NEXT MIDTERM EXAM
The second common hour exam will be held on Thursday NIGHT Nov 13, 9:5011:10 PM in 4 locations on 2 campuses. The assignment to exam room is based
on the first letters of your last name:
Allison Road Classroom 103 (Busch)
Physics Lecture Hall (Busch)
Lucy Stone Hall A102 Aud (Livingston)
Beck Hall 100 Aud (Livingston)
Aaa-Gzz
Haa-Lzz
Maa-Rnn
Roa-Zzz
There will be optional review sessions on Thursday, November 13; details will
be announced.
All exams will be no calculator, closed book exams with only 1 page of
equations. Types of questions include iclickers, only formulae, pure concepts,
simple numbers. For details go to
http://www.physics.rutgers.edu/ugrad/227/intro.html#Examinations.
If you have a conflict you have to contact Professor Cizewski
Cizewski@physics.rutgers.edu at your earliest opportunity but not later than
midnight on Sunday, November 9 to request a conflict exam.
1
Lecture 17. Magnetic Materials. Electromagnetic Induction.
Faraday’s Law
Outline:
Magnetic Materials.
Faraday’s Law.
Sign Convention: Lenz’s Law
The 3d Maxwell’s Equation (in
combination with the 4th one):
the basis for modern technology.
2
Dia-, Para-, and Ferromagnetics in Zero External B
Sources of local magnetic fields inside any material:
1. Magnetic moments associated with the electrons orbiting around nuclei.
2. The magnetic moments associated with spins of both electrons and nuclei.
1. Weak interactions between internal magnetic moments:
Diamagnets: No permanent magnetic moments (nonpolar
dielectrics.)
Paramagnets: Randomly oriented moments - the average
magnetic field is zero (polar dielectrics),
2. Strong interactions between internal moments: the moments
ordered in space.
Parallel ordering – ferromagnetics (iron, nickel, etc.), antiparallel
ordering – antiferromagnetics (chromium, etc.).
In ferromagnets, there is a strong local magnetic
field in the material. Large number of elementary
moments line up in the magnetic domains. The
global field depends on the orientation of
magnetic domains. In permanent magnets the
domains are preferentially oriented.
𝑩𝒆𝒆𝒆 = 𝟎
ferromagnetic
antiferromagnetic
3
Dia-, Para-, and Ferromagnetics in Non-zero External B
𝑩𝒆𝒆𝒆 ≠ 𝟎
𝐵 = 𝐵𝑒𝑒𝑒 + 𝜇0 𝑀
𝜒𝑚
𝐾𝑚 − 1
𝑀=
𝐵 =
𝐵𝑒𝑒𝑒
𝜇0 𝑒𝑒𝑒
𝜇0
𝐵 = 𝐵𝑒𝑒𝑒 + 𝜇0 𝑀 = 𝐾𝑚 𝐵𝑒𝑒𝑒
𝑀 - the magnetic moment per unit volume
𝐵 - the total field inside the material
𝜒 𝑚 - the magnetic susceptibility
𝐾𝑚 - the relative permeability
- “linear” magnetic materials
Paramagnetics and Diamagnetics:
In the external field, there is a (very small) preferential orientation of individual
moments (along the field in paramagnetics, against – in diamagnetics). The
“orientation” effect of 𝐵𝑒𝑒𝑒 is weak (thermal “disorientation” is much stronger), and, as
a result, 𝐾𝑚 is very close to unity, 𝐾𝑚 −1 = ± 10−5 − 10−4 .
Ferromagnetics:
External 𝐵 results in motion of domain walls in
such a way that the domains oriented along
the field grow while the domains oriented
against the field shrink. The global magnetic
field can be very strong, leading to 𝐾𝑚 up to
106 in weak external magnetic fields.
𝐵𝑒𝑒𝑒 = 0
4
Diamagnetic Levitation
𝐹=𝑚
𝑑𝐵𝑒𝑒𝑒
= 𝜌𝜌
𝑑𝑑
𝜒𝑚
𝑚=
𝐵
𝜇0 𝑒𝑒𝑒
Levitation condition (m – the induced-by-Bext magnetic
moment per unit volume, ρ - the substance density).
𝜒 𝑚 ≡ 𝐾𝑚 − 1 ≈ −10−5
for water
𝜒𝑚
𝑑𝐵𝑒𝑒𝑒
𝐵𝑒𝑒𝑒
= 𝜌𝜌
𝜇0
𝑑𝑑
𝑑𝐵𝑒𝑒𝑒 𝜌𝜌𝜇0 103 ∙ 10 ∙ 10−6
𝐵𝑒𝑒𝑒
=
≈
= 103 𝑇 2 /𝑚
−5
𝑑𝑑
𝜒𝑚
10
For graphite
𝜒 𝑚 ≈ −10−3
For superconductors
𝜒 𝑚 = −1 !
Magnetic levitation has no connection with
the electromagnetic induction!
5
Electromagnetic Induction: Experimental Observations
Reasonable guess: by moving a wire in a magnetic field, we can generate current.
Lorentz force 𝐹⃗ = 𝑄𝑣⃗ × 𝐵
Each set-up [(a) – (d)] produces an EMF:
(a) and (b) – EMF should be the same (only relative motion matters).
