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Physical therapy equipment
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Plan
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Magnetic Field
Flux Density
Magnetic Force on Moving Charge
Direction of Magnetic Force
Electromagnetic Fields
Biot-Savart Law
Electrophoresis
Electromagnetic Waves
Poynting Vector
Intensity of EM Wave
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Magnetism
Since ancient times, certain materials, called
magnets, have been known to have the property of
attracting tiny pieces of metal. This attractive
property is called magnetism.
S
Bar Magnet
S
N
N
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Magnetic Poles
Iron
filings
N
The strength of a magnet is
concentrated at the ends,
called north and south
“poles” of the magnet.
S
A suspended magnet:
N-seeking end and
S-seeking end are N
and S poles.
W
N
S
N
Bar magnet
S
N
E
Compass
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Magnetic Field Lines
We can describe
magnetic field lines by
imagining a tiny compass
placed at nearby points.
The direction of the
magnetic field B at any
point is the same as the
direction indicated by
this compass.
N
S
Field B is strong where
lines are dense and weak
where lines are sparse.
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Magnetic Field
A bar magnet has a magnetic
field around it. This field is 3D in
nature and often represented by lines
LEAVING north and ENTERING
south
To define a magnetic field you
need to understand the MAGNITUDE
and DIRECTION
We sometimes call the magnetic
field a B-Field as the letter “B” is the
SYMBOL for a magnetic field with the
TESLA (T) as the unit.
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Facts about Magnetism
Magnets have 2
poles (north and
south)
 Like poles repel
 Unlike poles attract
 Magnets create a
MAGNETIC FIELD
around them

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Magnetic Flux Density
• Magnetic flux lines are
continuous and closed.
• Direction is that of the B
vector at any point.
• Flux lines are NOT in
direction of force but ^.
When area A is
perpendicular to flux:

B
A
DA
Df
Magnetic Flux
density:

B  ;  = BA
A
The unit of flux density is the Weber per square meter.
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Calculating Flux Density When
Area is Not Perpendicular
The flux penetrating the
area A when the normal
vector n makes an angle of
q with the B-field is:
  BA cosq
n
A
q
a
B
The angle q is the complement of the angle a that the
plane of the area makes with the B field. (Cos q = Sin a)
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Origin of Magnetic Fields
Recall that the strength of an electric field E was
defined as the electric force per unit charge.
Since no isolated magnetic pole has ever been
found, we can’t define the magnetic field B in
terms of the magnetic force per unit north pole.
We will see instead that
magnetic fields result from
charges in motion—not from
stationary charge or poles.
This fact will be covered later.
E
+
+
v
B^v
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Magnetic Force on Moving Charge
Imagine a tube that
projects charge +q
with velocity v into
perpendicular B field.
F
B
v
N
S
Experiment shows:
F  qvB
Upward magnetic force F
on charge moving in B field.
Each of the following results in a greater magnetic
force F: an increase in velocity v, an increase in charge
q, and a larger magnetic field B.
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Magnetic Force on a moving charge
B
S
N
S
vo
-



FB  qv  B
FB  qvB sin q
N
If a MOVING CHARGE
moves into a magnetic field
it will experience a
MAGNETIC FORCE. This
deflection is 3D in nature.
The conditions for the force are:
•Must have a magnetic field present
•Charge must be moving
•Charge must be positive or negative
•Charge must be moving
PERPENDICULAR to the field.
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Direction of Magnetic Force
The right hand rule:
With a flat right hand, point F B
thumb in direction of
v
velocity v, fingers in
direction of B field. The
N
flat hand pushes in the
direction of force F.
F
B
v
S
The force is greatest when the velocity v is
perpendicular to the B field. The deflection
decreases to zero for parallel motion.
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Force and Angle of Path
N
N
N
S
S
S
Deflection force greatest
when path perpendicular
to field. Least at parallel.
F  v sinq
F
v sin q
q
v
B
v
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Definition of B-field
Experimental observations show the following:
F  qv sin q
or
F
 constant
qv sin q
By choosing appropriate units for the constant of
proportionality, we can now define the B-field as:
Magnetic Field
Intensity B:
F
B
qv sin q
or
F  qvB sin q
A magnetic field intensity of one tesla (T) exists
in a region of space where a charge of one
coulomb (C) moving at 1 m/s perpendicular to the
B-field will experience a force of one newton (N).
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Example 1. A 2-nC charge is projected with velocity 5 x 104
m/s at an angle of 300 with a 3 mT magnetic field as
shown. What are the magnitude and direction of the
resulting force?
Draw a rough sketch.
q = 2 x 10-9 C
v = 5 x 104 m/s
B = 3 x 10-3 T
q = 300
F
B
v sin f
300
B
v
v
Using right-hand rule,-9the force is4 seen to be -3upward.
F  qvB sin q  (2 x 10 C)(5 x 10 m/s)(3 x 10 T)sin 300
Resultant Magnetic Force: F = 1.50 x 10-7 N, upward
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Forces on Negative Charges
Forces on negative charges are opposite to those on
positive charges. The force on the negative charge
requires a left-hand rule to show downward force F.
Right-hand
rule for
positive q
N
F
B
v
S
Left-hand
rule for
negative q
N
B
F
v
S
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Indicating Direction of B-fields
One way of indicating the directions of fields perpendicular to a plane is to use crosses X and dots  :
A field directed into the paper
is denoted by a cross “X” like
the tail feathers of an arrow.
 
 
 
 








X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
A field directed out of the paper
is denoted by a dot “” like the
front tip end of an arrow.
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Practice With Directions:
What is the direction of the force F on the charge in
each of the examples described below?
X
X
X
X
F
X X X Up
X+ X v X
X X X
X X X
F
   
Up
v
   
  -  
negative q
   
X
X
Left
X
X
v
X X
FX X
+
X X
X X
X
X
X
X
   
  F 
Right
   
v
   
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Crossed E and B Fields
The motion of charged particles, such as electrons, can
be controlled by combined electric and magnetic fields.
Note: FE on electron
is upward and
opposite E-field.
+
But, FB on electron is
down (left-hand rule).
Zero deflection
when FB = FE
e-
x x x x
x x x x
v
-
FE
E
--
e
B
v
B
FB
v
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The Velocity Selector
This device uses crossed fields to select only those
velocities for which FB = FE. (Verify directions for +q)
When FB = FE :
qvB  qE
E
v
B
Source
of +q
+
x x x x
x x x x
+q
v
-
Velocity selector
By adjusting the E and/or B-fields, a person can
select only those ions with the desired velocity.
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Example 2. A lithium ion, q = +1.6 x 10-16 C, is projected
through a velocity selector where B = 20 mT. The E-field
is adjusted to select a velocity of 1.5 x 106 m/s. What is
the electric field E?
E
v
B
E = vB
Source
of +q
+
x x x x
x x x x
+q
v
V
E = (1.5 x 106 m/s)(20 x 10-3 T);
E = 3.00 x 104 V/m
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Circular Motion in B-field
The magnetic force F on a moving charge is always
perpendicular to its velocity v. Thus, a charge moving
in a B-field will experience a centripetal force.
Centripetal Fc = FB
2
mv
FC 
; FB  qvB;
R
The radius of
path is:
mv
R
qB
+
X
X +X
X
X
X
X
X
X FX
X
X
X
X
X
X
+
X
X X
RX X
c
X
+
X
X
X
X
X
X
X
X
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Mass Spectrometer
+q
-
slit
E
v
+ B
xx
Photographic
xx
plate
xx
R
xx
x x x x x x x x x
x x x x x x x x
x x x x x x x
m2
x x x x x x
x x x x
mv 2
 qvB
R
m1
Ions passed through a
velocity selector at
known velocity emerge
into a magnetic field as
shown. The radius is:
mv
R
qB
The mass is found by
measuring the radius R:
qBR
m
v
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Example 3. A Neon ion, q = 1.6 x 10-19 C, follows a
path of radius 7.28 cm. Upper and lower B = 0.5 T
and E = 1000 V/m. What is its mass?
+q
E
v
+ B
xx
Photographic
- xx
plate
xx
R
xx
x x x x x x x
slit x x x x x x x
x x x x x x x
m
x x x x x x
x x x x
E 1000 V/m
v 
B
0.5 T
v = 2000 m/s
mv
R
qB
(1.6 x 10-19C)(0.5 T)(0.0728 m)
m
2000 m/s
qBR
m
v
m = 2.91 x 10-24 kg
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Summary
The direction of forces on a charge moving in an electric
field can be determined by the right-hand rule for positive
charges and by the left-hand rule for negative charges.
Right-hand
rule for
positive q
N
F
B
v
S
Left-hand
rule for
negative q
N
B
F
v
S
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Summary (Continued)
F
v sin q
q
For a charge moving in a
B-field, the magnitude of
the force is given by:
v
B
v
F = qvB sin q
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Summary (Continued)
The velocity
selector:
E
v
B
The mass
spectrometer:
mv
R
qB
qBR
m
v
+
+
vq
x x x
x x x
x x
V
+q
-
xx
- xx +
xx
xx
x x x
slit x x x
x x x
x x
E
v
B
R
x
x
x
x
x
x
x
x
x x
x x
x x
x m
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Electromagnetic Fields


Electromagnetics is the study of the effect of
charges at rest and charges in motion.
Some special cases of electromagnetics:
 Electrostatics:
charges at rest
 Magnetostatics: charges in steady motion (DC)
 Electromagnetic waves: waves excited by charges
in time-varying motion
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Electromagnetic Fields

