Electric and Magnetic Fields
Copyright © by L.R.Linares, 2007, 2017, 2020
All rights reserved
Copyright© by L.R.Linares 2007, 2017
• In this Video …
• Concepts sequencing and storyboards
• L.R.Linares
• Scripts and Slide Production prototypes
• L.R.Linares
• Animation
• L.R.Linares and J.C.Linares
• Voice
• L.R.Linares
• Shooting, editing, postproduction and rendering
• L.R.Linares and J.C.Linares
Copyright© by L.R.Linares
Goals and Expectations
• Know the meaning, symbol and units for
•
•
•
•
Magnetic induction (flux density)
Magnetic field
Magnetic potential
Magnetic flux
• Know the parallelism between electric circuits and magnetic circuits
(cause effect chain for each)
Copyright© by L.R.Linares
(…)
• Know the concept of magnetomotive force and how to compute it.
• Numerical relationships between different units of magnetic flux, and
magnetic flux density.
• Know by heart the charge and mass of the electron, magnetic
permeability of air.
Copyright© by L.R.Linares
Foundations …
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Maxwell’s Equations
• Everything in EE spins around
Maxwell’s Equations
• Everything in EE spins around
Maxwell’s Equations
• Everything in EE spins around
Lorentz’s Forces
F  qE qv  B
So it begins …
Copyright© by L.R.Linares
Currents
create magnetic flux
Who creates
what?
f (Mx)
(lines)
(Wb)
(weber)
(kMx)
(kilolines)
i(A)
i(A)
1Wb  100 MMx=10 lines
8
Copyright© by L.R.Linares
Magnetic flux
• In the “old days” physicists used to measure the magnetic flux in…
• … lines.
• They would say f=3700 lines
• Or perhaps f=3.7 kilo lines
• That unit, the line, is now called maxwell
• So we would say f=3.7kMx
Copyright© by L.R.Linares
Magnetic flux (…cont…)
• The modern SI unit for magnetic flux is the weber, a huge unit:
• One weber = a hundred million lines.
• So we end having fluxes of some milliwebers, or even microwebers:
f  3.7 kMx
f  37 Wb
Copyright© by L.R.Linares
How closely packed the flux…
Some
times
is NOT
called
the magnetic
Magnetic
field
Induction
B
is aBconsequence
of the magnetic
field
• Very often, we are interested, not in how much flux we have …
• … but in how closely packed together those lines are …
• … the magnetic flux density … B
• …B is measured in …
•
Wb/m2 = T (tesla)
Mx/cm2 = G (gauss)
Cross section of the flux
Copyright© by L.R.Linares
Top Hat L2.a
• If one weber is a hundred million lines (maxwells), a gauss
is a density of one line per square cm, and a tesla is a
density of one weber per square meter, how many gauss
are equivalent to a tesla of flux density?
Copyright© by L.R.Linares
Summary so far
• Currents, amps, create flux f in webers.
• Flux density, B, is measured in teslas, T.
• B is not the magnetic field.
• B is a consequence of the magnetic field.
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Our good Old Friend
The Electric Field
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GUY MOVING CHARGE
ELECTRIC CHARGE (C) )
ELECTRIC FIELD
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Price Tag
• Everything in this universe has a price tag (with one or two
lonely exceptions)
• So … to “move” electric charge against an electric field …
we have to pay …
• Pay? Pay what? And to whom?
• Well, we pay Nature, if you will … through her proxy … the
Electric Field …
• We pay the only “currency” she accepts:
ENERGY … iN joulEs
Copyright© by L.R.Linares
In an Electric Field
ELECTRIC POTENTIAL
+q
J
V
C
J /C
E  30
m
V
E  30
m
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Copyright© by L.R.Linares
Units of the Electric Field
V N

m C
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?
Units of the Electric Field
V J / C N .m / C N



