Upload
sayan-kumar-khan
View
222
Download
0
Embed Size (px)
Citation preview
7/27/2019 4056
1/36
Module 3: Electromagnetism
Lecture - Magnetic Field
Objectives
In this lecture you will learn the following
Electric current is the source of magnetic field.
When a charged particle is placed in an electromagnetic field, it experiences a force, calledLorentz force.
Motion of a charged particle in a magnetic field and concept of cyclotron frequency.
Find the trajectory of a charged particle in crossed electric and magnetic field.
MAGNETIC FIELD
Electric charges are source of electric fields. An electric field exerts force on an electric charge, whether
the charge happens to be moving or at rest.
One could similarly think of a magnetic charge as being the source of a magnetic field. However, isolated
magnetic charge ( or magnetic monopoles) have never been found to exist. Magnetic poles always occur
in pairs ( dipoles) - a north pole and a south pole. Thus, the region around a bar magnet is a magnetic
field. What characterizes a magnetic field is the qualitative nature of the force that it exerts on an
electric charge. The field does not exert any force on a static charge. However, if the charge happens to
be moving (excepting in a direction parallel to the direction of the field) it experiences a force in the
magnetic field.
It is not necessary to invoke the presence of magnetic poles to discuss the source of magnetic field.
Experiments by Oersted showed that a magnetic needle gets deflected in the region around a current
carrying conductor. The direction of deflection is shown in the figure below
mywbut.com
1
7/27/2019 4056
2/36
Thus a current carrying conductor is the source of a magnetic field. In fact, a magnetic dipole can be
considered as a closed current loop.
Electric Current and Current Density
Electric current is the rate of flow of charges in electrical conductor. In a conductor the charges may
have random motion. However, the net drift velocity of the charges is zero, giving a zero net current. In
the presence of an external force field, the charges move with a net non-zero drift velocity, which gives
a current.
The direction of current has been defined, conventionally, as the direction in which the positive charges
move. In case of metallic conductors, the current is caused by flow of negatively charged electrons,
whose direction of motion is opposite to the direction of current. In electrolytes, however, the current is
due to flow of both positive and negative ions.
The current density at a position is defined as the amount of charge crossing a unit cross-
sectional area per unit time. In terms of the net drift velocity of the charges (taken opposite to the net
drift velocity of electrons), ,
where is the volume density of the mobile charges.
The integral of the current density over a surface defines electric current, which is a scalar.
The unit of electric current is Ampere (= coulomb/sec) and that of the current density is A/m . For a
thin wire with a small cross sectional area, the current density may be taken as uniform. In this
case,
where is the component of the area vector parallel to the direction of current.
mywbut.com
2
7/27/2019 4056
3/36
Lecture - Electric Current and Current Density
The Right Hand Rule
The direction of magnetic field due to a current carrying conductor is given by the right handrule.
If one clasps the conductor with one's right hand in such a way that the thumb points in thedirection of the current (i.e. in the direction opposite to the direction of electron flow) then, thedirection in which the fingers curl gives the direction of the magnetic field due to such aconductor.
The Lorentz Force
We know that an electric field exerts a force on a charge . In the presence of a
magnetic field , a charge experiences an additional force
where is the velocity of the charge. Note that
There is no force on a charge at rest.
A force is exerted on the charge only if there is a component of the magnetic fieldperpendicular to the direction of the velocity, i.e.the component of the magnetic field
parallel to does not contribute to .
mywbut.com
3
7/27/2019 4056
4/36
, which shows that the magnetic force does not do any work.
In the case where both and are present, the force on the charge is given by
This is called Lorentz force after H.E. Lorentz who postulated the relationship. It may be noted
that the force expression is valid even when and are time dependent
Unit of Magnetic Field
From the Lorentz equation, it may be seen that the unit of magnetic field is Newton-second/coulomb-meter, which is known as a Tesla (T). (The unit is occasionally written as
Weber/m as the unit of magnetic flux is known as Weber). However, Tesla is a very large unit
and it is common to measure in terms of a smaller unit called Gauss,
It may be noted that is also referred to as magnetic field of induction or simply as theinduction field. However, we will use the term ``magnetic field".