However, the interpretations are different.
(b) and (c) – EMF in one case
implies an EMF in the other: the
loop cannot tell whether the wire
is approaching or the current is
increasing.
current-carrying
wire (source of B)
wire
loop
(d) B is time-independent, but an
EMF is produced if the area of the
loop changes.
Faraday (and Henry) realized that all
these situations have one thing in
common: the flux of the magnetic
field through the loop changes with
time!
6
Faraday’s Law
𝑩
The e.m.f. is induced in a loop if the magnetic
flux changes, and is proportional to the rate of
change of the magnetic flux.
c
ℰ=−
ℰ = 𝐼𝐼
𝑑Φ𝐵
- the induced e.m.f.
𝑑𝑑
- if the self-inductance of
the loop can be neglected
Otherwise ℰ = 𝐼𝐼 + 𝐿
𝐼=−
Joseph Henry
1 𝑑Φ𝐵
𝑅 𝑑𝑑
𝑑𝑑
where L is the self-inductance.
𝑑𝑑
The flux can change for a variety of reasons:
a) B may be changing in time;
b) The loop may be altering in shape;
c) The orientation of the circuit with respect to B is changing;
d) Any combination of these effects.
7
Magnetically Induced Non-Conservative Electric Field
The magnetically induced e.m.f., similar to
the e.m.f. in a battery, is the result of nonelectrostatic (non-conservative*) forces:
The total electric field is the sum of
conservative and non-conservative terms:
ℰ = � 𝐸𝑁𝑁 ∙ 𝑑𝑙⃗ ≠ 0
𝑙𝑜𝑜𝑜
𝐸 = 𝐸𝐶 + 𝐸𝑁𝑁
conservative (due to charges at rest)
� 𝐸 ∙ 𝑑𝑙⃗ = � 𝐸𝐶 ∙ 𝑑𝑙⃗ + � 𝐸𝑁𝑁 ∙ 𝑑𝑙⃗ = � 𝐸𝑁𝑁 ∙ 𝑑𝑙⃗ = −
𝑙𝑜𝑜𝑜
𝑙𝑜𝑜𝑜
0
� 𝐸 ∙ 𝑑𝑙⃗ = −
𝑙𝑜𝑜𝑜
𝑑
𝑑𝑑
𝑙𝑜𝑜𝑜
�
𝑠𝑢𝑢𝑢𝑢𝑢𝑢
𝐵 ∙ 𝑑𝐴⃗
𝑙𝑜𝑜𝑜
ℰ=−
𝑑Φ𝑚
𝑑𝑑
𝑑Φ
𝑑
=−
𝑑𝑑
𝑑𝑑
�
𝑠𝑢𝑢𝑢𝑢𝑢𝑢
𝐵 ∙ 𝑑𝐴⃗
- the integral form of
Faraday’s Law.
Holds for any closed path, doesn’t require presence of the conducting loop
𝑑Φ
(the wire loop is just a “detector” of the electric field generated by ).
𝑑𝑑
Units. E[N/C]⋅dl [m] = (N ⋅ m=J) = J/C. Thus, the units of the e.m.f. are [J/C], which we
define as volt [V]. The right-hand side: [T⋅m2/s] = (T=N⋅s/(C⋅m)) = J/C.
8
* A conservative force = the work done is independent of the path
Sign Convention: Lenz’s Law
𝑑
� 𝐸 ∙ 𝑑𝑙⃗ = −
𝑑𝑑
𝑙𝑜𝑜𝑜
�
𝑠𝑢𝑢𝑢𝑢𝑢𝑢
𝐵 ∙ 𝑑𝐴⃗
Sign convention (~ Ampere’s Law): the
curled fingers along 𝑑𝑙⃗, the thumb in the
direction of 𝑑𝐴⃗.
It is often formulated in the form of Lenz’s Law:
An induced current has a direction such that the magnetic field due to the induced
current opposes the change in the magnetic flux that induces the current.
Let’s pick up some arbitrary direction of circulation around the loop (ccw in the figure):
𝑑Φ𝑚
ℰ=−
𝑑𝑑
“negative” (“positive”) means that 𝐸𝑁𝑁 is directed against (along) the initially
selected direction of circulation around the loop.
9
Multi-turn Coils
If we have a coil with N identical turns and the flux is the same through each turn, the
net e.m.f. will be N times the e.m.f. across a single turn:
𝑑Φ𝐵
ℰ = −𝑁
𝑑𝑑
Example: A 2-cm-radius loop of N=100 turns of wire lies in the plane of the paper. A
magnetic field of 0.15 T is introduced, pointing into the paper, in time 0.3 s. Find the
e.m.f. Does current tend to flow clockwise or counterclockwise?
ℰ=
−𝜋𝑟 2 𝑁
𝑑𝑑
= −3.14 × 0.02𝑚
𝑑𝑑
2
0.15𝑇
× 100 ×
= −0.063𝑉
0.3𝑠
counterclockwise
10
Experiment with a magnet falling inside a copper pipe
Please provide your explanation of the
experiment (for an extra credit ).
𝑩
𝑚
induced
currents
17
Next time: Lecture 18. Motional EMF
§§ 29.4-6
18