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Moving Charges And Magnetism
Concept of magnetic field :
The space around a current carrying conductor in which its influence can be
experienced is called magnetic field.
When a current is
passed through a conductor it produces a magnetic field
around it. If a magnetic pole is placed near the conductor carrying current it will
experience a force. Thus a moving charge or a current sets up a magnetic field in the
space around it. The magnetic field disappears as soon as the current is switched off or
the charges stop moving. It means a moving charge is a source of both electric and

magnetic field. The magnetic field B is a vector field. It obeys the principle of
superposition i.e. the magnetic field at a point due to several sources is the vector sum
of the magnetic field of each individual source. The magnetic field is represented by
field lines. But the magnetic field lines are closed paths without a starting point or
ending point. If the magnetic field is emerging out of the plane of the paper it is
denoted by a dot (.) If the magnetic field is going into the plane of paper it is denoted
by a cross (X).
Oersted’s experiment :
In 1820 a Danish physicist H.C.Oersted
discovered the magnetic effect of electric current.
He observed that when a conducting wire AB
carrying current was held over a magnetic needle
NS parallel to it.
When current flows from A to B, the N-pole of
the magnetic needle gets deflected towards west.
If the direction of current is reversed, so that it
flows from B to A, the N-pole of the magnetic
needle gets deflected towards east.
If the wire is placed below the needle, the direction of deflection of the needle is
again reversed.
Thus oersted’s experiment concluded that moving charges or current carrying
conductor produces magnetic field around it. If a strong current is passed through a
conductor then the magnetic field produced around the conductor is also very strong.
The direction of deflection of magnetic needle due to the current in a conductor is
given by Amper’s swimming rule. The direction of the magnetic field around a straight
conductor carrying current is given by right hand thumb rule or Maxwell’s cork screw
rule.
Biot – Savart’s law :
Biot – Savart’s law is an experimental law which is used to find
the magnetic field at a point due to current element. A current element
is the product of current I and the length dl of very small segment of
current carrying conductor.
Page : 1
Consider a conductor XY carrying current I. Let AB be a current element of length
dl. Let P be a point at a distance r from the mid point of the current element. Let  be
the angle between the current element and the line joining the point P to the current
element.
Statement : The magnetic field dB at a point P due to the current element is :
i. directly proportional to the current I through the conductor
ii. directly proportional to the length dl of the current element.
iii. directly proportional to the sine of the angle ‘  ’ between the direction of
current and the line joining the point P to the mid point of current element. and
iv. Inversely proportional to the square of the distance ‘r’ of the point from the
current element.
I dl sin 
Then dB 
2
r
KI dl sin 
dB 
r2
or
Where ‘K’ is a constant of proportionality.
The value of ‘K’ depends on the medium between the point P and the current
carrying conductor and the system of units used.
For free space and in SI, K 
0
 10 7 T m / A
4
Where  0 = 4 10 7 is called permeability of free space.
 dB 
 0 I dl sin 
4
r2
In vector form, Biot-Savart’s law is written as


