
loop
Example
What is the magnetic flux through the rectangular loop?
I
d
a
b
E
B
Faraday’s Law
AdB dtdsdEor
dtd
loop
Bloop
Notes:1.loop does not equal to 0 any more if dB/dt 02.There are two ways to make dB/dt 0:
(i) Changing B(ii) Changing A (loop shape)
E
B
Faraday’s Law
AdB dtdsdEor
dtd
loop
Bloop
Notes:1.loop does not equal to 0 any more if dB/dt 02.There are two ways to make dB/dt 0:
(i) Changing B(ii) Changing A (loop shape)
Electric Potential V
A
B
If E(r) is conservative, the potential difference V is defined as the negative work done by the force F(r) (which is path independent), divided by the charge (of the test charge).
1`1`
1`
rd)r(F- U f
i
Pay attention to the negative sign
rd)r(E- q UV
f
i
Unit of electric potential = J/C =V
Old slide from class 8
V=0 for closed loop
Warning
In the discussion here we will assume electric (force) field is a conservative (force) field. This will not be the case if there is a changing magnetic field. We will come to this point later in the semester.
Old slide from class 8
Faraday’s Law for changing magnetic field: Example I
AdB dtdsdEor
dtd
loop
Bloop
Notes:1.loop does not equal to 0 any more if dB/dt 02.There are two ways to make dB/dt 0:
(i) Changing B(ii) Changing A (loop shape)