Motion of a Charged Particle in a Uniform Magnetic Field
Let the direction of the magnetic field be taken to be z- direction,
we can write the force on the particle to be
The problem can be looked at qualitatively as follows. We can resolve the motion of the chargedparticle into two components, one parallel to the magnetic field and the other perpendicular to it.Since the motion parallel to the magnetic field is not affected, the velocity component in the z-direction remains constant.
mywbut.com
4
7/27/2019 4056
5/36
where is the initial velocity of the particle. Let us denote the velocity component
perpendicular to the direction of the magnetic field by . Since the force (and hence theacceleration) is perpendicular to the direction of velocity, the motion in a plane perpendicular to
is a circle. The centripetal force necessary to sustain the circular motion is provided by theLorentz force
where the radius of the circle is called the Larmor radius, and is given by
The time taken by the particle to complete one revolution is
The cyclotron frequency is given by
mywbut.com
5
7/27/2019 4056
6/36
Motion in a crossed electric and magnetic fields
The force on the charged particle in the presence of both electric and magnetic fields is given by
Let the electric and magnetic fields be at right angle to each other, so that,
If the particle is initially at rest no magnetic force acts on the particle. As the electric field exerts
a force on the particle, it acquires a velocity in the direction of . The magnetic force now actssidewise on the particle.
For a quantitative analysis of the motion, let be taken along the x-direction and along z-
direction. As there is no component of the force along the z-direction, the velocity of the particleremains zero in this direction. The motion, therefore, takes place in x-y plane. The equations ofmotion are
(1)
(2)
As in the earlier case, we can solve the equations by differentiating one of the equations andsubstituting the other,
which, as before, has the solution
with . Substituting this solution into the equation for , we get
mywbut.com
6
7/27/2019 4056
7/36
Since at , the constant , so that
The constant may be determined by substituting the solutions in eqn. (1) which gives
Since the equation above is valid for all times, the constant terms on the right must cancel, which
gives . Thus we have
The equation to the trajectory is obtained by integrating the above equation and determining theconstant of integration from the initial position (taken to be at the origin),
The equation to the trajectory is
which represents a circle of radius
mywbut.com
7
7/27/2019 4056
8/36
whose centre travels along the negative y direction with a constant speed
The trajectory resembles that of a point on the circumference of a wheel of radius , rolling
down the y-axis without slipping with a speed . The trajectory is known as a cycloid.
mywbut.com
8
7/27/2019 4056
9/36
Lecture - Force on a Current Carrying Conductor
Objectives
In this lecture you will learn the following
Calculate the force on a current carrying conductor in a magnetic field.
Find the torque on a current loop.
Define magnetic dipole and its magnetic moment.
Determine potential energy of a magnetic dipole
Force on a Current Carrying Conductor
A conductor has free electrons which can move in the presence of a field. Since a magnetic field exerts a
force on a charge moving with a velocity , it also exerts a force on a conductor carrying a
current.
Consider a conducting wire carrying a current . The current density at any point in the wire is given by'
where is the number density of electrons having a charge each and is the average drift velocity
at that point.
Consider a section of length of the wire. If is the cross sectional area of the section oriented
perpendicular to the direction of , the force on the electrons in this section is
where is the amount of charge in the section
mywbut.com
9
7/27/2019 4056
10/36
7/27/2019 4056
11/36
along the direction shown in the figure.
The plane of the loop (i.e. the normal to the loop) makes an angle to the direction of the field. Wetake the shorter sides PQ and RS (as well as the pivot axis OO') to be perpendicular to the field direction,
OO' being taken as the y-axis. The longer sides QR and QS make an angle with the field direction.
Since the force on a current segment is , the force is directed perpendicular to both and to
the direction of the current in these segments. The force has a magnitude and is directed
oppositely on the two sides. These forces are labelled and in the figure. (The forces are actuallydistributed along the lengths and the cancellation occurs for the forces acting on symmetrically placedelements on these two arms.)Further, since the lines of action of the forces acting on corresponding elements on these two sides arethe same, there is no torque.