 I dl  r
 dB  0
4
r3




The direction of dB is perpendicular to the plane containing dl and r . It is given
by right hand thumb rule.
Special cases :
1)
If  =00 , sin  =0
 dB=0
Thus the magnetic field is zero at points on the axis of the current element.
2)
If  =900 , sin  =1  dB is maximum
Thus the magnetic field due to a current element is maximum in a plane passing
through the element and perpendicular to its axis.
Magnetic field on the
axis of a circular current
loop
Consider a circular loop of
radius r with 0 as its centre carrying
Page : 2
a current I as shown in figure. Let this plane of the loop is perpendicular to the plane of
paper. Let P be a point on the axis of loop at a distance ‘x’ from o, at which the
magnetic field has to be determined consider a small element AB of length dl.
Join the line from point p to the midpoint of the current element Let ‘a’ be the
distance between these two points.
According to Biot-Savart’s law, the magnetic field at P due to the current element
AB is given by
dB 
 0 Idl sin 
4
a2
 dB 
 0 Idl
4 a 2
(    90 0 , sin   1 )
along PM
Similarly the magnetic field at P due to the current element A1B1 is
dB 
 0 Idl
.
4 a 2
along PN
dB along PM can be resolved into two components dB sin  along PX and dB
cos  along PY.
Similarly dB along PN can be resolved into two components dB sin along PX
and dB cos along PY1.
The components along py and py1 cancel each other and the resultant field at P
due to AB and A1B1 is 2dB sin  along PX.
Total field at P is obtained by integrating dB sin over the circumference of the
loop.
 Total magnetic field due to the entire loop is
B   2dB sin  =
=
But
 dl   r
a2
sin 
 0 2I
sin   dl
4 a 2
- half of the circumference of the loop.
from the figure, sin =
B
 0 Idl
 2. 4
r
a
and a2 = r2 + x2
0 2I r
.
r
4 a 2 a
 0 2Ir 2
B
.
4
a3
or B 
0
2Ir 2
.
4 r 2  x 2 3 2
For n turns of the coil
B
 0 2 n Ir 2
.
along PX.
4 r 2  x 2 3 2
Page : 3
The direction of the field is along the axis of the loop and towards the observer
facing the loop if the current in the loop is anticlockwise and away from the observer
facing the loop if the current is clockwise.
Special Case :
Magnetic field at the centre of the loop
At the centric of the loop x = 0
or
B 
 0 2 n I r 2
4
r3
B 
 0 2 n I
4 r
Ampere’s Circuital law :

Statement : The line integral of the magnetic field B around any closed path is
equal to µ0 times the total current I passing through this closed path.

Mathematically


B. dl   0 I
Proof : Consider a long straight conductor carrying a current I. The
magnetic field lines are produced around the conductor as
concentric circles.
The magnitude of the magnetic field due to the current carrying
straight conductor at point P is given by
B
0 I
.
2 r
Consider a circular path of radius r such that current carrying conductor passes
through its centre. Let PQ be a small element of length dl. The magnitude of field is
same at all points on the circle. According to right hand thumb rule the direction of B


at P is along the tangent to the circular path at that point. Hence B and dl are along
the same direction so that angle between them is zero.


 B . dl  B dl cos 0  Bdl
Hence the line integral of magnetic field along the circular path is given by.


0 I
 B . dl   B dl  B  dl  2 . r  dl
But  dl  2r the circumference of the circular path
 
 I
  B. dl  0 . .2r   0 I
2 r

Or

 B. dl  0 I
This proves the Ampere’s circuital law
Applications of Amperes circuital law :
1. Magnetic field due to infinitely long straight wire carrying
current :
Page : 4
Consider an infinitely long straight wire carrying current I. Let P be a point at a
distance ‘r’ from the wire. Draw a circle of radius ‘r’ passing through the point P.
The magnetic field at any point on the circular path is along the tangent to the circle

at that point. By symmetry the magnetic field B at any point on the circular path is
same.


