1.
AZAD SLEMAN HSENAZAD SLEMAN HSEN
ELECTRYCITY AND MAGNETICELECTRYCITY AND MAGNETIC
COLLEGE OF SCINCECOLLEGE OF SCINCE
physic departmentphysic department
magnitic fieldmagnitic field
2.
Lecture 41: FRI 01 MAY
Final Exam Review
Physics 2102
Jonathan Dowling
ABOUT ALL ELECTRYCITY
CLICK ON THE LUCTURE
ABOUT ALL ELECTRYCITY
CLICK ON THE LUCTURE
3.
Magnetic Field
• A magnetic field is a region in which a body
with magnetic properties experiences a force.
4.
Sources of Magnetic Field
• Magnetic fields are produced by electric
currents, which can be macroscopic currents in
wires, or microscope currents associated with
electrons in atomic orbits.
5.
Magnetic Field Lines
• A magnetic field is visualised using
magnetic lines of force which are
imaginary lines such that the tangent
at any point gives the direction of the
magnetic field at that point.
7.
The Earth’s Magnetic Field
• The Earth's magnetic field appears
to come from a giant bar magnet,
but with its south pole located up
near the Earth's north pole.
8.
Magnetic flux pattern due to current
in a straight wire at right angles to a
uniform field
Net flux is lesser
on this side of the
wire
Net flux is greater
on this side of the
wire
I
9.
Fleming’s Left Hand Rule
• If you point your left forefinger in the direction of
the magnetic field, and your second finger in the
direction of the current flow, then your thumb will
point naturally in the direction of the resulting
force!
10.
Magnetic Force
θ
θ
sin
sin
qv
F
B
qvBF
≡
= • A magnetic force acts on a
charge moving in a magnetic
field.
• Use to define magnetic field
B.
Direction of Force (Conductor in Field)
B
I
F
• The right-hand rule can
be used to obtain the
direction of the force on a
conductor placed in a
magnetic field.
11.
Helical Path of Charged Particle
qB
mv
r =
• Charged particle can undergo
circular motion perpendicular to
magnetic field.
• Velocity component parallel to field
results in helical motion.
• Example: earth’s radiation belts.
Field of a Long Straight Wire
r
I
B
π
µ
2
0
= + B
I
• Magnetic field
of a long
straight wire
depends on
current and
radial distance.
12.
Magnetic Field of a Point Charge
It is found that the magnetic field is perpendicular to both
the velocity of the charge and the unit vector from the
charge to the point in question
The magnitude of the field
is given by
2
0 sin
4 r
vq
B
φ
π
µ
=
where φ is the angle between
the velocity and unit vector
The magnetic field lines form concentric circles about the
velocity vector
13.
Force on a current-carrying
conductor
• The direction of magnetic force always
perpendicular to the direction of the
magnetic field and the direction of current
passing through the conductor.
→→→
×= BIF
θsinIF =
magnetic field: FB = qv × B
= |q|vB sin θ
14.
Magnetic Flux Density
• The magnetic flux density is defined as
the force per unit length per unit current
acting on a current-carrying conductor at
right angle to the field lines.
I
F
B =
Unit : tesla (T)
or gauss (G), 1 G = 10-4
T
or weber/m2
15.
Example: Helical particle motion
Proton (m=1.67*10-27
kg) moves in a uniform
magnetic field B=0.5 T directed along x-axis
at t=0 the proton has velocity components
5
5
1.5 10 /
0
2 10 /
x
y
z
v m s
v
v m s
= ×
=
= ×
a) At t=0 find the force on the proton
and its acceleration.
b) find the radius of the helical path and
the pitch of the helix
zv v⊥ = z zF qv B qv B j
∧
= × =
12 2
9.58 10 /
F
a m s
m
= = ×
7
7
4.18
4.79 10 /
2
1.31 10 s
19.7
z
x
mv
R mm
q B
q B
rad s
m
T
pitch v T mm
ω
π
ω
−
= =
= = ×
= = ×
= =
16.
The Earth’s Magnetosphere
Earth can be viewed as a gigantic bar magnet spinning in space.
• Its toroidal magnetic field encases the planet like a huge inner tube.
• This field shields Earth from the solar wind—a continuous stream of charged
particles cast off by the sun.
Produces a bullet-shaped cavity called the magnetosphere.
It also traps charged particles – leads to the radiation or Van Allen belts.
17.
The Earth’s Radiation Belts
Particles are trapped by the non-uniform geomagnetic field – much like a magnetic bottle
– they bounce back and forth from one hemisphere to another.
The trapped particles tend to congregate in distinct bands based on their charge, energy,
and origin.
Two primary bands of trapped particles exist: the one closer to Earth is
predominantly made up of protons, while the one farther away is mostly electrons.
18.
Magnetic Field Measurements
• Using a current balance (d.c.)
• Using a search coil (a.c.)
• Using a Hall probe (d.c.)