The forces acting on the sides PQ and RS (labelled and respectively) are also equal and opposite
mywbut.com
11
7/27/2019 4056
12/36
and have magnitude . However, these forces do not act along the same line. The force on PQ acts
parallel to axis while that on RS acts parallel to axis. Note that axis is not in the plane of theloop. The situation can be better visualized by redrawing the figure in the plane containing one of the
longer sides and .
The current enters the branch SR at S marked with and reenters from QP at P, marked with
. The directions of the forces on the branches SR and QP are shown. The magnitude of thetorque about the pivot is
where is the area of the loop.
The result above is independent of the shape of the loop. The following example for a circular
loop gives an identical result.
Potential Energy of a Magnetic Dipole
A current loop does not experience a net force in a magnetic field. It however, experiences atorque. This is very similar to the behaviour of an electric dipole in an electric field. A current
mywbut.com
12
7/27/2019 4056
13/36
loop, therefore, behaves like a magnetic dipole.
We define the magnetic dipole moment of a current loop to be a vector of magnitude
and direction perpendicular to the plane of the loop (as determined by right hand rule). If the
loop has turns, . In a magnetic field, the dipole experiences a torque
The form of torque suggests that in a magnetic field the dipole tends to align parallel to the field.
If the orientation of the dipole is at some angle to the field, there must be some potential
energy stored in the dipole. This is because, if we wish to bring the dipole from to somearbitrary angle , we have to oppose the torque due to the field and do work in the process.
The work done is given by
This amount of work is stored as the additional potential energy of the dipole. In analogy withthe case of electric dipole in an electric field, the potential energy of the magnetic dipole in amagnetic field is given by
The energy is lowest when and are along the same direction and is maximum when theyare anti-parallel.
mywbut.com
13
7/27/2019 4056
14/36
Lecture - Biot- Savarts' Law
Objectives
In this lecture you will learn the followingStudy Biot-Savart's law
Calculate magnetic field of induction due to some simple current configurations.
Define magnetic moment of a current loop.
Find force between two current carrying conductors.
Biot- Savarts' Law
Biot-Savarts' law provides an expression for the magnetic field due to a current segment. The field
at a position due to a current segment is experimentally found to be perpendicular to and
. The magnitude of the field is
proportional to the length and to the current and to the sine of the angle betweenand .
inversely proportional to the square of the distance of the point P from the current element.Mathematically,
In SI units the constant of proportionality is , where is the permeability of the free space.
The value of is
The expression for field at a point P having a position vector with respect to the current element is
mywbut.com
14
7/27/2019 4056
15/36
For a conducting wire of arbitrary shape, the field is obtained by vectorially adding the contributions due
to such current elements as per superposition principle, where the integrationis along the path of the current flow.
mywbut.com
15
7/27/2019 4056
16/36
Lecture - Ampere's Law
Objectives
In this lecture you will learn the following
Establish Ampere' law in integral form. Calculate the magnetic field for certain current configuration using Ampere's law. Derive the differential form of Ampere's law.
Ampere's Law
Biot-Savart's law for magnetic field due to a current element is difficult to visualize physically as
such elements cannot be isolated from the circuit which they are part of. Andre Ampereformulated a law based on Oersted's as well as his own experimental studies. Ampere's law states
that `` the line integral of magnetic field around any closed path equals times the currentwhich threads the surface bounded by such closed path. . Mathematically,
In spite of its apparent simplicity, Ampere's law can be used to calculate magnetic field of acurrent distribution in cases where a lot of information exists on the behaviour of . The fieldmust have enough symmetry in space so as to enable us to express the left hand side of (1) in afunctional form. The simplest application of Ampere' s law consists of applying the law to thecase of an infinitely long straight and thin wire.
Ampere's Law in Differential Form
mywbut.com
16
7/27/2019 4056
17/36
We may express Ampere's law in a differential form by use of Stoke's theorem, according to which theline integral of a vector field is equal to the surface integral of the curl of the field,
The surface is any surface whose boundary is the closed path of integration of the line integral.