  B. dl   B dl cos  B  dl  B.2 r
From Ampere’s circuital law, we have


 B. dl  
0
I
 B. 2r   0 I
or B 
0 I
.
2 r
Note :
1) The magnitude of the magnetic field at every point on a circle of radius ‘r’ is
same. It means the magnetic field due to a straight conductor of infinite length
carrying current has cylindrical symmetry.
2) The direction of the magnetic field at every point on the circle is tangential to it.
The lines of constant magnitude of magnetic field form concentric circles. These
lines are called magnetic field lines form closed paths.
3) Even though the wire carrying current is of infinite length, the magnetic field at
non zero distance is not infinite. Since B  1 , the magnetic field B decreases as
r
distance ‘r’ increases.
4) The direction of the magnetic field due to current carrying straight wire is given
by right hand rule.
Right hand rule :
“If we hold the wire in the right hand with the extended thumb pointing in the
direction of current the fingers curl around the wire in the direction of magnetic
field”.
The solenoid:
A solenoid consists of a long insulated wire wound closely in the form of helix.
Each turn of wire in the solenoid can be regarded as circular loop. When current flows
through the solenoid, the total magnetic field is the vector sum of the fields due to all the
turns. A long solenoid means that the length of the solenoid is large compared to its radius.
The magnetic field inside a long solenoid is uniform and strong. However, the
magnetic field at a point outside the long solenoid is non uniform and weak. For an ideal
solenoid, the magnetic field at a point outside the solenoid is practically zero.
Magnetic field at a point inside the solenoid (mention of expression) :Magnetic field at a point inside the solenoid is given by
B   0 nI
Page : 5
Where n=no. of turns per unit length of the solenoid
I=current passing through the solenoid
 0 =absolute permeability of free space
Note : The magnetic field at the end of the solenoid is half of that at its centre.
 nI
ie. Bend  0
2
The Toroid :
The toroid is a hollow circular ring on which a large
number of turns of insulated wire are closely wound. A
toroid is an endless solenoid in the form of a ring or a ring
shaped closed solenoid.
Magnetic field at a point inside the toroid is given by an
expression B   0 n I
Where n=no. of turns per unit length of the solenoid
I=current passing through the solenoid
 0 =absolute permeability of free space
Force on a moving charge in a magnetic field :

Consider a charge +q moving with a velocity v


in a magnetic field B and v makes an angle  with

B . The moving charge experiences a fore F is given
by

F  q v B sin 

This force is called magnetic Lorenz force, the direction of the force F is


perpendicular to both V and B and is given by right hand rule.


The magnitude of the force F depends on the charge q velocity V and the magnetic
field B.
In vector form F  qV  B 





Special cases :
1) If V=0 then F=0 ie stationary charged particle does not experiences a force in a
magnetic field.
Page : 6
2) If =00 or 1800 then F=0 ie a charged particle moving parallel to the magnetic
field does not experiences any force in the magnetic field.
3) If =900 then F = qVB ie a charged particle experiences a maximum force when
it moves perpendicular to the magnetic field.
Definition of magnetic field (B) :
We know that B=
F
qV sin 
If q=1, v=1, =900 then B = F
Thus magnetic field at a point is defined as the force acting on a unit charge moving
with unit velocity in a direction perpendicular to the field.
The SI unit of magnetic field is tesla(T).
If q = 1c, V=1m/s, F=1N, =900 then B= 1T
Tesla(T) : The strength of the magnetic field at a point is said to be one tesla if one
coulomb of charge moving with a velocity of 1m/s along a direction perpendicular
to the field experiences a force of one Newton.
Lorentz force :
The total force experienced by a charged particle moving in presence of both
electric and magnetic field is called Lorentz force.


The force experienced by a charge ‘q’ in an electric field is given by F  q E .
This force acts in the direction of field E and is independent of the velocity of the
charge.

The force experienced by a charge ‘q’ moving with a velocity V in a magnetic field
is given by
 
F  q V  B 




This force acts perpendicular to the plane of V and B and depends on the velocity

V of the charge.
The total force on the charge due to both electric and magnetic field is given by




F  q E  q(V  B ) .
or F  q  E  q(V  B )






Motion of a charged particle in a uniform magnetic field :
When a charged particle is moving in a magnetic field the
magnetic force on the charged particle is perpendicular to its
velocity. Therefore no work is done by the magnetic field on the
charged particle and there is no change in the magnitude of velocity.
However the direction of velocity may be changed.
Page : 7

Consider a charged particle of mass ‘m’ moving with a velocity V

in a uniform magnetic field B The magnetic force on the charged



particle is F  q (V  B ) .
1) When V is  to B : When the charged particle of charge q enters the region of

magnetic field with a velocity V r to the field. The magnitude of the force on it is
F=qvB sin900 or F = qvB.