A scalar is a quantity having only an amplitude
(and possibly phase).
Examples: voltage, current, charge, energy, temperature

A vector is a quantity having direction in addition
to amplitude (and possibly phase).
Examples: velocity, acceleration, force
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Electromagnetic Fields

Fundamental vector field quantities in
electromagnetics:
E )

Electric field intensity
units = volts per meter (V/m = kg m/A/s3)

D )
Electric flux density (electric displacement)
units = coulombs per square meter (C/m2 = A s /m2)

Magnetic field intensity

units = amps per meter (A/m)
Magnetic flux density
H )
B )
units = teslas = webers per square meter (T =
Wb/ m2 = kg/A/s3)
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
Universal constants in electromagnetics:
 Velocity
of an electromagnetic wave (e.g., light) in
free space (perfect vacuum)
c  3  108 m/s
 Permeability
 Permittivity
 Intrinsic
of free space
of free space:
0  4  10 H/m
7
 0  8.854  10 12 F/m
impedance of free space:
 0  120 
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Electromagnetic Fields

Relationships involving the universal
constants:
c
1
 0 0
0
0 
0
In free space:
B  0 H
D  0 E
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Electromagnetic Fields in
Materials

In a simple medium, we have:
D E
B  H
• linear (independent of field strength)
• isotropic (independent of position
within the medium)
• homogeneous (independent of
direction)
• time-invariant (independent of time)
• non-dispersive (independent of
frequency)
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Electromagnetic Fields in Materials




 = permittivity = r0 (F/m)
 = permeability = r0 (H/m)
s = electric conductivity = r0 (S/m)
sm = magnetic conductivity = r0 (/m)
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Biot-Savart Law



The Biot-Savart Law relates magnetic fields to
the currents which are their sources. In a similar
manner, Coulomb's law relates electric fields to
the point charges which are their sources.
Finding the magnetic field resulting from a
current distribution involves the vector product,
and is inherently a calculus problem when the
distance from the current to the field point is
continuously changing.
See the magnetic field sketched for the straight
wire to see the geometry of the magnetic field
of a current.
Magnetic field of currerent element
 0 Id L  r
dB 
2
4r
where dL- infinitesmal length of conductor carrying electric current
I, r - unit vector to specify direction of the vector distance r from the
current to the field point.
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
Electrophoresis is the motion of dispersed
particles relative to a fluid under the influence
of a spatially uniform electric field. This
electrokinetic phenomenon was observed for
the first time in 1807 by Reuss, who noticed
that the application of a constant electric field
caused clay particles dispersed in water to
migrate. It is ultimately caused by the presence
of a charged interface between the particle
surface and the surrounding fluid.
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The dispersed particles have an
electric surface charge, on which
an external electric field exerts an
electrostatic Coulomb force.
According to the double layer
theory, all surface charges in
fluids are screened by a diffuse
layer of ions, which has the same
absolute charge but opposite sign
with respect to that of the surface
charge.
The electric field also exerts a force on the ions in the diffuse
layer which has direction opposite to that acting on the surface
charge. This latter force is not actually applied to the particle,
but to the ions in the diffuse layer located at some distance
from the particle surface, and part of it is transferred all the
way to the particle surface through viscous stress. This part38of
the force is also called electrophoretic retardation force.

Electromagnetic Waves





It consists of mutually perpendicular and oscillating electric and
magnetic fields. The fields always vary sinusoidally. Moreover, the
fields vary with the same frequency and in phase (in step) with
each other.
The wave is a transverse wave, both electric and magnetic fields
are oscillating perpendicular to the direction in which the wave
travels. The cross product E  B always gives the direction in
which the wave travels.
Electromagnetic waves can travel through a vacuum or a material
substance.
All electromagnetic waves move through a vacuum at the same
speed, and the symbol c is used to denote its value. This speed is
called the speed of light in a vacuum and is:
The magnitudes of the fields at every instant and at any point are
related by
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Properties of the Wave





Wavelength λ is the horizontal distance between any two
successive equivalent points on the wave.
Amplitude A is the highest point on the wave pattern.
Period T is the time required for the wave to travel a distance
of one wavelength. Unit is second.
Frequency f : f=1/T. The frequency is measured in cycles per
second or hertz (Hz).
Speed of wave is v=λ/T= λf
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Poynting Vector



The rate of energy transport per unit area in EM wave is
described by a vector, called the Poynting vector
The direction of the Poynting vector of an electromagnetic wave
at any point gives the wave's direction of travel and the direction
of energy transport at that point.
The magnitude of S is
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Intensity of EM Wave

The time-averaged value of S is called the intensity I of
the wave
The root-mean-square value of the electric field, as
1 Em2

c 0 2
The energy associated with the electric field exactly equals
to the energy associated with the magnetic field.
•
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