In terms of the current density , we have,
where is the total current through the surface . Thus, Ampere's law isequivalent to
which gives
mywbut.com
17
7/27/2019 4056
18/36
7/27/2019 4056
19/36
mywbut.com
19
7/27/2019 4056
20/36
Lecture - Vector Potential
Objectives
In this lecture you will learn the following
Define vector potential for a magnetic field. Understand why vector potential is defined in a gauge. Calculate vector potential for simple geometries. Define electromotive force and state Faraday's law of induction
Vector Potential
For the electric field case, we had seen that it is possible to define a scalar function called the
``potential" whose negative gradient is equal to the electric field : . The existence ofsuch a scalar function is a consequence of the conservative nature of the electric force. It also
followed that the electric field is irrotational, i.e. .For the magnetic field, Ampere's law gives a non-zero curl
Since the curl of a gradient is always zero, we cannot express as a gradient of a scalarfunction as it would then violate Ampere's law.
However, we may introduce a vector function such that
This would automatically satisfy since divergence of a curl is zero. is knownas vector potential . Recall that a vector field is uniquely determined by specifying its divergence
and curl. As is a physical quantity, curl of is also so. However, the divergence of thevector potential has no physical meaning and consequently we are at liberty to specify its
mywbut.com
20
7/27/2019 4056
21/36
divergence as per our wish. This freedom to choose a vector potential whose curl is andwhose divergence can be conveniently chosen is called by mathematicians as a choice of agauge
. If is a scalar function any transformation of the type
gives the same magnetic field as curl of a gradient is identically zero. The transformation above
is known as gauge invariance . (we have a similar freedom for the scalar potential of theelectric field in the sense that it is determined up to an additive constant. Our most common
choice of is one for which at infinite distances.)
A popular gauge choice for is one in which
which is known as the ``Coulomb gauge". It can be shown that such a choice can always bemade.
Biot-Savart's Law for Vector Potential
mywbut.com
21
7/27/2019 4056
22/36
Biot-Savart's law for magnetic field due to a current element
may be used to obtain an expression for the vector potential. Since the element does not depend on theposition vector of the point at which the magnetic field is calculated, we can write
the change in sign is because .
Thus the contribution to the vector potential from the element is
The expression is to be integrated over the path of the current to get the vector potential for the system
The existence of a vector potential whose curl gives the magnetic field directly gives
as the divergence of a curl is zero. The vector identity
can be used to express Ampere's law in terms of vector potential. Using a Coulomb gauge in
which , the Ampere's law is equivalent to
which is actually a set of three equations for the components of , viz.,
mywbut.com
22
7/27/2019 4056
23/36
which are Poisson's equations.
Electromagnetic Induction :
We have seen that studies made by Oersted, Biot-Savart and Ampere showed that an electriccurrent produces a magnetic field. Michael Faraday wanted to explore if this phenomenon isreversible in the sense whether a magnetic field could be source for a current in a conductor.
However, no current was found when a conductor was placed in a magnetic field. Faraday and(Joseph) Henry, however, found that if a current loop was placed in a time varying magneticfield or if there was a relative motion between a magnet and the loop a transient current wasestablished in the conducting loop. They concluded that the source of the electromotive forcedriving the current in the conductor is not the magnetic field but the changing magnetic fluxassociated with the loop. The change in flux could be effected by (i) a time varying magneticfield or by (ii) motion of the conductor in a magnetic field or (iii) by a combined action of bothof these. The discovery is a spectacular milestone in the sense that it led to importantdevelopments in Electrical engineering like invention of transformer, alternator and generator.Shortly after Faraday's discovery, Heinrich Lenz found that the direction of the induced currentis such that it opposes the very cause that produced the induced current (i.e. the magnetic field
associated with the induced current opposes the change in the magnetic flux which caused theinduced current in the first place). Lenz's law is illustrated in the following.
In the figures the loops are perpendicular to the plane of the page. The direction of inducedcurrent is as seen towards the loop from the right. Note that the magnetic field set up by the
mywbut.com
23
7/27/2019 4056
24/36
induced current tends to increase the flux in the case where the magnet is moving away from theloop and tends to decrease it in the case where it is moving towards the loop,Mathematically, Faraday's law is stated thus : the electromotive force is proportional to the rateof change of magnetic flux. In SI units, the constant of proportionality is unity.
where is the flux associated with the circuit and the minus sign is a reminder of the direction
of the current as given by Lenz's law. If the loop contains turns, the equation becomes
Though the flux is a scalar, one can fix its sign by considering the sign of the area vector which
is fixed by the usual right hand rule. The dot product of and then has a sign.
mywbut.com
24
7/27/2019 4056
25/36
mywbut.com
25
7/27/2019 4056
26/36
Lecture - Electromagnetic Induction
Objectives
In this lecture you will learn the following
Study in detail the principle of electromagnetic induction. Understand the significance of Lenz's law. Solve problems involving motional emf.