This force is always r to the direction of V and B as shown in figure. This force
continuously deflects the particle sideways without changing its speed and the particle
will move along a circular path of radius ‘r’ r to the field. Thus the magnetic force
provides the centripetal force.
Now, centripetal force required by the particle to move in a circular path =
mv 2
r
This force is provided by magnetic Lorentz force = qvB .

mv 2
= qvB
r
or r 
mv
V
or r 
Bq
q
 B
m
Thus the radius of the circular path is inversely proportional to the specific charge and
to the magnetic field.
Period of revolution T 
2r 2 mv


V
V
Bq
or T =
2m
Bq
Thus the time period is independent of V and r. If the particle moves faster, the radius
is larger, It has to move along a larger circle so that the time taken is the same.
The angular frequency of the charged particle is
2 qB  2 qB
 


T
2m
m
This frequency is called cyclotron frequency.


2) When B and V are inclined to each other :


If V is inclined at an angle  with the direction of B as
shown in figure.

The velocity V can be resolved into two components Now


V11  V cos is the component of V along the direction of B

and V  VSin is the component of V perpendicular to the
direction of B .
Page : 8
Due to the  r component of velocity, the charged particle experiences a force


F=qVB which acts r to both V  and B . Thus the particle moves along a circular
mv
mv sin 
path in the y- z-plane. The radius of the circular path is r   =
.
qB
The period of the circular path is T 
qB
2r
2
mv sin 


V
V sin 
qB
T
or
2 m
.
Bq
Thus the charged particle moves along circular path in a
plane r to B due to velocity component V sin  . It is also

advances linearly in the direction of B due to the velocity
component Vcos  . Hence the resultant path of the charged

particle is helix with its axis along the direction of B .
The linear distance traveled by the charged particle in the
direction of magnetic field during its period of revolution is
called pitch of the helical path
Pitch = V11  T  V cos  
2m 2mv cos 

qB
qB
Note : Pitch of the helical path is equal to the spacing between its two consecutive
circular paths.
Cyclotron :
Cyclotron is a device used to accelerate charged particles like protons, deuterons, alpha
particles etc. to very high energies to carry out nuclear reactions. It was invented by
Lawrence and Livingston in 1934.
Principle : The charged particle is made to move in both electric and magnetic fields
which are perpendicular to each other. The magnetic field makes the charged particles
to move in circular paths and the electric field accelerates the charged particles to
acquire sufficiently large amount of energy.
Construction :
Cyclotron consists of two small hollow metallic
semicircular discs D1 and D2 called dees as shown in
figure. These dees are separated by a small gap where
the source of positively charged particles is placed near
the mid point of dees. These dees are insulated from
each other and enclosed inside another vacuum chamber
so that the air molecules may not collide with charged
particles. The dees are connected to a powerful high
frequency oscillator. As a result a varying electic field is
produced in the gap between the dees. This arrangement
is placed between two poles of a strong electromagnet.
The magnetic field is r to the plane of dees.
Page : 9
Working : If a positively charged particle say a proton enters the gap between the two
dees and at that instant when dee D2 is at negative potential and D1 is at positive
potential it will accelerate towards D2. As soon as it enters D2 it does not experiences
any electric field due to shielding effect of metallic chamber. Inside D2 it moves
perpendicular to the magnetic field and hence described a circular path inside it. After
completing the semicircle, it enters the gap between the dees at the time when the
polarities of the dees have been reversed. Now the proton is further accelerated towards
D1. Then it enters D1 and again describes a semicircle path due to the magnetic field
which is perpendicular to the motion of the proton. This process continues and the
proton will go on accelerating every time, it comes into the gap between the dees and
will go on describing semicircular paths of increasing speed and radius. Finally it
acquires sufficiently high energy when the radius is approximately equal to that of the
dees. Now the accelerated proton is ejected through a window using deflecting plate
and hits the target.

Theory : - Let a particle of charge ‘q’ and mass ‘m’enters a region of magnetic field B
with a velocity perpendicular to the field. The particle follows a circular path, the
necessary centripetal force is provided by the magnetic Lorentz force, then
Bqv =
mv 2
r
or
r
mv
Bq
Time taken by the charged particle to complete the semicircular path is
t=
 r  mv  m
= 
.

