This document discusses electricity and magnetism. It provides information on the fundamental properties of magnets and magnetic fields. Some key points include: - Magnetism and electricity are two aspects of a single phenomenon related to the motion of electric charges. - Magnetic fields can be produced by electric currents in wires, as discovered by Oersted in 1820. - Magnetic induction B is defined based on the force experienced by a moving charge in a magnetic field. - Materials can be classified as ferromagnetic, paramagnetic, or diamagnetic based on their behavior in magnetic fields. Permeability and susceptibility quantify a material's response to magnetic fields.

1. ELECTRICITY & MAGENETISM- MODULE 2
• The subject of magnetism now occupies an important
place in the realm of science. The practical applications of
magnetism are now of immense service in everyday life
the dynamo, electric motor, telephone and other
electrical equipment etc.
• For the clear understanding of applications of this subject
it is necessary to know the fundamental properties of
magnets and magnetic fields
• The fact that a magnetic phenomenon a magnetic field
is observed around a wire carrying current is very familiar

2. • It is also an established fact that an electric current which
produces magnetic field is simply a motion of electric
charges
• Hence magnetism and electricity are two aspects of a
single phenomenon
• The Magnetic Field magnetic effect of the current:
• Let us now consider the magnetic effect of electric
current. Before the discovery of this effect the electricity
and magnetism were treated as separate branches .
• But in 1820 Orested discovered that a current in a wire
can also produce magnetic effect

3. h
• This was demonstrated that when a compass needle is
brought near a wire carrying current , then the compass
needle shows deflection
• We define space around the magnet or a current carrying
conductor as the site of a magnetic field , just as we
define an electric field. The basic magnetic field vector
Bˆ is called magnetic induction
• It is represented by lines of induction. Bˆ is related to its
lines of induction as follows:
• 1. The tangent to the line of induction at any point gives
the direction of Bˆ at that point

4. • 2. The lines of induction are drawn so that the no. of lines
per unit cross-sectional area is proportional to the
magnitude of the magnetic field vector Bˆ where the lines
are close together, B is strong and where they are far
apart , B is weak
• THE DEFINITION OF Bˆ
• We have defined electric field Eˆ in terms of force
experienced by a test charge qo , which depend on the
situation of the charge qo The fact that force on an
electric charge depends not only on its situation , but also
on the velocity of the charge is considered for defining Bˆ

5. • Let us bring a positive test charge with velocity 𝑣 at a
point P situated in magnetic field. The test charge
experiences a deflecting force as shown in fig.
• Let us bring a positive test charge with velocity 𝑣 at a
point P situated in magnetic field. The test charge
experiences a deflecting force as shown in fig.
• Let us bring a positive test charge with velocity 𝑣 at a point
P situated in magnetic field. The test charge experiences a
deflecting force as shown in fig.
• Let us bring a positive test charge with velocity 𝑣 at a point
P situated in magnetic field. The test charge experiences a
deflecting force as shown in fig.
• Let us bring a positive test charge with velocity 𝑣 at a point
P situated in magnetic field. The test charge experiences a
deflecting force as shown in fig.

6. • The magnetic induction or the magnetic field vector Bˆis
defined in terms of this force and other measurable
quantities . This magnetic force has a strange directional
character. At any particular point in space, both the
direction of the force and its magnitude depend on the
direction of motion of the particle.
• At every instant the force is always at right angles to the
velocity vector, also at any particular point, the force is
proportional to the component of the velocity at right
angles to this unique direction. It is possible to describe
all these behavior by defining the magnetic vector Bˆ as
follows

7. • If we vary the direction of 𝑣 through point P, keeping
the magnitude of 𝑣 unchanged, though Fˆ will always
remain at right angles to 𝑣 , its magnitude F changes and
for particular orientation the force becomes zero we
define this direction as the of Bˆ
• Now if the test charge is again oriented such that it
moves at right angles to this direction of Bˆ for which the
force is zero, then the magnitude of force is maximum,
say this force denoted by F⊥ and we define the
magnitude of Bˆ from the measured magnitude of this
maximum force F⊥

8. • ie, B = F⊥/q0 𝑣
• Or in general , we can write F⃗X q0 𝑣⃗ X B⃗
• This equation is known as Lorenz force law
• Bˆ and 𝑣ˆ are right angles, and magnitude of deflection
force is given by
F = q0 𝑣B Sin𝞡
F⃗ is right angles to the plane formed by 𝑣⃗ and B⃗
Magnitude of B = F/q𝑣 Sin𝞡
= Force on the charged particle/charge X comp. of velocity ⊥ to B

9. SI unit of B is called tesla (T)
• Put F=1N, q=1C, 𝑣 = 1ms -1 and 𝞡 = 90,
• Then B= 1N/ 1C X 1ms -1 X Sin 90
= 1 N A-1m-1
It is given a special name as weber/m2 or tesla(T)

10. Magnetic induction B at a point is said to be one tesla if a
charge of 1C moving with velocity of 1ms-1 at right angles
to the magnetic induction field B at that point,
experiences a force of 1N
The direction of F is perpendicular to the plane containing
𝑣 and B. if is given by Fleming`s left hand rule
The fore finger, the central finger and thumb of the left
hand are stretched in such a way that they are mutually at
right angles to each other. If the fore finger points in the
direction of magnetic field vector B, and the central finger
points in the direction of current or velocity vector 𝑣, then

11. The thumb gives the direction of the deflecting force F on
the charge ( as in figure)
The direction of force on the negative charge is opposite to
that on the positive charge

12. Magnetic Flux 𝜙
The magnetic flux 𝜙 through a surface is defined as
𝜙 = ∫s B.dS
It represents the lines of induction crossing the surface S
The SI unit of magnetic flux is weber (Wb)
If B is uniform and normal to the area A, then 𝜙 = B A
If A= 1m2, 𝜙 = B ie, the magnetic induction is numerically
equal to normal flux per unit area
Therefore it is also called magnetic flux density

13. Lorenz Force on a moving charge :
Suppose a charge q moves with a velocity v through a
region where both electric field E and magnetic field B
are present
Then the resultant force F on a moving charge is
F = q E + q (v X B) = q ( E + v X B )
This equation is called Lorentz force equation

14. • Classification of Magnetic materials:
• In 1845 Faraday concluded that all substances were to a
greater or lesser extent affected by a magnet , a few being
attracted , but most are repelled. This gave rise to the
grouping of different substances into (a). Ferromagnetic
like iron, which are strongly attracted, (b). Paramagnetic
feebly attracted, and (c). Diamagneticwhich are repelled
by a magnet
• The two poles (north and south)of a magnet , which of
equal strength, can not be separated from each other if
the magnet be broken into two halves or even sub-divided

15. Thus it is impossible to produce an isolated magnetic pole – in
fact it does not exist at all. If we have an extremely short thin
permanent magnet, with poles exactly at the ends , we call it a
magnetic dipole, the smallest unit – the unit which retains the
properties of a magnet  is called a domain.
It is important to remember that the existence of magnetic
poles and magnetic fields is due to the electrical nature of
matter consisting of electrons, protons and other electric
charges, moving relatively to each other. Thus magnetism is a
manifestation of moving electric charges.
For example one of the faces of a circular orbit of a circling
electron about the nucleus is a north and the other a south pole

16. Behavior of magnetic substances in a magnetic field:
Consider a piece of iron placed in a
magnetic field. The lines of force
appear to be sucked into the iron as
in fig. and have a tendency to crowd
into the iron as if they find it easier to go through it than
through air. It is however, found that every substance
possesses, roughly speaking, a certain power of conducting
the lines of forces through it,
This power of conducting the magnetic lines of force is
called permeability

17. Relative permeability is measured by the ratio of number of
lines of force per unit area in the medium to the number of
lines per unit area if the medium were replaced by vacuum
Air and other paramagnetic substances offer a resistance to
the passage of lines of force and are, therefore , said to
have a reluctance.
The permeability and reluctance of different substances are
different under different conditions
When a ferro or paramagnetic substance is placed in a
magnetic field, it behaves as a magnet . This process of its
becoming a magnet is called magnetic induction

18. Associated with it the poles produced are called induced poles.
This phenomenon is most pronounced in soft iron and is widely
used for producing intense magnetic fields , or for completely
screening a point from magnetic influence
Magnetic Intensity and Magnetic Induction:
Considering the atomic structure of matter suppose we have a
cylindrical bar of a magnetic substance of face area A, placed
inside a solenoid, a coil of insulated wire (round the cylindrical
magnetic material), through which a current 𝒊 is passed. A
magnetic field is produced threading through the material . Due
to this field the dipoles, or current loops, of the magnetic
material, align themselves in a particular direction

