Magnetic effect of current - 3 PPT Physics Class 12

 Page 1


MAGNETIC EFFECT OF CURRENT - III
1. Cyclotron
2. Ampere’s Circuital Law
3. Magnetic Field due to a Straight Solenoid
4. Magnetic Field due to a Toroidal Solenoid
Page 2


MAGNETIC EFFECT OF CURRENT - III
1. Cyclotron
2. Ampere’s Circuital Law
3. Magnetic Field due to a Straight Solenoid
4. Magnetic Field due to a Toroidal Solenoid
N
S
D
1
D
2
+
W
B
Cyclotron:
D
1
, D
2
– Dees          N, S – Magnetic Pole Pieces           
W       – Window          B - Magnetic Field    
H F 
Oscillator
D
2
D
1
Working: Imagining D
1
is positive and D
2
is negative, the + vely charged 
particle kept at the centre and in the gap between the dees get accelerated 
towards D
2
.  Due to perpendicular magnetic field and according to Fleming’s 
Left Hand Rule the charge gets deflected and describes semi-circular path.
When it is about to leave D
2
, D
2
becomes + ve and D
1
becomes – ve.  
Therefore the particle is again accelerated into D
1
where it continues to 
describe the semi-circular path.  The process continues till the charge 
traverses through the whole space in the dees and finally it comes out with 
very high speed through the window.
W
B
Page 3


MAGNETIC EFFECT OF CURRENT - III
1. Cyclotron
2. Ampere’s Circuital Law
3. Magnetic Field due to a Straight Solenoid
4. Magnetic Field due to a Toroidal Solenoid
N
S
D
1
D
2
+
W
B
Cyclotron:
D
1
, D
2
– Dees          N, S – Magnetic Pole Pieces           
W       – Window          B - Magnetic Field    
H F 
Oscillator
D
2
D
1
Working: Imagining D
1
is positive and D
2
is negative, the + vely charged 
particle kept at the centre and in the gap between the dees get accelerated 
towards D
2
.  Due to perpendicular magnetic field and according to Fleming’s 
Left Hand Rule the charge gets deflected and describes semi-circular path.
When it is about to leave D
2
, D
2
becomes + ve and D
1
becomes – ve.  
Therefore the particle is again accelerated into D
1
where it continues to 
describe the semi-circular path.  The process continues till the charge 
traverses through the whole space in the dees and finally it comes out with 
very high speed through the window.
W
B
Theory:
The magnetic force experienced by the charge provides centripetal force 
required to describe circular path.
mv
2
/ r  =  qvB sin 90°
(where m – mass of the charged particle,    
q – charge, v – velocity on the path of 
radius – r, B is magnetic field and 90° is the 
angle b/n v and B)
v =
B q r
m
If t is the time taken by the charge to describe the semi-circular path 
inside the dee, then
t =
p r
v
or t =
p m
B q
Time taken inside the dee depends only on 
the magnetic field and m/q ratio and not on 
the speed of the charge or the radius of the 
path.
If T is the time period of the high frequency oscillator, then for resonance,
T = 2 t or
T =
2pm
B q
If  f is the frequency of the high frequency oscillator (Cyclotron Frequency), 
then
f =
2pm
B q
Page 4


MAGNETIC EFFECT OF CURRENT - III
1. Cyclotron
2. Ampere’s Circuital Law
3. Magnetic Field due to a Straight Solenoid
4. Magnetic Field due to a Toroidal Solenoid
N
S
D
1
D
2
+
W
B
Cyclotron:
D
1
, D
2
– Dees          N, S – Magnetic Pole Pieces           
W       – Window          B - Magnetic Field    
H F 
Oscillator
D
2
D
1
Working: Imagining D
1
is positive and D
2
is negative, the + vely charged 
particle kept at the centre and in the gap between the dees get accelerated 
towards D
2
.  Due to perpendicular magnetic field and according to Fleming’s 
Left Hand Rule the charge gets deflected and describes semi-circular path.
When it is about to leave D
2
, D
2
becomes + ve and D
1
becomes – ve.  
Therefore the particle is again accelerated into D
1
where it continues to 
describe the semi-circular path.  The process continues till the charge 
traverses through the whole space in the dees and finally it comes out with 
very high speed through the window.
W
B
Theory:
The magnetic force experienced by the charge provides centripetal force 
required to describe circular path.
mv
2
/ r  =  qvB sin 90°
(where m – mass of the charged particle,    
q – charge, v – velocity on the path of 
radius – r, B is magnetic field and 90° is the 
angle b/n v and B)
v =
B q r
m
If t is the time taken by the charge to describe the semi-circular path 
inside the dee, then
t =
p r
v
or t =
p m
B q
Time taken inside the dee depends only on 
the magnetic field and m/q ratio and not on 
the speed of the charge or the radius of the 
path.
If T is the time period of the high frequency oscillator, then for resonance,
T = 2 t or
T =
2pm
B q
If  f is the frequency of the high frequency oscillator (Cyclotron Frequency), 
then
f =
2pm
B q
Maximum Energy of the Particle:
Kinetic Energy of the charged particle is
K.E. = ½ m v
2
= ½ m (
B q r
m
)
2
= ½
B
2
q
2
r
2
m
Maximum Kinetic Energy of the charged particle is when r = R (radius of the D’s).
= ½
B
2
q
2
R
2
m
K.E. 
max
The expressions for Time period and Cyclotron frequency only when             
m remains constant. (Other quantities are already constant.)
m =
m
0
[1 – (v
2
/ c
2
)]
½
If frequency is varied in synchronisation with the variation of mass of the 
charged particle (by maintaining B as constant) to have resonance, then the 
cyclotron is called synchro – cyclotron.
If magnetic field is varied in synchronisation with the variation of mass of 
the charged particle (by maintaining f as constant) to have resonance, then 
the cyclotron is called isochronous – cyclotron.
NOTE: Cyclotron can not be used for accelerating neutral particles.  Electrons can 
not be accelerated because they gain speed very quickly due to their lighter mass 
and go out of phase with alternating e.m.f. and get lost within the dees.
But m varies with v according to 
Einstein’s Relativistic Principle as per
Page 5