V V Bq
Bq
This shows that, time is independent of the speed and radius and is same to complete
any semicircle.
Period of revolution :
If T is the time period of the electric field , then the polarities of the dees will change
after time T 2 .
The particle will be accelerated if time taken by it to complete a semicircle is equal to
T
.
2
ie.
T
m
t 
2
Bq
or T=
2 m
.
Bq
The frequency of revolution of the charged particle is :
1
T
 
Bq
.
2m
This frequency is known as cyclotron frequency or magnetic resonance frequency.
Maximum energy of the charged particle :
The charged particles will attain maximum velocity when it moves along the largest
circular path of radius R.
Page : 10
Then Bq Vmax =
 Vmax 
2
mVmax
R
BqR
m
 Maximum K.E. of the particle is given by
Emax
1
1  BqR 
= mv 2 max = m

2
2  m 
E max 
2
B2 q2 R2
2m
Uses of cyclotron :
1. It is used to produce radio active isotopes which are used in hospitals to diagnosis
and treatment.
2. It is used to improve the quality of solids by adding ions.
3. It is used to bombard the atomic nuclei with highly accelerated particles to study
the nuclear reactions.
4. It is used to synthesize fresh substances.
Magnetic force on a current carrying conductor :
Consider a conductor of length l and uniform cross
sectional area A carrying a current I placed in an external

magnetic field B . A current carrying conductor contains

electrons moving with velocity V d in a direction opposite to
the direction of current in a conductor.
An electron moving in a uniform magnetic field experiences a force which is
given by f  e v d  B 







This force acting perpendicular to the plane containing V d and B .
If n is the no. of free electrons per unit volume of the conductor, then total no.
of electrons in the conductor is N = n × volume = nAl
Total force on the conductor is F  N f  nAl  e v d  B 






  
F  nAe  l v d  B 



If I l represents a current element vector in the direction of current, then

vectors l and V d will have opposite directions and we can take

 l V d  Vd l .

 
 F  nAeVd l B
But nAeVd =I
Page : 11

 
 F  I l  B


The magnitude of the force on the current carrying conductor is given by
F = BIl sin 
Where ѳ is the angle between the direction of magnetic field and direction of flow of
current.
Special cases :
1. If   0 0 or 1800 then F  0
Thus a current carrying conductor placed parallel to the direction of magnetic field
does not experience any force.
2. If   90 0 then Fmax = BIl
Thus a current carrying conductor placed perpendicular to the direction of
magnetic field experiences a maximum force.
The direction of force is given by Fleming’s left hand rule.
Fleming’s left hand rule :
Statement : If the first three fingers of the left hand are
stretched mutually perpendicular to each other. Such that
forefinger points in the direction of magnetic field(B), the central
finger in the direction of current (I) then the
thumb points in the direction of force(F) on the conductor
Velocity selector (velocity filter)
Velocity selector is an arrangement used to select charged particles of a particular
velocity from a beam passed through a space having crossed electric and magnetic
fields. Mutually perpendicular electric and magnetic fields are called crossed fields.
Consider two equal and oppositely charged plates such that electric field is setup
from +ve to –ve plate. Let a magnetic field be perpendicular to electric field and
directed inwards.
Let a charged particle of charge q enters a region with a velocity v where the
crossed electric and magnetic field exists.
The force acting on the charged particle due to the electric field is given by


Fe  q E .
This force acts in the downward direction (i.e. along the direction of E )
The force acting on the charged particle due to the magnetic field is given by



Fm  q(V  B) or Fm  qvB sin 90 0  qvB

This force deflects the charged particle in the upward direction. The values of E

