PPT No. 16
* Magnetic Flux
* Magnetic Induction/ Magnetic Flux Density B
* Lorentz Force
* Force between Current Elements
Magnetostatics
Electrostatics is the study of electric fields
which are static i.e. constant in time.
They are produced by stationary charges.
Magnetostatics is the study of Magnetic fields
which are static / stationary i.e. constant in time.
They are produced by steady or dc (direct) current.
Electric current
Electric current (I) is defined as the rate of flow of
electric charge (Q) through
any cross-sectional area of a conductor
i.e. I = Q/ t.
SI unit of current is Ampere.
Force due to a Current Element
Electric current can also be defined in terms of
the current density vector J as
the current passing through per unit cross sectional area
I = J. A
(A is the area vector normal to the area A)
SI unit of Current density is Ampere/ Meter2
Magnetic Field
A stationary electric charge produces only
an electric field E in the space surrounding it.
A moving charge, (e.g. in a current carrying wire),
in addition, produces a magnetic field.
This field, similar to the field of a bar or horseshoe magnet
can be detected experimentally by
the movement of the needle in a compass.
Magnetic Field
Magnetic field is the region around a magnet,
moving charges (i.e. electric current), or
changing electric field
in which Magnetic effects are observable i.e.
in which another permanent magnet or moving charge
experiences a side way deflecting (mechanical) force.
Some examples of sources of magnetic fields:
Bar/horseshoe magnet, the Earth, solenoid etc.
Magnetic Field
Magnetic Field
Ampere Model of Magnetism
Hans Oersted (1777-1851) was the first person
to observe the link
between electricity and magnetism (1820).
Andre Marie Ampère carried out
experiments on current carrying wires and
their mathematical analysis.
Thus the subjects of electricity and magnetism
were brought together and
electromagnetism emerged
Ampere Model of Magnetism
Magnetic fields are produced by electric currents &
can be due to
1) Macroscopic electric currents
(motion of charges) in wires or
2) Microscopic currents
associated with elementary particles (e.g. electrons)
spinning around themselves or
revolving in atomic orbits.
This approach is called as
the “Ampere Model of Magnetism”.
Magnetic Pole Approach
Another approach is called as
the “Magnetic Pole Approach”
It conceives magnetic field due to
a magnet as having
two distinct (North and South) magnetic poles.
e.g. A Bar or Permanent magnet
It is called as the Gilbert Model.
Magnetic Pole Approach
Though it is useful to explain some concepts of magnetism
(especially related to the bar/ permanent magnets)
at elementary level and convenient for
professional magneticians to design permanent magnets,
the Gilbert model fails
to explain many magnetic phenomena
which can be explained effectively by the Ampere model
(e.g. Absence of magnetic monopole, connection between
magnetic moment and angular momentum etc.).
Magnetic Induction or B-field
The magnetic field at a point is expressed
in terms of a quantity called
magnetic induction or the B-field
(to distinguish it from the H-field explained later).
It is denoted by the symbol B.
In electrostatics, the electric field at a point is defined
as the force experienced by
the unit positive charge placed at that point;
Force due to a Current Element
Magnetic field is defined in a manner similar
to that of the electric field.
Magnetic field B at a point is defined in terms of
the magnetic force experienced by
a moving test charge at that point i.e.
based on the effects B-field has on
its environment or surroundings.
Lorentz Force
An electric charge, q, moving with a velocity v,
through a point P situated in a magnetic field B
experiences a (deflecting) force, F.
Then the magnetic B-field is defined by the relation
It is called as the (magnetic component of)
the Lorentz Force
Lorentz Force
•This equation of Lorentz Force
•defines the strength and direction of
the magnetic B field at any point in terms of
1) a force proportional to the strength of the magnetic field,
2) the component of the velocity that is ┴
to the magnetic field and
3) the charge of the particle.
•The vector (cross) product implies that
The force is ┴ to both
the velocity v of the charge q and
the magnetic field B
Lorentz Force
The magnitude of the force is F = qvB sinθ where
Angle θ< 1800 between the velocity v and the magnetic field B.
•This implies that
the magnetic force on a stationary charge (v=0) or
a charge moving parallel to the magnetic field (θ=0) is zero.
The B-field can be defined alternatively in terms of
the torque on a magnetic dipole placed in a B-field.
(Explained later in Magnetic dipoles & Magnetic dipole moments).
The Unit of Magnetic Induction or B-field
The unit of magnetic B-field in SI system is called Tesla .
It is derived from magnetic part of the Lorentz force law,
Fmagnetic =Fm = qvB
A particle passing through a magnetic field of 1 tesla at
1 meter per second carrying a charge of 1 coulomb
experiences a force of 1 newton.