Motional Emf :
Consider a straight conductor AB moving along the positive x-direction with a uniform speed .
The region is in a uniform magnetic field pointing into the plane of the page, i.e. indirection.
mywbut.com
26
7/27/2019 4056
27/36
The fixed positive ions in the conductor are immobile. However, the negatively charged
electrons experience a Lorentz force , i.e. a force along the
direction. This pushes the electrons from the end A to the end B, making the former positivewith respect to the latter. Thus an induced electric field is established in the conductor along the
positive direction. The acceleration of electrons would stop when the electric field is built to a
strength which is strong enough to annul the magnetic force. This electric field isthe origin of what is known as motional emf . The motion of charges finally stops due to theresistance of the conductorIf the conductor slides along a stationary U- shaped conductor, the electrons find a path and acurrent is established in the circuit. The moving conductor thereby becomes a seat of themotional emf. We may calculate the emf either by considering the work that an external agency
has to do to keep the sliding conductor move with a uniform velocity or by direct application ofFaraday's law.
If the induced current is , a force acts on the wire in the negative x direction. In order tomaintain the uniform velocity, an external agent has to exert an equal and opposite force on the
sliding conductor. Since the distance moved in time is , the work done by the externalagency is
where is the amount of charge moved by the seat of emf along the direction of thecurrent. The emf is an electric potential difference.
Thus, the emf is equivalently the work done in moving a unit charge. Thus,
This emf corresponds to the potential difference beteen the ends A and B.An alternate derivation of the above is to consier the flux linked with closed circuit. Taking the origin
at the extreme left end of the circuit, the area of the circuit in the magnetic field is where is thedistance of the sliding rod from the fixed end. The flux linked with the circuit is, therefore,
. The rate at which flux changes is therefore given by
mywbut.com
27
7/27/2019 4056
28/36
If is not perpendicular to the plane of the circuit, we will need to take the perpendicular
component of in the above formula.Note that, in the illustration above, the magnetic flux linked with the circuit is increasing with time inthe negative z direction. The direction of the induced current is, therefore, such that the magnetic fielddue to the current is along the positve z-direction, which will oppose increase of flux. Hence thecurrent, as seen from above, is in the anticlockwise direction.
mywbut.com
28
7/27/2019 4056
29/36
Lecture - Time Varying Field
Objectives
In this lecture you will learn the following
Relate time varying magnetic field with emf generated. Define mutual inductance and calculate it in simple cases. Define self inductance. Calculate energy stored in a magnetic field.
Time Varying Field
Even where there is no relative motion between an observer and a conductor, an emf (andconsequently an induced current for a closed conducting loop) may be induced if the magneticfield itself is varying with time as flux change may be effected by change in magnetic field withtime. In effect it implies that a changing magnetic field is equivalent to an electric field in whichan electric charge at rest experiences a force.
Consider, for example, a magnetic field whose direction is out of the page but whose
magnitude varies with time. The magnetic field fills a cylindrical region of space of radius .
Let the magnetic field be time varying and be given by
Since does not depend on the axial coordinate as well as the azimuthal angle , theelectric field is also independent of these quantities. Consider a coaxial circular path of radius
which encloses a time varying flux. By symmetry of the problem, the electric field at
every point of the cicular path must have the same magnitude and must be tangential to thecircle.
Thus the emf is given by
mywbut.com
29
7/27/2019 4056
30/36
By Faraday's law
Equating these, we get for ,
For , the flux is , so that
and the electric field foir is
Mutual Inductance
According to Faraday's law, a changing magnetic flux in a loop causes an emf to be generated in
that loop. Consider two stationary coils carrying current. The first coil has turn and carries a
mywbut.com
30
7/27/2019 4056
31/36
current . The second coil contains turns. The current in the first coil is the source of a
magnetic field in the region around the coil. The second loop encloses a flux
, where is the surface of one turn of the loop. If the current in
the first coil is varied, , and consequently will vary with time.