and B are so selected that electric force and magnetic force become equal and
opposite.
If the particle continues to move in a straight line without any deflection then
Upward force=downward force
i.e. qvB=qE
V 
E
B
Page : 12
Only those particles move with this velocity will pass undeflected through the
slit S. This arrangement is called velocity selector. This method can be used to
measure the specific charge (e/m).
Force between two parallel conductors carrying
current :
Consider two long parallel conductors x and y separated by a distance ‘r’
carrying current I1 and I2 in the same direction. The magnetic field produced by
current I1 at any point P on the conductor Y is given by
B1 =
 0 I1
.
2 r
-------- (1)
This field is perpendicular to Y and directed inwards according to right hand
thumb rule Now the conductor Y carries a current I2 lies in the magnetic field B1.
Hence the force acting on the length l of the conductor Yis given by
F2 = B1I2l =
 0 I1 I 2l
.
---- (2)
2 r
By Fleming’s left hand rule, the force F2 acts  r to Y and
directed towards X.
Similarly the magnetic field B2 is produced by the current
I2 at any point Q on the conductor X is given by.
B2 =
0 I 2
.
2 r
Since conductor X carries current I1 lies in this magnetic field B2 it experiences a
force F1 given by.
F1 = B2I1l =
 0 I1 I 2 l
--------(3)
2 r
This force is perpendicular to X and directed towards Y. Hence two parallel
conductors carrying current in the same direction attracts each other.
Note : If two parallel conductors carrying current in opposite direction repels each
other.
Definition of ampere :
 II l
From equations (2) and (3) F1 = F2 = F = 0 . 1 2
2 r
F  0 I 1I 2
 Force per unit length is

.
l 2 r
F 4 10 7 11

 2 10 7 N
If I1 = I2 = 1A, r=1m then
=
m
l
2
1
“Thus one ampere is that steady current, which when flowing through two
infinitely long straight parallel conductors placed one metre apart in free space
produces between them a force of 2 10-7
newton per metre of their length”.
Torque on a current loop placed in a magnetic
field :
Page : 13
Consider a rectangular coil PQRS of length ‘l’ and breadth ‘b’ is placed in a
uniform magnetic field B with its axis  r to the field.
According to Fleming left hand rule the magnetic forces on sides PS and QR are
equal and opposite and along the axis of the loop so their resultant is zero.
The side PQ experiences a force perpendicular to PQ and directed inward which
is equal to BIl. The side RS also experiences an equal force but it is directed
outwards. These two forces constitute a couple which tends to rotate the coil.
 Torque = Force   r distance
  BIl  b sin 
  BIA  sin 
If the loop has ‘n’ turns then
  nBIA sin 
But nIA = m - the magnetic moment of the loop
   mB sin 
Where  is the angle between the direction of B and normal to the plane of the
loop.



In vector from   m  B
The direction of torque  is such that it rotates the loop in clockwise direction
about the axis of the loop.
Special Cases :
1) If  =00,  =0 ie when the plane of the loop is  r to the magnetic field, the
torque is minimum.
2) If  =900,  =nBIA ie when the plane of loop is parallel to the magnetic
field the torque is maximum.
Moving Coil Galvanometer :
A Moving coil galvanometer is a device used to measure very small currents of the
order of 10-10 A.
Principle : when a current loop is placed in a magnetic
field it experiences a torque.
Construction : It consists of rectangular coil of several
turns of insulated copper wire wound on a non magnetic
frame. The coil is suspended by phosphor – bronze wire in
between two poles of a permanent magnet. The upper end
of the wire is fixed to movable torsion head TH and
connected to terminal T2,. The other end of the coil is
connected to a light spring ‘S’ which is finally connected
to the terminal T1. A plane circular mirror (m) is attached
to the suspension wire to measure the deflection of the coil
Page : 14
using lamp and scale arrangement.
A soft iron cylinder is placed within the frame of the
coil without touching it. It helps to intensify the field.
The pole pieces are made concave to provide the radial
field. The plane of the coil rotating in such a field
remains parallel to the field in all positions.
Theory of moving coil Galvanometer : experiences a
torque given by   nIAB sin  .
At any position of the coil, the magnetic field is parallel to
its plane then  =900.
   nIBA
This torque tends to deflect the coil and is known as
deflecting torque. As the coil gets deflected, the suspension
wire gets twisted and
Let ‘n’ be the number of turns in the coil ABCD. Let ‘l’ and ‘b’ be the length
and breadth of the coil and I be the current flowing through it. Let ‘B’ is the strength
of the magnetic field in which the coil is placed.
When the current flows through the coil it
a restoring torque is developed in it, which tries to restore the coil to its original
position.
If ‘k’ is the restoring torque per unit twist and  be the twist produced in the
suspension wire.
Then,  r K
For equilibrium of the coil,    r
ie.
nIAB = K 
or
I= 
 K 