19. ie, they are polarized and produce surface or bound
currents. The surface current produces surface density of
magnetization Im . The end faces of the magnetic material
thus give rise to N-pole and S-pole each of strength ±m;
from N-pole arise lines of force and end on S-pole, making a
complete circle outside and then through the material as
if there is neither a beginning nor and end).
The no. of these lines of force per unit area = m/A = Im
Besides these magnetic lines of force inside the material
there are magnetic lines of force produced by the solenoid
coil,
the no.of which per unit area = H (the magnetizing field)

20. The magnetizing field due to free current in the coil and
outside the material.
Thus we have two sets of lines of force both in the same
direction through the material ie,
(1). due to the bound currents producing polarization of
magnetic diploes of the material resulting in Im and
(2).due to the solenoid producing the magnetizing field H;
The induction due to its magnetization produced in empty
space = µ0 Im
Hence total no.of lines of force per unit area (or lines of
induction)  µ0H + µ0 Im

21. This total number of lines per unit area is called magnetic
induction or flux density , is represented by B
Thus B = µ0H + µ0 Im
If the magnetic material is absent, then B0 = µ0H , because
there are no dipoles to be aligned
µ0 permeability of free space or space constant
and B = µ0 (H + Im)
The total no. of lines of force crossing any area at right
angles to the surface is called total induction or
magnetic flux , and is represented by 𝜙

22. B is the flux per unit area. If A is the area of the surface and B
is the induction , then 𝜙 = B. A
The unit of magnetic induction is called the gauss, and the unit
of magnetic flux is Maxwell; 1Maxwell is equal to 1line of
induction
Magnetic intensity and magnetic induction should not be
confused the former has to do with the property of the field
expresses in terms of force , while the latter with the condition
of the medium
Just as H describes the magnetic/magnetizing field at a point,
B describes the field at the point provided in this case the
presence of the medium is included in the expression for B in
the formulae

23. On the S I system of measurement the unit of flux is 1
weber, and the unit of magnetic flux density is 1Tesla ,
where 1weber = 108 maxwells ; 1 Tesla = 104 gauss
Im  weber/meter 2
H , Im , B are vector quantities
Since the intensity of the magnetic field at a point is
measured by the no. of lines of force crossing unit area
surrounding that point or by the magnetic flux round the
point, we find that:
H is the flux density before and B is the flux density after
the introduction of the material.

24. Each measures the force on the unit pole, the former in the
absence of the magnetic material, while latter inside the
magnetic material
The magnetic flux over a surface dS is usually written as
𝜙 = ∫ B cos 𝞡 dS
where B cos 𝞡 is the normal induction
In vector notation 𝜙 = ∫ B⃗. dS⃗
We can now define a line of induction :
Thus a line of induction is a curve drawn so that it
everywhere indicates the direction of induction

25. ie, of the resultant of the field causing the induced
magnetism and that of magnetization of the material
The lines of induction grouped together form tubes of
induction
Thus the lines and tubes of force refer to lines (and tubes)
which occupy a space if there were no magnetizable
medium in the field, while lines and tubes of induction refer
to resultant of magnetizing /inducing and the induced
fields inside the medium
In the case of a permanent magnet, since there is no
inducing field B = µ0 Im

26. The following few points are to be noted in connection with
H and B
1. H is the cause and B is the effect. Both measure the
intensity of magnetic field and are vectors
B includes the presence of a medium ie, (B=µH & B0=µ0H)
2. B and H refer to external fields and not to magnet`s own
field. A magnet can not exert a torque on itself.
In a magnetic field the torque may be due to
(a). Equivalent surface currents or (b) forces on poles of
the magnet

27. 3. The above magnetic phenomenon is analogous to
electrostatic phenomenon. Just as P, D, E in electrostatic
are related to bound charges, free charges, total charges
respectively, Im, H, B are related to bound currents,
free currents , all currents
Im, is analogous to dielectric polarisation P
4. In S I system B and H are measured in weber/m2 and
amp-turns respectively
5. Since magnetic lines of force are closed loops , the
magnetic field has no divergence
∮ B⃗ dA⃗ = 0 OR to say ∇. B⃗ =0

