Transcript K L University MAGNETOSTATICS • Introduction to Magneto statics – Magnetic field, Magnetic force, Magnetic flux • Biot-Savat’s law --Applications of Bio-Savart’s law • Ampere’s Circular.

K L University
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MAGNETOSTATICS
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• Introduction to Magneto statics
– Magnetic field, Magnetic force, Magnetic flux
• Biot-Savat’s law
--Applications of Bio-Savart’s law
• Ampere’s Circular Law
• Cyclotron
• Hall Effect & Applications
• LCR Series Resonance Circuit
Worked Problem
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Magnet and Magnetic Field:
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– The magnitude of dB is inversely proportional to
r2, where r is the distance from the element ds to
the point P.
– The magnitude of dB is proportional to the
current I and to the length ds of the element.
– The magnitude of dB is proportional to sin ϕ,
where ϕ is the angle between the vectors ds and
rhat.
ˆ
μ

I

dl
x
r
o
• Biot-Savart law: dB 

4 πr

2
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• µo is a constant called the permeability of free space;
µo =4· x 10-7 Wb/A·m (T·m/A)
• Biot-Savart law gives the magnetic field at a point
for only a small element of the conductor ds.
• To determine the total magnetic field B at some point
due to a conductor of specified size, we must add up
every contribution from all elements ds that make up
the conductor (integrate)!

μo  I
ds x rˆ 
dB


2


4π
r
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• The direction of the magnetic field due
to a current carrying element is
perpendicular to both the current
element ds and the radius vector rhat.
• The right hand rule can be used to
determine the direction of the
magnetic field around the current
carrying conductor:
– Thumb of the right hand in the
direction of the current.
– Fingers of the right hand curl
around the wire in the direction of
the magnetic field at that point.
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Lorentz Force:
Charges moving in a magnetic field
experience an electromagnetic
force.
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Magnetic Field of a Thin Straight Conductor:
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Magnetic Field of a Thin Straight Conductor:
• The magnetic field lines are
concentric circles that surround the
wire in a plane perpendicular to
the wire.
• The magnitude of B is constant on
any circle of radius a.
• The magnitude of the magnetic
field B is proportional to the
current and decreases as the
distance from the wire increases.
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Magnetic Field on the Axis of a Circular Current Loop
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Electric currents
create
magnetic fields.
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What Is The Hall Effect?
According to Hall effect When a magnetic field is
applied perpendicular to a current carrying conductor,
a potential difference is developed between the points
on opposite side of the conductor.
http://www.nikhef.nl/pub/linde/MEDIA/ANIMATIONS/FLASH/RemcoBrantjes/hall-effect.swf
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 When a current-carrying conductor is placed in a MF, a voltage is
generated in a direction perpendicular to both the current and the MF.
 The Hall Effect results from the deflection of the charge carriers to one
side of the conductor as a result of the magnetic force experienced by the
charge carriers.
The arrangement for observing the Hall Effect consists of a flat conducting
strip carrying a current I in the x-direction.
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• A uniform magnetic field B is applied in the y-direction.
• If the charge carriers are electrons moving in the negative
x-direction with a velocity vd, they will experience an
upward magnetic force FB.
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 The electrons will be deflected upward, making the
upper edge negatively charged and the lower edge
positively charged.
The accumulation of charge at the edges continues until the
electric field and the resulting electric force set up by the
charge separation balances the magnetic force on the charge
carriers (Fmag = Felectric).
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When equilibrium is reached, the electrons are no longer deflected
upward.
A voltmeter connected across the conductor can be used to measure
the potential difference across the conductor, known as the Hall
voltage VH.
When the charge carriers are positive, the charges experience an
upward magnetic force q·(v x B).
The upper edge of the conductor becomes positively charged,
leaving the bottom of the conductor negatively charged.
The sign of the Hall voltage generated is opposite the sign of the
Hall voltage resulting from the deflection of electrons.
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The sign of the charge carriers can be determined from the
polarity of the Hall voltage.
When equilibrium is reached between the electric force q·E and
the magnetic force q·vd·B, the electric field produced between the
positive and negative charges is referred to as the Hall field, EH,
therefore, q·EH = q·vd·B.
EH = vd·B
If d is taken to be the width of the conductor, then the Hall
voltage VH measured by the voltmeter is:
VH  E H  d  v d  B  d
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The measured Hall voltage gives a value for the drift velocity of
the charge carriers if d and B are known.
The number of charge carriers per unit volume (charge density),
n, can also be determined by measuring the current in the
conductor:
I
vd 
n qA
Area A = thickness t·d,
VH
I Bd
VH 
n qA
therefore:
IB

nq t
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Hall coefficient, RH =
n q
The Hall coefficient can be determined from
IB
RH  I  B
VH 

n qt
t
The sign and magnitude of RH gives the sign of the
charge carriers and their density.
In most metals, the charge carriers are electrons.
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Principle of operation
Particle acceleration is achieved using an RF field between
“dees” with a constant magnetic field to guide the particles
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Construction:
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Working :
 +ve ions emiited (source) --- accelerated in the gap towards the
dee (which is –ve at that time) say D2
 Since there is no electric field inside the dees , the +ve ions move
with constant velocity along the circles of constant radius ( under
the influence of magnetic field)
 If by the time the ions emerge from D2 , the polarity of the
applied potential is reversed (D1 is negative).
 +ve ion again accelerated by the field in the gap.
 Since their velocity is increased they will move through D1 along
circular arc of greater radius.
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Cyclotron
Working :
 The +ve ions move faster and faster moving in everexpanding circles until they reach the outer edge of the dees
where they are deflected by deflector plate and strike the target.
 The time required for the positive ions to make one complete
turn within dees is the same for all speeds and is equal to the
time period of oscillator.
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Top View
Side View
Injected ions
Uniform
Alternating
B-field
region
E-field
DEMO Video
Ejected ions
www.hyperphysics.com
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Limitations:
The maximum available particle energy is limited due to
the following factors:
1)Due to the limited power and frequency of the
oscillator.
2)Due to the maximum strength of the magnetic field
which can be produced
3)The energy of charged particle emerging from
cyclotron is limited due to variation of mass with
velocity.
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