 nAB 
I = G
Where G =
or I  
I

=
 k 

 is a constant for a given galvanometer called
 nAB 
galvanometer constant. Thus deflection of the coil is directly proportional to the
current flowing through it.  can be measured by lamp and scale arrangement.
Sensitivity of a galvanometer :
A Galvanometer is said to be sensitive if it produces a large deflection even when
a small current flows through it .
Current sensitivity : It is defined as the deflection produced in the galvanometer
when a unit current flows through it.
Current sensitivity Is=

I

nAB
K
Page : 15
Sensitivity of a galvanometer can be increased by (i) increasing the number of turns
(n) (2) increasing the area (A) of the coil (3) increasing the magnetic field(B) and by
(4) decreasing the restoring torque per unit twist(K).
Voltage Sensitivity : It is defined as the deflection produced in the galvanometer
when a unit potential difference is applied across its ends.
Voltage sensitivity Vs =

V


IR

nAB current sensitivity

KR
Re sis tan ce
Advantages of moving coil galvanometer :
1) It can be made extremely sensitive so that even a minute current in the circuit can
be detected.
2) Since   I , the galvanometer can be used linear scale
3) As the magnetic field is very strong, its deflection is not affected by external
magnetic field.
4) As the coil is wound over a metallic frame, the eddy current produced in the
frame bring the coil to rest quickly.
Disadvantage of MCG :
1) Its sensitive ness cannot be changed at will
2) It is not portable and very delicate
3) It is not suitable for measuring a large current
4) Current is not obtained directly. It can be calculated by measuring the
displacement of the image using lamp and scale arrangement.
Pointer Galvanometer :
In order to overcome the disadvantage of moving coil
galvanometer some modifications are necessary. Instead of being
suspension the coil is pivoted between two jeweled bearings.
Two spiral springs are attached to the coil at its ends which
provides the necessary restoring torque. A large pointer is attached to
the coil moves over a graduated scale and indicates current. The whole arrangement
is enclosed in an ebonite case with a glass window. This is known as pointer
galvanometer and is compact and portable.
Conversion of Galvanometer into an ammeter :
An ammeter is a device used to measure current in the
circuit. It is always connected in series in the circuit So that the
current which is to be measured actually passes through it. Its
introduction in the circuit does not alter the current. So ammeter
is designed to have a low resistance. The resistance of an ideal
ammeter is zero.
Since the galvanometer has an appreciable resistance. Hence its effective
resistance should be made very small. This is done by connecting a low resistance(s)
called shunt in parallel with the galvanometer. With this modification the
galvanometer is converted into an ammeter.
Page : 16
Let I be the main current Ig be the current required to produce full scale
deflection in the galvanometer of resistance G. (I-Ig) be the current through S. The
value of S to be connected in parallel depends upon the maximum current to be
measured.
Since G and S are in parallel
p.d across S= p.d across G.
(I-Ig)S = Ig G
S 
I gG
(I  I g )
Also Ig =
IS
or Ig  I
GS
So the scale can be graduated to read the value of current I directly.
Conversion of a Galvanometer into a voltmeter :
A Voltmeter is an instrument used to measure
potential difference across any two points in a circuit. It is
always connected in parallel to the circuit element across
which the potential difference is to be measured. Its
introduction in the circuit does not alter the p.d and hence it
should not draw more current. There fore voltmeter should be designed to have a
very high resistance. The resistance of an ideal voltmeter is infinity.
A Galvanometer has certain resistance. Its effective resistance is increased by
connecting high resistance in series with it. With this modification the galvanometer
is converted into voltmeter.
Let Ig be the current required to produce full scale deflection in the
galvanometer of resistance G. The value of R to be connected in series depends upon
the maximum p.d to be measured.
By ohm’s law , V= Ig(G+R)
R =
Also Ig =
V
G
Ig
V
GR
 G and R are constants for a given voltmeter, then Ig  V Hence the scale can be
graduated to read the value of p.d directly.
Current loop as a magnetic dipole :
We know that magnetic field produced at a large distance x from the centre of
the circular loop along its axis is given by
B=
 0 2Ir 2
.
4
x3
or
B=
 0 2 IA
.
-----(1)
4 x 3
Where A =  r 2 is its area.
Page : 17
On the other hand the electric field of an electric dipole at a point on the axial
line far away from it is given by
E=
1 2p
----(2)
4 0 x 3
Where P is electric dipole moment of the electric dipole.
On comparing equations (1) and (2), we note that E and B are distant dependent
 1 
 3  and they have the same direction. This suggests that a circular current loop
x 