19.
Magnetic flux density due to a
straight wire
• Experiments show that the magnetic flux
density at a point near a long straight wire is
r
I
B ∝
r
P ×•TThis relationship is valid as long as r, the
perpendicular distance to the wire, is much
less than the distance to the ends of the wire.
20.
Calculation of B near a wire
r
I
B o
π
µ
2
=
)m(HAmT104 -1-17−
×= πµo
Where µo is called the permeability of free space.
Permeability is a measure of the effect of a material
on the magnetic field by the material.
21.
Magnetic Field due to a Solenoid
• The magnetic field is strongest at the
centre of the solenoid and becomes
weaker outside.
22.
Crossed magnetic and electric fields
r
F=qE+q
r
vx
r
B
y
23.
Magnetic Flux Density due to a
Solenoid
• Experiments show that the magnetic flux
density inside a solenoid is
IB ∝
N
B ∝and
So we have
NI
B oµ
=
or nIB oµ=
where
N
n =
24.
Variation of magnetic flux density
along the axis of a solenoid
• B is independent of the shape or area of
the cross-section of the solenoid.
• At a point at the end of the solenoid, nB oµ
2
1
'=
B
Distance from the
centre of the
solenoid0
nIB oµ=
nB oµ
2
1
'=
2
1
2
1
−
25.
Magnetic Flux Density due to Some
Current-carrying conductors(1)
• Circular coil
r
NI
B o
2
µ
=
• Helmholtz coils
r
NI
r
NI
B oo µµ
72.0
125
8
≈=
26.
Magnetic Flux density due to Some
Current-carrying Conductors (2)
27.
Force on a moving charge in a
magnetic field
• The force on a moving charge is proportional
to the component of the magnetic field
perpendicular to the direction of the velocity of
the charge and is in a direction perpendicular
to both the velocity and the field.
http://webphysics.davidson.edu/physlet_resources/bu_semester2/c12_force.html
θsinqvBF =
BvqvBF ⊥= formax
BvF //for0=
28.
Right Hand Rule
• Direction of force on a positive charge given
by the right hand rule.
→→→
×= BvqF
29.
Free Charging Moving in a
Uniform Magnetic Field
• If the motion is
exactly at right
angles to a uniform
field, the path is
turned into a circle.
• In general, with the
motion inclined to
the field, the path is
helix round the lines
of force.
30.
Mass Spectrometer
• The mass spectrometer is used
to measure the masses of atoms.
• Ions will follow a straight
line path in this region.
• Ions follow a circular
path in this region.
qvBqE =
r
mv
qvB
2
'=
31.
Aurora Borealis (Northern Lights)
• Charged ions
approach the Earth
from the Sun (the
“solar wind” and are
drawn toward the
poles, sometimes
causing a
phenomenon called
the aurora borealis.
32.
Causes of Aurora Borealis
• The charged particles from
the sun approaching the
Earth are captured by the
magnetic field of the Earth.
• Such particles follow the
field lines toward the poles.
• The high concentration of
charged particles ionizes
the air and recombining of
electrons with atoms emits
light.
http://www.exploratorium.edu/learning_studio/auroras/selfguide1.html
33.
Hall Effect
• When a current carrying conductor is held firmly
in a magnetic field, the field exerts a sideways
force on the charges moving in the conductor.
• A buildup of charge at the sides of the conductor
produces a measurable voltage between the two
sides of the conductor.
• The presence of this
measurable transverse
voltage is called the
Hall effect.
34.
Hall Voltage
• The transverse voltage builds up until the
electric field it produces exerts an electric
force on the moving charges that equal and
opposite to the magnetic force.
• The transverse voltage produced is called the
Hall voltage.
35.
Charge Carriers in the Hall Effect
• The Hall voltage has a different polarity
for positive and negative charge carriers.
• That is, the Hall voltage can reveal the
sign of the charge carriers.
36.
Hall Probe
• Basically the Hall probe is a small piece
of semiconductor layer.
• When control current IC
is flowing through the
semiconductor and
magnetic field B is
applied, the resultant
Hall voltage VH can be
measured on the sides
• Four leads are connected to the midpoints
of opposite sides.
37.
Force between two parallel
current-carrying straight wires (1)
1. Parallel wires with current flowing in the
same direction, attract each other.
2. Parallel wires with current flowing in the
opposite direction, repel each other.
38.
Force between two parallel
current-carrying straight wires (2)
• Note that the force exerted on I2 by I1 is equal
but opposite to the force exerted on I1 by I2.
a
II
F o
π
µ
2
21
=
39.
Definition of the ampere
• The ampere is the constant current
which, if maintained in two parallel
conductors of infinite length, of negligible
cross-section, and placed one metre
apart in a vacuum, would produce
between these conductors force of 2 x 10-
7
N per metre of length.
40.