m
m
m
C
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Volt = electric potential
• To move one positive coulomb from A to B
• … we spend 37 joules
• We say that the difference of electric potential
between A and B is 37 J/C
• Or we simply say the “voltage” between A and B is 37 volts.
• Or that B is 37 volts higher than A
• Do remember that J/C = V
Copyright© by L.R.Linares
From a source…
• A “battery” of 100V, applied between
two points separated by 5 metres …
• … creates an electric field between those points:
100 V
V
N
E
 20  20
5m
m
C
• We call batteries and generators, electromotive forces, emf,
measured in volts.
Copyright© by L.R.Linares
Summary so far
• Currents, amps, create flux f in webers.
• B is flux density, in Wb/m2, in teslas.
• B is not the magnetic field.
• B is a consequence of the magnetic field.
• Electric fields, E in V/m …
• … are created by …
• … electromotive forces, emf, in volts
Copyright© by L.R.Linares
Our good New Friend
The Magnetic Field
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In a Magnetic Field
+UMC
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Copyright© by L.R.Linares
Copyright© by L.R.Linares
In a Magnetic Field
MAGNETIC POTENTIAL
+UMC
H  30
J / UMC
H  30
m
Vmag
MAGNETIC
VOLT, Vmag
m
Copyright© by L.R.Linares
Copyright© by L.R.Linares
In a Magnetic Field
MAGNETIC POTENTIAL
J / cooper
H  30
m
+cooper
J
 Vmag
cooper
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J
A
cooper
Two jobs for the ampere
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By day, the ampere is
the unit of currents!
Copyright© by L.R.Linares
By night, the ampere is
the unit for magnetic potential!
… a magnetic “volt” of sorts!
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So … what’s ampere?
• Ampere is the unit for electric current
C
A
s
• Ampere is also the unit for magnetic potential
JJ
A
cooper
UMC
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Units of E
V
E 30
m
Copyright© by L.R.Linares
Units of H
A
H 30
m
Copyright© by L.R.Linares
Every effect has a cause!
Chain of cause effect for…
S
f (Wb)   B  d S
Copyright© by L.R.Linares
S

emf (V )  E  J   E  I   J  d S

mmf ( A)  H  B   H  f   B  d S
So … we apply volts (emf), and we get current?
So … we apply amps (mmf), and we get flux?
Copyright© by L.R.Linares
V    A 
S
emf (V )  E    J  2    E 
 I ( A)   J  d S
m
m 
S
Unit of electric
potential
S
 A    Wb 
mmf ( A)  H    B  2    H  f (Wb)   B  d S
m
m 
S
Copyright© by L.R.Linares
V    A 
S
emf (V )  E    J  2    E 
 I ( A)   J  d S
m
m 
S
Unit of magnetic
potential
S
 A    Wb 
mmf ( A)  H    B  2    H  f (Wb)   B  d S
m
m 
S
Copyright© by L.R.Linares
V    A 
S
emf (V )  E    J  2    E 
 I ( A)   J  d S
m
m 
S
We get amps
S
 A    Wb 
mmf ( A)  H    B  2    H  f (Wb)   B  d S
m
m 
S
Copyright© by L.R.Linares
V    A 
S
emf (V )  E    J  2    E 
 I ( A)   J  d S
m
m 
S
We get webers
S
 A    Wb 
mmf ( A)  H    B  2    H  f (Wb)   B  d S
m
m 
S
Copyright© by L.R.Linares
OHM’S
LAW
V    A 
S
emf (V )  E    J  2    E 
 I ( A)   J  d S
m
m 
S
OHM’S
LAW
S
 A    Wb 
mmf ( A)  H    B  2    H  f (Wb)   B  d S
m
m 
S
Copyright© by L.R.Linares
Magnetomotive Force
MMF … in amps!
Copyright© by L.R.Linares
Hmm… what is an MMF?
• The (CW) MMF applied to a (any) closed path is computed as follows:
• Imagine a surface limited by that path
• Add all the amps that (from our perspective) enter into the surface …
• … subtract all the amps that (from our perspective) come out of the
surface …
• The resulting amps is the total CW MMF applied to that closed path!
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What is the MMF (CW)?
• … on this path
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Imagine a surface limited
• … by the path
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Add current entering …
• … the surface
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subtract currents leaving …
• … the surface
MMFcw 725A
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What’s the CW emf in the left loop?
+
_ 5V
12V
+
_
2V
_+
emf=5  12  2  15 V
Copyright© by L.R.Linares
+
_ 7V
What’s the CW MMF in the left window?
I1
N
1
N
3
I2
I
3
N
5
N2
I
4
N
4
Copyright© by L.R.Linares
I5
What’s the MMF in the left window?
I1
N
1
N
3
I2
I
3
N
5
N2
I
4
N
4
Copyright© by L.R.Linares
I5
What’s the MMFCW in the left window?
MMFaCW  N1 I1  N 2 I 2  N 4 I 4  N3 I 3
I1
NN11
I3
NN33
I2
I5
N
5
N
N22
I
4
N
4
MMFCW 
Copyright© by L.R.Linares

entering
surface
i
i
leaving
surface
Top Hat (L2.1, L2.2)
What is the total clockwise magnetomotive force applied to the right window.
I1
N
1
I2
N
3
I
3
I5
N
5
N2
I
4
N
4
N1 500
I1 3 A
N 2 700
I 2 5 A
N 3 1000 I 3 7 A
N 4 50
I 4  200 A
N 5 750
I 5  2.7 A
What’s the MMFCW in the left window?
MMFaCW  N1 I1  N 2 I 2  N 4 I 4  N3 I 3
I1
NN11
I3
NN33
I2
N
5
N
I5
N22
I
4
N
4
MMFbCW  N 5 I 5  N3 I 3
Copyright© by L.R.Linares
What’s the MMF in the left window?
I a  N1I1  N 2 I 2  N 4 I 4  N3 I 3
What is the CW-MMF
applied to the left
window? … to the
right window?
Ia
Ib
I b  N5 I 5  N3 I 3
Copyright© by L.R.Linares
Magnetic Potential Drop!
(For Wednesday, 2023 Sep 21)
Copyright© by L.R.Linares
Top Hat.3
• If the electric field is constant, 7.5 V/m, in a segment of an
electric circuit of 3.2 metres long, what is the total voltage drop
from one end (A) to the other (B) of that segment, in volts?
B
A
Copyright© by L.R.Linares
Top Hat.4
• If the electric field is constant, 7.5 V/m, in the segment of an
electric circuit shown, what is the total current, in amps?
B
A
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Top Hat.5
• If the magnetic field is constant, 9.2 A/m, on a segment of an
magnetic circuit of 0.25 metres long, what is the total magnetic
potential drop from one end to the other of that segment, in
amps?
B
A
Copyright© by L.R.Linares
Top Hat.6
• If the magnetic field is constant, 9.2 A/m, on a segment of an
magnetic circuit shown, what is the flux from A to B, in Wb?
B
A
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Top Hat.7
• If the magnetic field is constant, 200 A/m, on a segment of an
magnetic circuit shown, what is the flux from A to B, in Wb?
B
A
Copyright© by L.R.Linares
Voltage drop … Amps drop?
V  El
U  H l
Copyright© by L.R.Linares
Electric Potential Drops
• If that electric field is constant along a segment of the circuit of length
l(m)…
• … the electric potential drop along that segment is computed simply
as
V  E.l
Copyright© by L.R.Linares
KVL
• In any loop, the sum of emf, equates the voltage drops
n
E l
 emf  

1
Copyright© by L.R.Linares
Magnetic Potential Drops
• If that magnetic field is constant along a segment of the circuit of
length l(m)…
• … the magnetic potential drop (amps) along that segment is
computed simply as
U  H .l
Copyright© by L.R.Linares
The KVL of magnetic circuits: Ampere’s Law
• In any window, the sum of mmf, equates the magnetic potential drops
n
H l
 mmf  