The Unit of Magnetic Induction or B-field
One tesla is equal to
the magnitude of the magnetic field vector
necessary to produce a force of one newton
on a charge of one coulomb
moving perpendicular to the direction of the magnetic field vector
with a velocity of one meter per second.
Therefore the unit is Newton seconds /(Coulomb meter), or
Newton/ Ampere meter.
The Unit of Magnetic Induction or B-field
Tesla is equivalent to one Weber per square meter (Old unit).
Weber is the unit of magnetic flux.
The tesla (T) can also be expressed as
V = volt, s = second , m = meter
Wb = weber, m = meter
N = newton, A = ampere, m = meter
kg = kilogram, C = coulomb, s = second
kg = kilogram, A = ampere, s = second
The Unit of Magnetic Induction or B-field
,
Tesla
is used in the work involving strong magnetic fields
Smaller unit of magnetic field is the Gauss
(1 Tesla = 10,000 Gauss).
The gauss is more useful with small magnets or
for small fields like the Earth's magnetic field.
1 Tesla is equivalent to:
10,000 (or 104) gauss (G), used in the CGS system.
Thus, 10 G = 1 mT (1 millitesla) and 1 G = 10-4 T.
1,000,000,000 (or 109) gammas (γ), used in geophysics
1 γ = 1 nT (nanotesla)
The Unit of Magnetic Induction or B-field
For those concerned with
low-frequency electromagnetic radiation in the home,
the following conversions are needed often:
1000 nanotesla = 1 microtesla = 10 milligauss (mG)
1,000,000 microtesla = 1 Tesla
Tesla is named after Nikola Tesla(1856–1943),
who was a Serbian physicist and an electrical engineer,
inventor of the radio and the electric motor and generator.
Magnetic Induction or B-field
For studying the magnetism in matter,
the concept of H-field needs to be introduced.
According to modern understanding magnetic B-field
is considered to be a fundamental entity and
H-field as Auxiliary field
(though historically H-field was developed first).
Magnetic Flux and Magnetic Flux Density
Magnetic fields are commonly represented by
continuous lines of force, that emerge from
North magnetic pole and enter South pole
(outside of magnet)
from the South to the North pole of a magnet
(within the magnet).
They are in the form of circular closed loops
around a current carrying wire,
(their direction given by right hand rule).
They constitute the magnetic flux and are denoted by φ.
Magnetic Field due to a Current carrying Wire
The magnetic field B due to a straight wire carrying current I
circles the wire and its magnitude (or strength) decreases
with radial distance r from the wire.
Magnetic Flux and Magnetic Flux Density
The magnetic flux φ through a surface is
proportional to the areal density i.e.
Net number of Magnetic field lines through the surface.
The direction of lines corresponds to the direction of B.
Magnetic flux φ is given by
the product of the average Magnetic field B times
the perpendicular area A that it penetrates.
Magnetic Flux
Fig. Magnetic Flux φ- Curved surface split into
elements for taking surface integral
Magnetic Flux
If the area is not perpendicular to the flux,
then the component of area perpendicular to flux
is to be taken into account i.e.
Magnetic Flux is given by the scalar product of
the magnetic field and the area element vector
Magnetic Flux
If the area is a curved surface then
the surface is split into small surface elements.
Each element is associated with vector dS
of magnitude equal to the area of the element and
with direction normal to the element and pointing outward.
The Magnetic Flux
The magnetic flux
through a surface S is defined as
the integral of the magnetic field B
over the area of the surface:
where dS is an infinitesimal vectorits direction is the surface normal
Magnetic Flux Density
Due to this relation between B and
the magnetic field B is also called “Magnetic flux density".
The density of the lines indicates the magnitude of the field.
The lines are crowded together if the magnetic field is strong
The strength of the flux is determined by the number of
magnetic domains that are aligned within a material.
Magnetic Flux Density
The unit of the magnetic flux is
tesla meter2 (T·m2)
also called the Weber symbolized as Wb.
The older unit for the magnetic flux was
the maxwell (equivalent to 10-8 Wb).
Lorentz force
The Lorentz Force is the Force on a point charge
due to Electromagnetic fields.
It is given in terms of the Electric and Magnetic fields as
The total force F (called as Lorentz Force) on
a charge q moving with velocity v in an electric field E and
magnetic field B is given by the Lorentz Force Law
Lorentz force
Lorentz Force Law
(FE = Electric Force, FM = Magnetic Force)
Lorentz force
Electric Force FE of magnitude qE is
parallel to the local electric field.