Torque on a Rectangular Current-
carrying Coil in a Uniform Magnetic Field
• Let the normal to the coil plane make an
angle θ with the magnetic field.
• The torque τ is given by θτ sinNBAI=
41.
Moving Coil Galvanometer
• A moving coil galvanometer consists of a coil of
copper wire which is able to rotate in a magnetic
field.
• The magnetic field is produced in the narrow air
gap between concave pole pieces of a
permanent magnet and a fixed soft-iron cylinder.
• The coil is pivoted on jewelled bearings and its
rotation is resisted by a pair of spiral hair springs.
42.
Radial Magnetic Field
• In order to have a meter with a linear scale,
the field lines in the gap should be always
parallel to the plane of the coil as it rotates.
• This could be achieved if we have a radial
magnetic field. The soft iron cylinder gives
us this field shape.
43.
The Principle of a Moving Coil
Galvanometer
• The torque due to the current in the coil is
given by NBAI=τ
• The coil rotates until
• The resisting couple due to the hair springs
is ατ k='
'ττ =
• Then we have
k
NBAI
=α
Where α is the angle of deflection
and k is the torsion constant.
Iei ∝α..
44.
Current Sensitivity
• The current sensitivity of a galvanometer is
defined as the deflection per unit current.
k
NBA
I
=
α
• High current sensitivity can be achieved by
• A coil of large area,
• A coil of large number of turns,
• Large value of B which could be achieved by
using strong magnet and narrow air gap.
• Hair springs with small torsion constant k.
Unit : Rad A-1
or mm A-1
45.
Limitation of High Current Sensitivity
• If the coil is too large, the moment of inertia is
also large and hence the coil would swing
about its deflection position before a reading
can be taken.
• If the coil has a large number of turns, the air
gap needs to be wide.
• If the hair springs have small torsion
constant, the restoring torque would become
weak and the coil would swing before coming
to rest.
46.
Voltage Sensitivity
• The voltage sensitivity of a galvanometer is
defined as the deflection per unit voltage
across the galvanometer.
kR
NBA
V
=
α Unit : rad V-1
or mm V-1
Where R is the resistance of the galvanometer.
• High voltage sensitivity is desirable in
circuits of relatively low resistance.
47.
Example:
A point charge q = 1 mC moves in the x direction with v = 108
m/s. It
misses a mosquito by 1 mm. What is the B field experienced by the
mosquito?
108
m/s
90o
rˆ 2
0
4 r
v
qB
π
µ
=
26
83
2
7
10
1
101010
m
CB s
m
A
N
−
−−
×××=
TB 4
10=
48.
To find the E field of a charge distribution use:
Use:
2
ˆ
r
rkdq
Ed =
To find the field of a current distribution use:B
Note vdq
dt
ds
dqds
dt
dq
ids
===
2
0 ˆ
4 r
rsid
Bd
×
=
π
µ
49.
Topic: Biot – Savart Law
Use to find B field at the center of a loop of
wire.
2
0 ˆ
4 r
rlid
B
×
= ∫
π
µ
R
i Loop of wire lying in a plane. It has radius R and total current i
flowing in it.
First find rld ˆ×
rld ˆ×
is a vector coming out of the paper at the same angle
anywhere on the circle. The angle is constant.rˆ
rˆ
ld
ld
R
R
i
dl
R
i
R
idl
dBB π
π
µ
π
µ
π
µ
2
444 2
0
2
0
2
0
==== ∫∫∫
R
i
B
2
0µ
=
k
R
i
B ˆ
2
0µ
=
Magnitude of B field at
center of loop. Direction is
out of paper.
R kˆ
i
50.
Example: Find magnetic field inside a long, thick wire of radius a
Cross-sectional
view
Path of integral
dl
a
r
rBdlBBdlldB
r
π
π
2
2
0
===⋅ ∫ ∫∫
CIurB 02 =π
IC = current enclosed by the
path
i
a
r
i
a
r
IC
2
2
2
2
==
π
π
r
i
a
r
B
π
µ
2
1
2
2
0=
2
0
2 a
ir
B
π
µ
=
Example: Find field
inside a solenoid. See
next slide.
51.
Toroid
∫ =⋅ CIldB 0µ
∫ = NiBdl 0µ
NirB 02 µπ =
r
Ni
B
π
µ
2
0
=
a < r <
b
a
b ld
BdlldB =⋅
10cos =o
Tokamak Toroid at
Princeton
I = 73,000 Amps for 3
secs
N is the total
number of turns
53.
Power in Electric Circuits
A battery sets up a current
in a circuit containing an
unspecified conducting
device
The battery maintains a voltage V across its
own terminals, and thus across the terminals
of the unspecified device. The amount of
charge dq that moves between those terminals
in time interval dt is equal to i dt. This charge
moves through a potential decrease in
magnitude by the amount
The moving charge loses potential at rate of dU/dt = i V
The power associated with this transfer of energy is the rate of transfer, which is
The unit of power that follows
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