28. Magnetic susceptibility and permeability:
Im acquired by a magnetic substance depends on the
strength of magnetizing field, the nature and condition of
the substance.
If H = strength of the inducing field and Im is the resulting
intensity of magnetization, the ratio  Im/H measures the
magnetic susceptibility, denoted by k of the material
ie, k = Im/H or, Im = kH
Thus the susceptibility of a material is measured by the
intensity of magnetization induced in it by unit magnetizing
field

29. The magnetic susceptibility is a measure of the ease with
which the substance is magnetized, and varies with the
magnitude of magnetizing field.
The ratio of magnetic induction B produced in a material to
the magnetizing field H (in the absence of any magnetic
material) is called the absolute permeability  µ of the
material
ie, µ = B/H or, B = µ H = µr µo H
and in vacuum Bo = µo H in Teslas or weber/m2
where µ depends on the magnetizing force, nature and
condition of the material

30. The permeability of an isotropic homogeneous
medium is measured by the flux density
induced in it by unit magnetizing field.
µ is sometimes slightly less and sometimes
greater than unity ; based on which substances
in former cases are called diamagnetic and in
the latter case are called paramagnetic
In short the differences in the values of k and µ
lead us to classify substance into these three
groups

31. 1. Paramagnetic: substances are those for which µ >1 and k
has small +ve value. Examples are Plattinum, solutions of
salts of iron, oxygen, manganese, palladium and osmium
These have feeble magnetic properties.
For platinum, µ=1.00002 and k= 1.71X10-6
Paramagnetic substances move towards stronger parts of a
magnetic field ,or lines of force crowd towards such
substances. Hence they are attracted by a magnet
Also k not only decreases with increase of magnetic force,
but it also depends on temperature. Currie discovered that
the susceptibility of some paramagnetics varies inversely as
absolute temperature

32. 2. Ferromagnetic: substances are those which can be
magnetized to great extent and have abnormally high
positive value of k, eg, iron, steel, nickel, cobalt, alloys of
these substances and gadolinium
In these substances as in paramagnetism, magnetization is
not proportional to the magnetizing force , hence k and µ
vary with magnetizing force considerably. Also k varies with
temperature
When a ferromagnetic substance is heated, its k varies
inversely as the absolute temperature this is called Currie
law, and is expressed as kT = constant , where k is
susceptibility and T is absolute temperature

33. Thus k decreases steadily with rise of temperature , until
critical temp, called Curie point temperature is reached. At
this temperature ferromagnetism disappears and substance
become paramagnetic.
The susceptibility of a ferromagnetic substance above its
Curie point is inversely proportional to the amount its
temperature is above the Curie point  This is called Curie
Weiss law.
The Curie point is about 100 0C for cobalt, 400 0C for nickel,
and 770 0C for iron
A ferromagnetic substance has most pronounced magnetic
properties

34. 3. Diamagnetic: substances are those which µ <1, k is constant
and has –ve value. eg..bismuth, antimony, Zn, Ag, Cu, Sb,
Au,Pb, water, alcohol ,hydrogen and inert gases
When placed in a magnetic field , they have a tendency to move
away from the field. The lines within the space occupied by a
diamagnetic body are fewer than there would be , in this space,
in the absence of this body. Outside the body the lines of force
will be more closely packed than they would be in the absence
of the body, because outside the body the tubes of force due to
induced magnetism of diamagnetic substance are in the same
direction as the tubes of force in the field. Hence the polarity in
a diamagnetic substance is opposite to that created in a
paramagnetic substance

35. A diamagnetic medium reduces the flux density B much as
a dielectric reduces the electric field intensity E between
the plates of a condenser.
Im is also called magnetic polarization and is analogous to
dielectric polarization
HALL EFFECT:
We have learnt that a metallic conductor carrying current
(or a beam of electrons), placed in a magnetic field , is
deflected in a direction perpendicular to both to the
direction of current and to that of the of the field.

36. • Hall argued that if the conductor were fixed, slowly
drifting charged particles, constituting the current, should
be displaced sidewise within the conductor under the
action of a transverse magnetic field, and thus it should be
possible to find the number of charged particles.
• He , therefore passed a current along a strip of metal foil
and found that when a magnetic field , perpendicular to
the plane (and length) of the foil , was applied, a
measurable transverse e.m.f called  Hall voltage was
set up between the surfaces A and B , on the opposite
edges of the foil ( ref fig.)