1
m
n
H l
N I 

k 1
k k
1
A window with ‘m’ coils, each one with Nk turns, and a current Ik
It has ‘n’ segments with constant H.
Copyright© by L.R.Linares
KCL of magnetic circuits
• In electric circuits, we relied on three laws to solve them: Ohm’s, KVL,
and KCL.
• In magnetic circuits: for any Gauss surface, the sum of all the
In any
Gauss
surface,
magneticKCL:
flux going
into the
surface,
equals thethe
sumsum
of all of
the
magnetic flux
out of the
surface.
all coming
the current
going
into the surface
equates the sum of all the currents
coming out of the surface
 B  0
i f  
 fi
in in
Gauss
Gauss
out
out
Gauss
Gauss
Copyright© by L.R.Linares
What about Ohm’s Law?
reluctance or magnetic resistance
• If we can assume that “mu” is constant
• … across a segment of the magnetic circuit with a given geometry…
• … we can approximate the magnetic potential drop (amps) across the
segment as proportional to the flux (webers) through the segment
U  f
amps
Copyright© by L.R.Linares
webers
Magnetic Resistance?
• If we want a flux f webers from A to B … what is the potential drop from A to
B, in amps?
B
f
A
f
H
A
f
l
U 
l
f  Rf
A A
RELUCTANCE
Copyright© by L.R.Linares
A word about units
• The first unit for E that we learned came from Coulomb’s law, about
the force that an electric field applies a charge q
F  qE
• So the unit for E were given as N/C
• Yes, force per unit of charge.
• Now we have seen it can also be V/m
• But, H is given in A/m (mag. Pot. Per metre)
Copyright© by L.R.Linares
A word about H units
• Originally, physicists in Europe used the cgs system
(not the metric, SI, system).
• In cgs, the force unit was the dyne (10-5N)
• The unit of magnetic charge was the “unit-pole”, so
the magnetic field H was given in
dyne
1 Oe  1
unitpole
• 1 oersted equals (1000/4p) A/m
Copyright© by L.R.Linares
Unit pole
• The concept of the unit pole as an integral part of the presentation of
magnetic fields was present in Physics’ books until the 1930s, and
even later for some authors.
Copyright© by L.R.Linares
Flux Density, B
• What creates the flux density is H, the magnetic field.
• The same way that what creates
the current density J is E, the electric field.
• B and H are related, but are not the same.
• B depends on H, but also on the magnetic
permeability of the material or medium.
Copyright© by L.R.Linares
THE END
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Caboose
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Copyright© by L.R.Linares
Copyright© by L.R.Linares
So … the number of joules per metre and per
kilogram isGravitational
a measure
Field
of how strong the gravitational field is
The eagle
is pulling up
the fish!
J / kg
G  30
m
For every metre
the eagle raises 1kg
of fish, the eagle
spends “so many”
joules.
On Jupiter it would
Need to spend more
Joules per metre per
Kilogram!
AGAINST
THE FIELD!
Copyright© by L.R.Linares
Units of the Electric Field
J /C N

m
C
Copyright© by L.R.Linares
B, Wb/m2
H, A/m
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Ridiculosity is forever!
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Ridiculosity is forever!
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B, flux density!
LOW B
hairs
2
cm
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B, flux density!
HIGH B
hairs
2
cm
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Example, a cathode ray tube
Copyright© by L.R.Linares
10000 V
V
N
E
 100, 000  100, 000
0.1 m
m
C
The force on an electron:
- 10,000V +
- - 10cm
F  qE
N
 1.6022 10 C 100, 000
C
14
19
 1.602 10
N
The acceleration on an electron:
F 1.602 1014 N
15 m
F  ma  a  
 17.6 10 2
31
m 9.1110 kg
s
The time to travel the cannon:
2
at
d  vot 
Yes, 302, not 289, not 2the standard!
m
17.6
10015to
t 2 100km/h in 6s)
aMustang  4.63 2 (3020.1
cu.in.
V8

 t  3.37ns
s
2
Copyright© by L.R.Linares
TOP HAT FLD.1
What is the time it takes an electron to
travel between the plate on the left and
the plate on the right, in nano seconds?
- 10,000V +
- - 10cm
Copyright© by L.R.Linares
In a Magnetic Field
MAGNETIC POTENTIAL
J / UMC
H  30
m
+UMC
J
A
UMC
A
H  30
m
Copyright© by L.R.Linares

emf (V )  E (V / m)  J ( A / m )   E
2
S
 I ( A)   J  d S
Every effect has a cause!
S
Unit of electric
Chain
potential
of cause
effect for…
To current…
From voltage…

mmf ( A)  H ( A / m)  B(Wb / m )   H
2
S
f (Wb)   B  d S
Copyright© by L.R.Linares
S
In a Magnetic Field
MAGNETIC POTENTIAL
J / cooper
H  30
m
+cooper
J
 Vmag
cooper
Copyright© by L.R.Linares
A
H  30
m