behaves as a magnetic dipole of magnetic moment m = IA In vector form m  I A .
Thus “the magnetic dipole moment of any current loop is equal to the product of
the current and the area of the loop.”
The direction of the dipole moment is given by right hand thumb rule.
“If we curl the fingers of the right hand in the direction of
current in the loop the extended thumb gives the direction of
magnetic moment associated with the loop.”
Since the upper face of the current loop behaves as N-pole
and the lower face as S-pole. Then a current loop behaves like
magnetic dipole. The magnetic moment acts perpendicular to the plane of the loop in
the upward direction.
If the current carrying circular loop has ‘n’ turns then m=nIA.
The factor nI is called ampere turns of current loop.
Magnetic moment of current loop = Ampere turns  area of the loop.
SI unit of magnetic dipole moment is Am2.
Magnetic dipole moment of a revolving electron :
In an atom, negatively charged electrons revolve around
positively charged nucleus in circular orbits. These circular
orbits of electrons may be considered as small current loops.
Due to this an atom possesses magnetic dipole moment and
hence behaves as a magnetic dipole.
Consider an electron of mass me and charge ‘e’ revolves in
a circular orbit of radius ‘r’ around the nucleus in anticlockwise
direction. Let V be the velocity and T is its time period.
The orbital motion of electron is equivalent to a current
I =
ch arg e e
e
 
time
T 2 r

v
ev
2r
Area` of electron orbit ( area of current loop) A =  r 2
 Orbital magnetic moment of the electron is M = IA
ie. M =
eV
evr
 r 2 =
------ ( 1)
2r
2
Page : 18
As the negatively charged electron is revolving anticlockwise, the associated
current flows in clockwise direction. According to right hand thumb rule, the
direction of magnetic dipole moment of the revolving electron is  r to the plane of
its orbit and in the downward direction.
The angular momentum of the electron due to its orbital motion is given by
L = meVr –-----(2)

The direction of L is normal to the plane of the electron orbit and in the upward
direction.
Dividing equation (1) by eqn. (2) we get.
evr
M
2 e

L me vr 2me
The above ratio is called gyromagnetic ratio. It is a constant and its value is 8.8  1010
c
kg
 e 
 L
 M  
 2me 

 e 

 L
Vectorially, M   
2
m
 e


The negative sign shows that direction of L is opposite to that of M .
According to Bohr’s quantization condition, the angular momentum of an
electron in a stationary orbit is equal to integral multiple of
h
nh
ie L 
where
2
2
n=1,2,3, …..
 e  nh

 M  
2
m
 e  2
 eh 

or M  n
 4me 
This equation gives the orbital magnetic moment of an electron revolving in nth
orbit.
Bohr magneton :
It is defined as magnetic moment associated with an electron due to its orbital
motion in the first orbit of hydrogen atom.
 eh 
 is the least value of magnetic
From the above equation it follows that 
4

m
e 

moment. This natural unit of magnetic moment is called Bohr magneton. It is
denoted by  B .
B 
eh
4me
1.6 10 c6.63  10 Js  9.27 10
4  3.14  9.1 10 kg
19
=
34
31
24
Am2
In addition to orbital motion, electron has got spin motion also. The magnetic
moment possessed by an electron due to its spin motion is called spin magnetic
moment.
Page : 19
The total magnetic moment of the electron is the vector sum of these two
momenta.
Page : 20