37. fig. (a) fig. (b)
• A moving electron itself is surrounded by its own magnetic
field which is superimposed upon the external field ; thus
referring to fig. (a) the magnetic field above the electron
path is increased, and below the path is reduced resulting
in the motion of the electron in a curved path.

38. Hence, in the interior of the metal conductor, the
movement of electrons gives rise to a p.d or voltage
between the surfaces P and Q
Charges build up on the faces of the conductor as in fig. (a)
until the resulting electric field strength due to these
charges is large enough to oppose further drift of electrons.
When this happens a steady state is reached. In the steady
conduction the voltage is set up, between P and Q , is called
the Hall voltage or transverse Hall electric field.
Again referring to fig. (a) if the current is due to the motion
of electrons moving upwards, (ie in a direction opposite to
the conventional current), then by applying Flemings left

39. hand rule, to the conventional current as shown in fig.(b),
the force on the charge carriers is towards the edge Qand
the electrons are urged towards Q making it –ve , while the
edge A becomes +ve , which could be detected and
measured.
Hall thus concluded that the current in metal is the result
of –vely charged particles (electrons)
Hall also found that if the foil is a semi-conductor, in which
current id carried by both holes and electrons, depending on
whether it is p-type or n-type semi-conductor, the hole is
drift toward the edge P ; latter acquires +ve charge and Q
a –ve charge.

40. Thus the Hall effect for hole movement is opposite to that for
electron movement.
From the above discussion it follows that the Hall voltage is due
to transverse force on the carriers in the sample.
So calculation of Hall voltage;
let e = charge on carrier
𝑣a =vel. of drift of charge in the conductor
n = no. of charged particles per unit volume
B= magnetic field intensity / flux density
EH = electric field set up by Hall voltage between P and Q
I = strength of current in the foil

41. When the sideways magnetic deflecting force on the
charge carriers is just opposed by the Hall electric
field
ie, for steady state/ for equilibrium
eEH = Be𝑣, so EH = B𝑣
If d = thickness, l = length, and b= breadth of the
foil
VH = Hall voltage between P and Q on the two
surfaces
Therefore EH = VH /b = B𝑣

42. THE BIOT-SAVARAT LAW:
The first significant relationship between a current and its
magnetic field was discovered by Oerstead and in same time,
Biot and Savert formulated the equation for the field due to
current in a long straight wire. Finally they produced a
mathematical formula for field due to a single current element
rather than a finite length of wire , the equation can be used to
calculate the field due to many different current configurations
It was found experimentally that the magnetic induction
resulting from a charge q moving with velocity 𝑣⃗ at a distance r
away from the charge where r is a vector pointing from charge
to the point where the field being found, is related by

43. B 𝞪 q𝑣 Sin𝞡/ r2
where B is the magnitude of induction and 𝞡 the angle
between 𝑣⃗ and r⃗ as shown in fig.
The above equation can be written in terms of equality by
inserting a constant µo/4π , therefore B = (µo/4π)q𝑣 Sin𝞡/ r2

44. In fact , we are often interested not in the field of moving
charge , but in that of an element of current , as length dl
of the wire carrying current ί . We can easily set up an
equivalence with equation above
Suppose the cross-section of wire is A and 𝝆 is the charge
density of charge in wire . Assume this charge moves with a
velocity 𝑣
The charge crossing any cross-section per second = 𝝆 𝑣 A= ί
If we multiply both sides by dl, we get
𝝆 𝑣 Adl = ί dl
where 𝝆Adl is the total charge contained in the wire of
length dl, and so we can write q 𝑣 = ί dl

45. Consider a conductor carrying current ί as in fig. with an
element of length dl . P is a point at a distance r from the
midpoint of the element dl and𝞡 the angle between dl and r.

46. Then the magnetic induction dB at the point P is given by
dB = (µo/4π)q𝑣 Sin𝞡/ r2
Substituting for q 𝑣 = ί dl in this equation
dB = (µo/4π) ί dl Sin𝞡/ r2
Since rˆ is a unit vector along r , we can write
dB = (µo/4π) ί (dl X rˆ) / r2
The direction of dB is that of vector dl X rˆ
The total magnetic induction B at point P due to the
current flowing in entire length of the conductor is then
given by B = ∫ dB = (µo/4π) ∫ ί (dl X rˆ) / r2

47. In vacuum, B is related to H (magnetic field intensity) by the
formula B = µo H
where µo is a constant , called the permeability of free
space
Magnetic induction at a point due to a straight conductor
carrying current:
Consider a straight conductor XY carrying a current 𝒊 in the
direction Y to X as in fig. P is a point at a perpendicular
distance a from the conductor. Consider an element AB of
length dl. Let Bp =r and ⦟OBP = 𝞡

49. Magnetic induction at P due to the element AB = dB
dB = (µo/4π) ί dl Sin𝞡/ r2
From B, draw BC perpendicular to PA
Let ⦟OPB = 𝜙, ⦟BPA = d𝜙
Then BC = dl Sin 𝞡 = r d𝜙
So dB = (µo/4π) ί r d𝜙 / r2 = µo/4π) ί d𝜙 / r
In △OPB, Cos 𝜙 = a/r or r= a/ Cos 𝜙
Therefore dB = (µo/4π) ί Cos 𝜙 d𝜙 /a

50. The direction dB will be perpendicular to the plane
containing dl and r. it will be directed into the page at P as
shown by right hand rule
{right hand clasp rule: Clasp a conductor in the right hand with
thumb pointing in the direction of the current, then the direction of
bend of the rest of the fingers gives the direction of the magnetic
lines of force}

51. Let 𝜙1, 𝜙2 be angles made by the ends of the wire at P
Then magnetic at P due to the whole conductor is
B = - 𝜙1∫ 𝜙2 (µo/4π) ί Cos 𝜙 d𝜙 /a = (µo/4π) ί/a〔sin 𝜙〕 - 𝜙1
𝜙2
= (µo/4π) ί/a〔sin 𝜙2 – sin(- 𝜙1) 〕
= (µo/4π) ί/a 〔sin 𝜙2 + sin( 𝜙1) 〕
If the conductor is infinitely long , 𝜙1= 𝜙2= 90o
Then B = (µo/4π) ί/a 〔1 + 1〕 = (µo ί /2πa )
ie, the magnitude of B depends on 𝒊 and a
Or to say B 𝞪 1/a

52. The lines of B form concentric circles around the wire as
shown in fig. below

53. Magnetic Field at the centre of a Circular Coil carrying
current:
Consider a circular coil of one turn of radius r and centre O,
carrying current 𝒊 . To find the flux density or magnetic
induction of the magnetic field produced at the centre,
consider an element dl of the circular conductor as in fig.

54. According to Biot-Savart`s law,
dB = (µo/4π) ί dl Sin𝞡/ r2
Here, 𝞡= 90o Sin 90o = 1
dB = (µo/4π) ί dl / r2
The whole circular conductor is made up of a large no. of
such elements of length dl
Then the total magnetic induction at the centre due to the
entire coil is
B = 𝜮(µo/4π) ί dl / r2 = (µo𝒊/4πr2 ) 𝜮dl

55. But 𝜮dl = 2πr , so B = (µo𝒊/4πr2 ) 2πr
= µo𝒊/2r weber/ sq. metre
If the circular coil has n turns of mean radius r, the
magnetic induction of the field produced at the
centre is given by
B = n µo𝒊/2r
Here B is in Tesla, r in metres and l in amperes

56. AMPERE`S CIRCUTAL LAW:
Statement the line integral ∮ B.dl for a closed curve is
equal to µo times the net current 𝒊 through the area
bounded by the curve.
That is ∮ B.dl = µo 𝒊
where µo is the permeability constant
Proof:
Consider a long straight conductor carrying a current 𝒊
perpendicular to the page directed outwards as in fig.

58. According to Biot- Savart law , the magnitude of magnetic
induction at a distance r from it is given by B = µo 𝒊/2πr
At each point on this circle , B has a constant magnitude,
which is always tangential to the path of integration ,
points in the same direction as B
Thus ∮ B.dl = ∮ Bdl = B ∮ dl = (B) (2πr)
Here ∮ dl = 2πr is the circumference of the circle
Substituting the value of B,we get ∮ B.dl =(µo𝒊/2πr)(2πr)= µo𝒊
Thus the integral ∮ B.dl is µo times the current through the area
bounded by the circle. This is Ampere`s law

59. Differential form of Ampere`s Law:
Let j be the current density in an element dSof the surface
bounded by the closed path . Then total current
𝒊 = ∫s j .dS
Therefore ∮ B.dl = µo ∫s j .dS
Using Stokes theorem, we can write ∮ B.dl = ∫s curl B.dS
Hence ∫s curl B.dS = µo ∫s j .dS
So curl B = µo j
This is the differential form of Ampere`s Law

60. Divergence of Magnetic Field vector B:
Divergence of magnetic field vector B is defined as
the flux through a surface S enclosing a unit volume
Hence Div B = Flux/𝑣 limit 𝑣0 = ∫s B.dS/ 𝑣 limit 𝑣0
We know that the magnetic lines of induction are closed
curves. If we construct any closed surface in a magnetic
field, every line that enters this closed surface must also
leave it.
In other words, the net flux is equal to the net efflux of
lines of force. Thus the total normal magnetic flux over a
closed surface is zero

61. ∫s B.dS = 0
or div B =0 everywhere
comparison of Electrostatic and magnetic fields
Electrostatic field Magnetic field
1 Line integral for a closed path is zero
∮ E.dl =0
Line integral for a closed path is
given by ∮ B.dl = µo 𝒊
2 Curl E = 0 Curl B = µo j
3 div E = 𝝆/𝞊0 div B = 0

62. APPLICATIONS OF AMPERE`S LAW:
1. Magnetic induction due to long straight current carrying
wire  consider a long straight wire carrying a current I as
in figure

63. The magnetic lines of force are concentric circles centred on
the wire. Let P be joint at distance r from the wire
The magnetic field at P is required
Consider a circular path of radius r passing through P. By
symmetry , the value of magnetic field B is same at each
point on the circular path.
Consider a small element dl of a line of magnetic field at P
 B and dl are always directed along the same direction,
ie, the angle between B and dl is zero
line integral of B along the boundary of the circular path ,

64. given by ∮ B.dl = ∫B.dl Cos 0o = B ∫c dl = B
2πr
From Ampere`s theorem
∮ B.dl = µo X current enclosed by the path
B 2πr = µo I
Therefore B = (µo I)/ 2πr

65. 2. Magnetic field inside a long Solenoid:
Consider a long straight solenoid having n turns per unit
length.
Let 𝒊 be the current flowing in the solenoid. It is experimentally
noted that magnetic field outside the solenoid is very small in
comparison with the field inside. The lines of induction inside the
solenoid are straight and parallel as in fig.

66. Consider a closed path pqrs. The line integral of the
magnetic field B along the path pqrs is
∮pqrs B.dl = ∫pq B.dl + ∫qr B.dl + ∫rs B.dl + ∫sp B.dl
Let pq =l, for path pq, B and dl are in e same direction.
Therefore ∫pq B.dl = ∫ B.dl = Bl
For paths qr and sp , B and dl are mutually perpendicular .
Therefore ∫qr B.dl = ∫sp B.dl = ∫ B dl Cos 90o = 0
For paths rs, B = 0 (since field is zero outside solenoid)
Therefore ∫rs B.dl =0

67. So the 1st equation becomes,
∮pqrs B.dl = ∫pq B.dl = Bl
By ampere`s law,
∮ B. dl = µo X current enclosed by the path
Bl = µo (nl) 𝒊
ie, B = µo n 𝒊

68. 3. Magnetic Induction due to a toroid (endless solenoid)
Consider a toroid carrying a current 𝒊o as in figure below

69. Point P is within the toroid while the point Q is inside and point
R outside. By symmetry , the direction of B at any point is
tangential to a circle drawn through that point with same centre
as that of toroid . The magnitude of B on any point of such a
circle will be constant. Let us consider a point P within the
toroid
Let us draw a circle of radius r through it .
Applying Amperes law to this circle , we haveπ
∮ B. dl = µo 𝒊
where 𝒊 is the net current enclosed by the circle. Now
∮ B. dl = B(2πr)
And 𝒊 = N𝒊o , where N is the total no, of turns in the toroid

70. Therefore the 1st equation becomes
B (2πr) = µo N 𝒊o
Or B =(µo /2π) (N 𝒊o/r)
Thus the field B varies with r
If l be the mean circumference of the toroid, the l = 2πr, so that
B = = µo No /l
The field B at an inside point such as Q is zero, because there is no
current enclosed by the circle through Q
The field B at an outside point such as R is also zero because the net
amount of current enclosed in the circle through R will be zero. This
is because each turn of the winding passes twice through this area
enclosd by the , carrying equal currents in opposite diections,