Lorentz force
Magnetic Force FM of magnitude qvBsinө
┴ to both v and B away from the observer
Lorentz force
Directions of B, v, F
Lorentz force
The electric force on a charged particle is
parallel to the local electric field.
It is in the direction of the electric field
if the charge q is positive.
however,
The magnetic force is perpendicular to both
the local magnetic field and the particle's direction of motion,
given by the Right Hand Rule.
No magnetic force is exerted on
a stationary charged particle.
Biot Savart’s Law
Right Hand Rule for conductors
Right Hand Rule for conductors:
.
The direction of the flow of
current i.e. Positive charges
is indicated by
the Direction in which the Thumb points
while the Direction of the Magnetic field is
indicated by the Curl of
the fingers of the Right Hand
Right Hand Rule for conductors
Right Hand Rule can be used
to find the direction of the current or
the direction of the magnetic field
due tp the flow of current in a conductor
if one of the factors are known.
Lorentz Force Law as the Definition of E and B
If the electromagnetic force F,
a test charge would experience is expressed as
a certain function of its charge q and velocity v as follows
then the electric and magnetic fields E and B respectively
are said to be defined by the equation of Lorentz Force Law.
Lorentz Force Law as the Definition of E and B
The electric field E is 1 volt/meter if
a charge of 1 coulomb
experiences a force of 1 newton
The magnetic field induction B is 1 tesla if
a charge of 1 coulomb moving at right angle (i.e. θ =900)
to the magnetic field with a velocity of 1 meter /sec
experiences a force of 1 newton at the point
F = qvB sinθ
Lorentz Force Law as the Definition of E and B
Since magnetic force and velocity are perpendicular,
the magnetic force does no work i.e.
Magnetic force cannot change the speed of the particle
and its kinetic energy remains constant while
a charged particle may gain (or lose) energy
from an electric field.
One interesting application is in particle accelerators, where
motion of charged particles is guided in a circle using
magnetic fields but they are accelerated by electric fields
Force due to a Current Element
Ampère carried out experiments on
long straight wire/s carrying electric current/s.
He detected the force exerted by wires
using compass or short test wire carrying a current.
He found that a current carrying wire exerts a force
on another current carrying wire parallel to it.
The direction of the force is reversed if
the direction of current is reversed.
Force due to a Current Element
The experimental observations of Ampere
can be analyzed as follows
The motion of electrically charged particles (e.g. electrons)
constitutes a flow of an electric current in a conducting wire.
The force exerted on the current carrying wire placed
in a magnetic field (of another current carrying wire)
is actually the resultant of the forces
exerted on the moving charges in the wire.
Force due to a Current Element
Consider a straight conducting wire carrying a current I
having (uniform) cross-sectional area A and length L.
n = the number density (number per unit volume)
of electrons in the wire.
Each electron has charge q and velocity v.
According to the Lorentz Force the magnetic force f on a charge
(e.g. electron) in the presence of magnetic field B is
f =qvBsinθ
θ is the angle between magnetic field B and velocity vector v.
N, the total number of electrons is given by
N=nAl
Force due to a Current Element
dB is the magnetic field at P due to the current element “Iđℓ ”
Force due to a Current Element
Total magnetic force F on the wire of length "L"
is equal to sum of forces on an electron
F = ∑f = nALxf = nALqvBsinθ
Velocity v is given by
v = I / nqA
Substituting for v in the expression for total force:
F = nALqIBsinθ/ nqA
= ILBsinθ
F = ILxB (Using vector notation)
The direction of length vector is same as that of
the direction of current in wire.
Force due to a Current Element
Net magnetic force on the wire is the arithmetic sum of
individual magnetic forces on conduction electrons
Force due to a Current Element
In the case of a nonlinear or curved wire,
an infinitesimally small length of wire element đℓ
can be considered to be straight wire and
above equation for straight wire can be applied.
Magnetic Field due to a Current Element
Magnetic field due to small
thin current element
Force due to a Current Element
The magnetic force on
an infinitesimally small length đℓ of wire is:
F=I đℓ xB
In the case of a moving charge q,
the magnetic force is
F = q (v x B)
The magnetic force on
a small current carrying wire element
F = I đℓ xB
Force due to a Current Element
Magnetic field acts
perpendicular to
the plane formed by
current element and
displacement vectors
A Current Element
If đℓ is an element of a wire
carrying a current I,
which is due to the charge q
moving with velocity v
Then the term “Iđℓ ” plays the role
equivalent to “qv”
in the case of experiencing magnetic force.
It is called as the Current element