The variation of causes an emf to be developed in the second coil. Since is
proportional to , so is . The emf, which is the rate of change of flux is, therefore,
proportional to ,
where is a constant, called the mutual inductance of the two coils, which depends ongeometrical factors of the two loops, their relative orientation and the number of turns in eachcoil.
Analogously, we can argue that if the second loop carries a current which is varied with time,
it generates an induced emf in the first coil given by
mywbut.com
31
7/27/2019 4056
32/36
For instance, consider two concentric solenoids, the outer one having turns per unit length
and inner one with turns per unit length. The solenoids are wound over coaxial cylinders of
length each. If the current in the outer solenoid is , the field due to it is ,which is confined within the solenoid. The flux enclosed by the inner cylinder is
If the current in the outer solenoid varies with time, the emf in the inner solenoid is
so that
If, on the other hand, the current in the inner solenoid is varied, the field due to itwhich is non-zero only within the inner solenoid. The flux enclosed by the outer solenoid is, therefore,
If is varied, the emf in the outer solenoid is giving
One can see that
mywbut.com
32
7/27/2019 4056
33/36
This equality can be proved quite generally from Biot-Savart's law. Consider two circuits shownin the figure.
The field at , due to current in the loop (called theprimary ) is
where . We have seen that can be expressed in terms of a vector potential ,where
, by Biot-Savart's law
The flux enclosed by the second loop, (called thesecondary ) is
mywbut.com
33
7/27/2019 4056
34/36
Clearly,
It can be seen that the expression is symmetric between two loops. Hence we would get an
identical expression for . This expression is, however, of no significant use in obtaining themutual inductance because of rather difficult double integral.Thus a knowledge of mutual inductance enables us to determine, how large should be the changein the current (or voltage) in a primary circuit to obtain a desired value of current (or voltage) in
the secondary circuit. Since , we represent mutual inductance by the symbol .
The emf in the secondary circuit is given by , where is the variablecurrent in the primary circuit.
Units of is that of Volt-sec/Ampere which is known as Henry (h)
Self Inductance :
Even when there is a single circuit carrying a current, the magnetic field of the circuit links with the
circuit itself. If the current happens to be time varying, an emf will be generated in the circuit to oppose
the change in the flux linked with the circuit. The opposing voltage acts like a second voltage source
connected to the circuit. This implies that the primary source in the circuit has to do additional work to
overcome this back emf to establish the same current. The induced current has a direction determined
by Lenz's law.
If no ferromagnetic materials are present, the flux is proportional to the current. If the circuit contains
turns, Faraday's law gives
mywbut.com
34
7/27/2019 4056
35/36
where is known as Self Inductance of the circuit. By definition, is a positive quantity. From the
above it follows, on integrating,
Since when , the constant is zero and we get
The self inductance can, therefore, interpreted as the amount of flux linked with the circuit for unit
current. The emf is given by
Energy Stored in Magnetic Field
Just as capacitor stores electric energy, an inductor can store magnetic energy. To see this consider an L-R
circuit in which a current is established. If the switch is thrown to the position such that the battery gets
disconnected from the circuit at , the current in the circuit would decay. As the inductor providesback emf, the circuit is described by
With the initial condition , the solution of the above is
As the energy dissipated in the circuit in time is , the total energy dissipated from the time thebattery is disconnected is
mywbut.com
35
7/27/2019 4056
36/36
Thus the energy initially stored must have been . If an inductor carries a current , it stores
an energy . Thus the toroidal inductor discussed earlier stores an energy
when it carries a current . We eill now show that this is also equal to the volume integral of
Consider the magnetic field in the toroid at a distance from the axis. We have seen that the magnetic
field is given by . Thus the value of at this distance is .
Considering the toroid to consist of shells of surface area and thickness , the volume of the
shell is . The volume integral of is therefore,
which is exactly the expression for the stored energy derived earlier
mywbut.com