MAGNETIC EFFECT OF CURRENT - III
1. Cyclotron
2. Ampere’s Circuital Law
3. Magnetic Field due to a Straight Solenoid
4. Magnetic Field due to a Toroidal Solenoid
N
S
D
1
D
2
+
W
B
Cyclotron:
D
1
, D
2
– Dees          N, S – Magnetic Pole Pieces           
W       – Window          B - Magnetic Field    
H F 
Oscillator
D
2
D
1
Working: Imagining D
1
is positive and D
2
is negative, the + vely charged 
particle kept at the centre and in the gap between the dees get accelerated 
towards D
2
.  Due to perpendicular magnetic field and according to Fleming’s 
Left Hand Rule the charge gets deflected and describes semi-circular path.
When it is about to leave D
2
, D
2
becomes + ve and D
1
becomes – ve.  
Therefore the particle is again accelerated into D
1
where it continues to 
describe the semi-circular path.  The process continues till the charge 
traverses through the whole space in the dees and finally it comes out with 
very high speed through the window.
W
B
Theory:
The magnetic force experienced by the charge provides centripetal force 
required to describe circular path.
mv
2
/ r  =  qvB sin 90°
(where m – mass of the charged particle,    
q – charge, v – velocity on the path of 
radius – r, B is magnetic field and 90° is the 
angle b/n v and B)
v =
B q r
m
If t is the time taken by the charge to describe the semi-circular path 
inside the dee, then
t =
p r
v
or t =
p m
B q
Time taken inside the dee depends only on 
the magnetic field and m/q ratio and not on 
the speed of the charge or the radius of the 
path.
If T is the time period of the high frequency oscillator, then for resonance,
T = 2 t or
T =
2pm
B q
If  f is the frequency of the high frequency oscillator (Cyclotron Frequency), 
then
f =
2pm
B q
Maximum Energy of the Particle:
Kinetic Energy of the charged particle is
K.E. = ½ m v
2
= ½ m (
B q r
m
)
2
= ½
B
2
q
2
r
2
m
Maximum Kinetic Energy of the charged particle is when r = R (radius of the D’s).
= ½
B
2
q
2
R
2
m
K.E. 
max
The expressions for Time period and Cyclotron frequency only when             
m remains constant. (Other quantities are already constant.)
m =
m
0
[1 – (v
2
/ c
2
)]
½
If frequency is varied in synchronisation with the variation of mass of the 
charged particle (by maintaining B as constant) to have resonance, then the 
cyclotron is called synchro – cyclotron.
If magnetic field is varied in synchronisation with the variation of mass of 
the charged particle (by maintaining f as constant) to have resonance, then 
the cyclotron is called isochronous – cyclotron.
NOTE: Cyclotron can not be used for accelerating neutral particles.  Electrons can 
not be accelerated because they gain speed very quickly due to their lighter mass 
and go out of phase with alternating e.m.f. and get lost within the dees.
But m varies with v according to 
Einstein’s Relativistic Principle as per
Ampere’s Circuital Law:
The line integral     B . dl for a closed curve is equal to µ
0
times the net 
current I threading through the area bounded by the curve.
?
?
B . dl = µ
0 
I
?
B . dl =
?
B . dl cos 0°
?
B . dl =  B
=
?
dl
=  B (2p r)    = ( µ
0
I / 2p r) x 2p r
?
B . dl = µ
0 
I
I
B
B
r
O
dl
I
Current is emerging 
out and the magnetic 
field is anticlockwise.
Proof: