Download THE UNIVERSE AND ENERGY The entire universe is composed of

THE UNIVERSE AND ENERGY
The entire universe is composed of Energy and Energy alone. However, this energy comes in two forms in the known
universe. (i) Matter and (ii) Electromagnetic Radiation. We all know what matter is… You and me, dogs, cats, birds, the
planets, the sun, the stars, asteroids, the earth, the bus, the car the table, the buildings and every material object that you
can think about is matter! They all have an energy stored in them given by the equation E = mc 2 where m is the mass of
the object that you are talking about and c the speed of light.
Electromagnetic Radiation is the second form of energy it happens as a result of the pulsing forward of oscillating Electric
and Magnetic Fields. Quantum Mechanics has now proved that waves sometimes behave like a particle and such a particle
that represents a wave is called a photon1. One photon has an energy given by E = hν which of course is the where 𝜈 is
the frequency of the wave and h is the Planck’s constant2.
TRANSVERSE NATURE OF ELECTROMAGNETIC RADIATION
As of now, we are going to deal with the wave properties of EM Radiations hence the name Wave Optics. The figure
below shows how an Electromagnetic wave propagates in space. In the figure, we see clearly that the electric field
vibrates along the x-y plane as 𝐸(𝑥, 𝑡) = 𝐸0 𝑠𝑖𝑛(𝑘𝑥 + 𝜔𝑡) and the Magnetic Field vibrates along the x-z plane 𝐵(𝑥, 𝑡) =
𝐵0 sin(𝑘𝑥 + 𝜔𝑡) and the wave itself propagates forward perpendicular to these two vibrations, along the x axis. Here,
𝐸(𝑥, 𝑡) and 𝐵(𝑥, 𝑡) are the instantaneous values of Electric Field and Magnetic Field3 at given values of x and t, E0 and B0
are the maximum values of 𝐸(𝑥, 𝑡) and 𝐵(𝑥, 𝑡), 𝜔 = 2𝜋𝑓 where f is frequency of oscillation and k is the wave number4
given 𝑘 = 2𝜋/𝜆. When the vibrating particles of the medium are perpendicular to the direction of propagation, such a
wave is called a Transverse wave. Our job now is to prove that Light is indeed a transverse wave.
1 Photon is a particle with zero rest mass consisting of a quantum of em radiation. The photon may also be regarded as a unit of energy hν
2 Remember that this is only Energy of one photon. The total Energy of the wave depends on its intensity. More the intensity, more the number of photons and
hence the total energy will be the sum of the individual energies of all the photons.
3 This representation of E(x,t) is just like putting f(x) showing that the value of Electric field E(x,t) is a function of the x space coordinate and time or in other words, E
depends on x and t. If you want the value of E at time say 𝑥 = 5 𝑚𝑒𝑡𝑒𝑟𝑠 𝑎𝑛𝑑 𝑡 = 2 𝑠𝑒𝑐𝑜𝑛𝑑𝑠, put it in the equation and we get the value E(5,2) = E0 sin(5k +
2ω). Similar is the case for B(x,t)
4 Wave Number is the number of waves in unit length times 2𝜋
1
qed_amk_gesXII1011
The Proof…
Let us consider a plane electromagnetic wave propagating along the X axis. The wave front will be a plane one parallel to
YZ plane (see figure below). For any position of the wave front, the electric and magnetic fields E and B will be zero to
the right of the wave front because the wave has not reached there yet, while to the left of it, it will depend on x (space –
where we are talking about) and t (time – when we are talking about) but not on y and z as we are considering a pane
wave. To establish the transverse nature of EM waves, we shall show that E and B are perpendicular to the direction of
propagation of the wave.
Consider
an
elementary
cuboid
ABCDPQRS with sides𝐴𝑃 = 𝑑𝑥, parallel
to the X axis, 𝐴𝐵 = 𝑑𝑦, parallel to the Y
axis, and 𝐵𝐶 = 𝑑𝑧, parallel to the Z axis.
Suppose the wave front reaches the face
PQRS at a given time, say t. The fields E
and B are zero to the right of PQRS
(because the wave has not reached there
yet) while to the left of PQRS, the values
depend only on x (because we have just
now define that we are looking at this
matter at a value of time = t). Since the
cuboid does not enclose any charge,
according to Gauss’ Law, for electricity,
the total electric flux through all its six
faces must be zero. Mathematically speaking we have,
∮ 𝑬 ∙ 𝑑𝑨 = 0
⟹
∫ 𝑬 ∙ 𝑑𝑨 +
𝐴𝐵𝐶𝐷
[(𝐵𝑎𝑐𝑘)
∫ 𝑬 ∙ 𝑑𝑨 +
𝑃𝑄𝑅𝑆
(𝐹𝑟𝑜𝑛𝑡)
∫ 𝑬 ∙ 𝑑𝑨 +
𝐴𝐵𝑄𝑃
(𝑅𝑖𝑔ℎ𝑡)
∫ 𝑬 ∙ 𝑑𝑨 +
𝐷𝐶𝑅𝑆
(𝐿𝑒𝑓𝑡)
∫
𝐴𝐷𝑆𝑃
(𝐵𝑜𝑡𝑡𝑜𝑚)
𝑬 ∙ 𝑑𝑨 + ∫ 𝑬 ∙ 𝑑𝑨 = 0
𝐵𝐶𝑅𝑄
(𝑇𝑜𝑝)
]
Since the value of E does not vary with y, the electric flux through the top (BCRQ) and bottom (ADSP) faces normal to
the Y axis are equal and opposite and hence the contribution of these two integrals in the above equation cancel each
other. Similarly, the flux through the two side faces ABQP (Right) and DCRS (Left) also cancel each other as the E does
not vary with the value of z.
What is the meaning of E does not depend on y and z?. Strictly speaking, the Electric Field Vector E must be written
as 𝑬(𝑥, 𝑦, 𝑧, 𝑡), telling us that the value of E is a function of space coordinates (x, y and z) and of the time coordinate t. If you
still don’t understand, remember that to specify the position of an object, we need 4 coordinates x, y, z, and t. Say for example,
how do we spot a plane flying somewhere above the surface of the Earth? We need the Latitude (x axis), Longitude (y axis),
Altitude (z axis) and also the time of the day (t axis). Hence we say that the position of that plane is a function of space (x, y
and z) and time (t). So we write, say, Position of Plane is 𝑃(𝑥, 𝑦, 𝑧, 𝑡).
Similarly, here we are talking about the instantaneous value of Electric Field of an Electromagnetic wave of frequency f given
by 𝐸(𝑥, 𝑦, 𝑧, 𝑡). But since the Wave is oscillating along the x axis, it is independent of y and z values and so reduces to 𝐸(𝑥, 𝑡).
Same is the case for 𝐵(𝑥, 𝑡)
Hence the equation reduces to the following…
∫ 𝑬 ∙ 𝑑𝑨 +
𝐴𝐵𝐶𝐷
∫ 𝑬 ∙ 𝑑𝑨 = 0
𝑃𝑄𝑅𝑆
[(𝐵𝑎𝑐𝑘)
]
(𝐹𝑟𝑜𝑛𝑡)
′
Let 𝐸𝑥 and 𝐸𝑥 be the values of the x component of E at the faces ABCD (Back) and PQRS (Front) respectively. Then we
will have,
2
qed_amk_gesXII1011
∫ 𝑬 ∙ 𝑑𝑨 = −[𝐸𝑥 (𝑑𝑦. 𝑑𝑧)]
∫ 𝑬 ∙ 𝑑𝑨 = +[𝐸𝑥′ (𝑑𝑦. 𝑑𝑧)]
&
𝐴𝐵𝐶𝐷
(𝐵𝑎𝑐𝑘)
𝑃𝑄𝑅𝑆
(𝐹𝑟𝑜𝑛𝑡)
The negative sign in the first equation only indicates that the E and dA are in opposite direction. From the above three
equations, we conclude that
∫ 𝑬 ∙ 𝑑𝑨 +
𝐴𝐵𝐶𝐷
[(𝐵𝑎𝑐𝑘)
∫ 𝑬 ∙ 𝑑𝑨 = 0
𝑃𝑄𝑅𝑆
(𝐹𝑟𝑜𝑛𝑡)
⟹
−[𝐸𝑥 (𝑑𝑦. 𝑑𝑧)] + [𝐸𝑥′ (𝑑𝑦. 𝑑𝑧)] = 0
]
⟹ [𝐸𝑥′ (𝑑𝑦. 𝑑𝑧)] − [𝐸𝑥 (𝑑𝑦. 𝑑𝑧)] = 0
⟹ [𝐸𝑥′ ] − [𝐸𝑥 ] = 0
This relation says that 𝐸𝑥′ − 𝐸𝑥 = 0. This leaves us with two possibilities… 𝐸𝑥′ = 𝐸𝑥 or 𝐸𝑥′ = 0 & 𝐸𝑥 = 0. The first
possibility of 𝐸𝑥′ = 𝐸𝑥 would mean that the electric field is uniform. But a uniform field cannot propagate a wave of finite
wavelength according to Maxwell’s Theory. Hence the only possibility is that both 𝐸𝑥′ & 𝐸𝑥 are separately zero. This
proves that there is no component of the electric field parallel to the direction of propagation. A very similar argument
will hold true for the magnetic field B also.
Hence we have proved that in an electromagnetic wave, both Electric and Magnetic field are perpendicular to the direction
of propagation, that is, Electromagnetic Wave is a Transverse Wave
Remember the following relation where c is the speed of light in vacuum, v is the speed of light in a medium, μ0 is the permeability of free
space, μr is the relative permeability, ε0 is the permittivity of free space and εr is the relative permittivity of the medium or the Dielectric
constant.
In vacuum,
c=
1
√μ0 ε0
In any medium,
v=(
v=(
1
√(μ0 μr )(ε0 εr )
)=(
1
)
√με
1
1
c
)(
)=(
)
μ
ε
μ
√ r εr
√μ0 ε0 √ r r
c
c
v=(
) ⟹ √μr εr = = n
v
μ
ε
√ r r
Where n is the absolute refractive index (with respect to vacuum) of the medium under consideration so,
n = √μr εr
3
qed_amk_gesXII1011
MODULATION AND DEMODULATION
(Syllabus Says… Modulation and Demodulation – quantitative only, Amplitude Modulation (AM), Frequency Modulation (FM). Production and Detection, Uses)
Modulation is a word often used in Musical terminology. When you say that a song modulates from one pitch to another, it means that the
song changes the key or the scale (say G major scale to D major scale). For those of you who don’t understand this, it means that you
change you range of frequencies from one to another. Say if I was using a range of 250Hz to 500Hz, I shift to using 350Hz to 700Hz
instead. Such a change of frequency is called as modulation. One can also modulate the amplitude (loudness) of ones singing. In the
context of ISC, (although this topic is a pain in the neck), we have to talk about the idea of modulation of EM waves and why we do it.
When a teacher speaks in a class room, he or she is heard because the sound waves produced in her vocal chords will disturb the air around
her mouth and that disturbance will travel in air just like waves on a water surface. However, if you are far away you don’t hear the teacher
as loudly as you do when you are close by. This happens as a result of the conservation of Energy. WHAT? Yes… This because the wave
spreads out, the same energy has to be spread over a larger area, so the energy at a given area reduces. So, the wave ‘fizzles out’ just as a
water wave fizzles out at distances far away from its source. So naturally, communication over large distances is not possible without the
use of technology. You cannot stand in an open ground and shout out “I LOVE YOU” and expect your girlfriend or boyfriend to hear
you! Rajnikanth may do that… but surely not humans like us. So let us explore how technology is used to achieve communication over
large distances.
As we all know, any audible audio
signal is only of the order of 20Hz to
20,000Hz. To transmit this over a large
distance, the following setup is made5.
The input analogue signal is received
by a microphone that coverts the
spoken sound or music or whatever it
is into a digital signal (voltages and
currents). This is fed into an amplifier
that amplifies the signal (the voltage or
current) and sends it to an antenna that
transmits
the
signal
as
an
electromagnetic wave into space. The
arrangement so far is called a Transmitter. The electromagnetic signal travelling in free space at the speed of light is
intercepted by another antenna at the receiving end. It is then fed into an amplifier and finally to a loud speaker which
converts the electrical signal back into the original sound signal. This arrangement constitutes a Receiver.
This kind of communication system suffers from two drawbacks…
(i) The Audio Frequency (20Hz – 20000Hz) signals have small power and so cannot be effectively radiated to very long
distances.
(ii) There is not just one transmitter and one receiver in the world. So when different transmitters send signals of different
frequencies simultaneously, confusion is created at the receiver due to interference of the different signals.
These drawbacks are overcome by making some modifications…
(i) The Audio Frequency (AF) signal or message or wave 6 is superimposed on a Radio Frequency Wave 7 before
transmission. This process is known as MODULATION. The high frequency signals can be radiated to very long
distances with comparatively less power).
(ii) Different transmitting stations are allotted different frequencies for transmission. A single receiver can be tuned to
pick up any desired frequency.
5 Remember that in electronics and communication science, we use some terminology that may confuse you. For example, we may say ‘Audio Signal’ or ‘Input
Signal’ or ‘Audio Information’, or ‘Input Information’. All this means the same thing… and what is it? When you speak on the phone, what you speak is converted to
bits of information and this IS the ‘Audio Signal’ or ‘Input Signal’ or ‘Audio Information’, or ‘Input Information’. Similarly we have Output Signal of Output
Information. So don’t get confused…
6 Signal or Wave or Message all means the same thing… So don’t get confused. It is in general called as Information… Audio information, video information etc.
7 This Radio Frequency Wave as you know is an electromagnetic wave and s also known as a CARRIER WAVE and the AF Signal (information… the digital version of my
voice and Rehman’s voice) is also known as Modulating signal or Modulating wave
4
qed_amk_gesXII1011
Ok what does all this
mean? Due to our
time constraint, you
need to understand
only what happens
and not how it
happens… Let us say
that I am singing on
the radio in Kerala
and it is transmitted
all around the world
(Ya rite!). But there is
also AR Rehman
singing in Chennai
and that is being aired
at the exact same time all around the world. But you see, surely, people love to hear me more that they love to hear Mr.
Rehman. So how does one go about sorting this confusion? Surely you don’t want my sound to be mixed with his… This
can be quite an insult for him (or is it for me?)!
I am singing at a radio station in Trivandrum and Rehman at Chennai. So what the communication department of the
country does is to allot specific radio frequencies to the Trivandrum Tower (say 93.5MHz FM) and to the Chennai Tower
(say 103.5 MHz FM8). The idea now is to mix (modulate) my sound signal with this 93.5 MHz and to mix Rehman’s
voice with 103.5MHz in a certain manner so that when it reaches your radio, (which has a tunable amplifier) you have the
choice of tuning to 93.5 or to 103.5. If you choose 93.5, (which of course is the smart thing to do) then the radio will catch
onto this mix of 93.5 Hz and my voice and ‘unmix’ (demodulate9) it. This ensures that the amplifier in your radio can
catch onto only my voice and amplify it so that you can hear my sweet, lovely voice! If your radio on the other had is
tuned to 103.5 (not a very smart thing to do though) then Rehman’s voice is demodulated and you hear that in your radio.
So here we have the definitions…
Modulation: It is the super imposing of an Audio Frequency (AF) Input Signal with a Radio Frequency (RF) Wave10
before transmission of the wave over large distances. This is done so that there is efficient transmission of the signal
without loss of information.
Demodulation: It is the reverse process of Modulation where the super imposed RF Wave and AF Wave are separated
after it is received by a receiver antenna. The audio signal on being amplified and fed into a loud speaker (or headphone)
reproduces the original speech or music.
Now, this superimposing of input AF signal with RF Signal called as modulation is of two types…
(i) Amplitude Modulation (AM)11: In amplitude modulation, the AF signal is superimposed on an RF carrier wave in
such a manner that the frequency of the modulated wave is the same as that of the carrier wave, but its amplitude
varies in accordance with the instantaneous amplitude of the modulating wave (message signal)
(ii) Frequency Modulation (FM): In Frequency Modulation, the modulating wave is superimposed on a carrier wave in
such a manner that the amplitude of the carrier wave remains constant, but its frequency varies in accordance with the
instantaneous amplitude of the modulating wave.
8 FM stands for Frequency modulation… We will see what that is shortly
9 The Modem in your computer is nothing but a Modulator – Demodulator.
10 Remember that A Radio Frequency Wave is an Electromagnetic Wave that travels at the speed of light.
11 This is the AM and FM in your radio.
5
qed_amk_gesXII1011
AMPLITUDE MODULATION
FRQUENCY MODULATION
Audio Frequency Modulating Wave (Signal)
Audio Frequency Modulating Wave (Signal)
Radio Frequency Carrier Wave
Radio Frequency Carrier Wave
Amplitude Modulated Wave
Frequency Modulated Wave
The need for Modulation…
1. There is a consequence of all this communication science that for efficient transmission and reception, the
transmitting antenna should have a height roughly equal to a quarter of the wave length of the wave itself. Thus, for
AF 10kHz (say), the required height of the antenna is about 7.5km!
𝜆 1 𝑐
3 × 108 𝑚/𝑠
= ( )=
= 7500𝑚 = 7.5𝑘𝑚
(10 × 103 )𝑠 −1
4 4 𝜈
Now, if a carrier wave of frequency 1MHz is used, it turns out that the height of the tower needs to be only 75m
which can be easily setup.
2. The energy carried by an audio (low frequency) signal is too small and so the power radiated from transmission is
insignificant. The Power radiated from an antenna fed by a high frequency signal is quite large and the signal can
therefore travel long distances.
3. All audio signals are of the range of 20Hz to 20,000kHz. Hence audio signals from different transmitting stations
would overlap and cause confusion. The problem of overlapping of the signals from different transmitting stations is
overcome by allotting a band of frequencies of the carrier wave to each station.
6
qed_amk_gesXII1011
CLASSIFICATION OF MAGNETIC MATERIALS
In this section, I will tell you everything I know about magnetic classification of materials. That way, you will have the
concepts very clear. However, you don’t have to study the whole thing as far as your exam is concerned. Check with the
syllabus and learn what is required. Call me if you have doubts. As far as possible, I will try and indicate which is important
and which is not! This may seem long, but a single reading will give you an idea as to what is happening... So it will be worth...
Read it when you are taking a break... read it like a story.
All substances show magnetic properties. Even materials
that are not attracted by magnets show magnetic properties
only they are so feeble that we cannot notice it. In order to
understand how materials behave in the presence of an
external magnetic field, you need to know three things…
1. You already know that a current carrying loop is like a
tiny bar magnet. If you have a current loop (of ANY
shape), then the current either goes clockwise or counter
clockwise and that loop will have an area. If the Area is
treated as a vector here and the direction of the vector
given by the right hand cork screw rule (We often did
this in class… rotate your right hand clockwise in the
horizontal plane and then push it up). This will give you
the direction of the Area Vector A. If the current
flowing is ‘i’, then the Magnetic Dipole Moment or the
Magnetic Moment ‘µ’ (your text book gives the variable
The loop has current clockwise as seen from below
‘M’ for this) is given by the formula 𝝁 = 𝑖𝑨. If there
are ‘N’ such loops (like in the case of a solenoid), then we have 𝝁 = 𝑁𝑖𝑨. This magnetic moment vector gives us the
strength of the magnetic field. Higher the value of 𝝁 = 𝑁𝑖𝑨, stronger is the magnetic field in the vicinity. As the
formula clearly indicates, the Magnetic Moment vector points in the direction of the Area Vector12. This Magnetic
moment 𝝁 = 𝑁𝑖𝑨 can be said to be like a tiny bar magnet as shown in the figure. The 𝝁 vector always points in the
direction from South to North Pole of this Magnet as shown in the figure. So in short, a current loop can be considered
as a tiny bar magnet and this loop for mathematical purposes can be represented by the Magnetic Dipole moment
vector that points from South to North Pole of the equivalent bar magnet. Remember that the loop need not always be
circular. It can be of any shape (rectangular, square or any irregular shape) as long as it will have a defined Area
Vector!
2. When such a Magnetic Moment 𝝁 is brought in the presence
of an external magnetic field B, the moment vector tries to
align itself in the direction of the field as the north side of the
magnetic moment (the arrow head of the vector) goes in the
direction of the field and the south side of the magnetic
moment (the tail of the vector) goes opposite to the direction
of the external magnetic field B and how successful this
aligning is depends on the strength of the field (See figure).
This is just like how an electric dipole aligns itself in the
presence of an external Electric field. The positive charge (the
arrow head of the electric dipole moment vector) goes in the
direction of the Electric field while at the same time the
negative charge (the tail of the electric dipole moment vector) goes in the direction opposite to that of the field.
3. As you know, all materials are made up of atoms and molecules which are made up of electrons spinning about its
own axis and revolving around the nucleus. Each such electron constitutes a tiny bar magnet (as it goes around, it
constitute a current in a loop of very tiny area!). Or in other words, electrons contribute to a Magnetic dipole moment
12 This Magnetic Dipole Moment 𝝁 = 𝑖𝑨 is analogous to the electric dipole moment that we defined as 𝒑 = 2𝑞𝑳, where q is the charge and 2L is the vector distance
between the two charges. Here, the vector p takes the direction of L.
7
qed_amk_gesXII1011
within the atom or the molecule. Therefore how materials behave in an external magnetic field will depend on the
electronic configuration of these all these billions and billions of atoms and molecules that make up a material. We do
not see the atoms and molecule; we only see the material and how it behaves.
Now that you have an idea as to what a Magnetic dipole is, and how it behaves in the presence of an external magnetic
field, and of how all substances are made of billions and billions of such very tiny bar magnets, we are ready to talk about
how these materials behave in an external magnetic field. Depending on how materials behave in the presence of an
external magnetic field, they are divided into three categories namely Diamagnetic, Paramagnetic and Ferromagnetic
materials. We will discuss them qualitatively as well as quantitatively here.
You also need to know that all materials have thermal energy and depending on the temperature of the substance, the
amount of thermal energy will vary. Due to the thermal energy the atoms and molecules in the material are in very
constant thermal agitation. More the temperature, more the thermal energy and hence more the thermal agitation of the
atoms and molecules and electrons of the material. Therefore, when a material is subject to an external magnetic field B,
how far the field will be successful in aligning the magnetic moment vectors (tiny bar magnets) of each of those atoms
and molecules in the direction of the field will naturally depend on the temperature of the material and on the strength of
the applied external field B itself. Sometimes, if the field is not strong enough, the thermal agitation will overcome the
external Magnetic field. As you make the field stronger, the chances of success will increase.
Diamagnetism
In nature, we see that all materials, when you introduce them to an external magnetic field will to some degree oppose the
field. All material will produce on an atomic scale an emf that opposes the applied external magnetic field. This is
contrary to what we have discussed above but it is true. Why this is so can only be understood through the laws of
Quantum Mechanics and hence we make no attempt to deal with it here but we will accept it13. As a result, what will
happen is that the field inside the diamagnetic material is always a little smaller than that of the external field as the
dipoles will oppose the external field. Examples of diamagnetic materials are Zinc, Copper, Silver, Gold, Lead, water,
Mercury, Sodium Chloride, Nitrogen, Hydrogen etc.
Paramagnetism
There are many substances where the atoms and the molecules themselves have a magnetic dipole moments. So you can
think of them as being little magnets. If you have no external magnetic field, then these dipoles are extremely chaotically
oriented and so the total net magnetic field is zero and so they are not permanent magnets. But the moment you expose
them to an external magnetic field, this magnetic field will try to align them and the degree of success will depend on the
strength of the magnet field and the temperature. The lower the temperature, the easier it is. The tiny magnetic moments
within the material will try and align itself to the external magnetic field as we have discussed in the previous page. As a
result, what will happen is that the field inside the paramagnetic material is always a little higher than that of the external
field as the dipoles will add on to the external field. The moment we remove the external field, all of the tiny bar magnets
goes back to being chaotic and therefore there is no permanent magnetism left. In the presence of a non uniform magnetic
field, the paramagnetic material will always attract to the stronger side of the field. Examples of paramagnetic substances
are Aluminium, Sodium, Platinum, Manganese, Antimony, Copper Chloride, Liquid Oxygen, solution of salts of Iron,
nickel etc.
Ferromagnetism
We have a third form know as ferromagnetism which is also the most interesting. There are substances that we know in
nature that have just in the case of paramagnetic substances inherent permanent dipole moment. They also have a net zero
magnetic moment (means they are not permanent magnets). But however, unlike in the case of paramagnetic substances,
within the materials, there are small spaces or domains within the material of the order of 0.1 mm to 0.3mm where the
dipoles are 100% aligned in certain given directions14. So within a certain small area (domain), the magnetic moment is
13 You may say Lenz Law is the reason (faradays law of electromagnetic induction)… WRONG! It has nothing to do with Lenz law; Lenz law is the production of
opposing emf by the free electrons in a conductor in the presence of a changing magnetic field. This is a permanent external magnetic field that we are talking
about.
14 The reasons for this can only be understood using Quantum Mechanics
8
qed_amk_gesXII1011
non zero, meaning that they are permanent magnets within that area only. Now if you move to another nearby domain, it
will have a different orientation but will have 100% orientation in that direction. Like this the entire material is made of
large number of domains consisting of 1017 to 1021 atoms that are permanent magnets but the material as a whole may not
be magnetic.
Now if the material is exposed to
an external magnetic field, the
domains as a whole will try to
align itself to the external field
and the degree of success will
depend on the strength of the
field and the temperature; lower
the temperature, the lower the
thermal agitation and then the
easier it is to orient these
domains. As a result, what will happen is that the field inside the ferromagnetic material can be up to thousands of times
higher than that of the external field as the dipoles will aid the external field. A diagrammatic representation of these
domains before and after the introduction of an external magnetic field is given in the diagrams above (In the second
diagram the field is applied from below to above as can be seen). Now unlike in the case of paramagnetic substances, once
the external field in introduced and then removed, some of the domains may continue to remain aligned in the direction in
which the field was applied. If we very carefully remove the external field, undoubtedly some of the domains will flip
back to its original configuration because of thermal agitation while some will remain oriented in the direction the field
was applied and thereby giving the material a net magnetic moment and hence it acts like a magnet (we know from daily
experience that that when paper clips are attached to a magnet for a long time, even after the magnetic is removed, it
continues to attract other paperclips. That’s the idea!!!). In order to lose this permanent magnetism, we can either hit the
material hard on some object or with some object or heat it! When this is done, we distort the orientation and force it to
come back to how it was originally and hence it loses its magnetism!
In the presence of a non uniform magnetic field, like in the case of the paramagnetic material, the ferromagnetic material
will also always attract itself to the stronger side of the field only in this case the attraction is thousands of times stronger.
This is why a paper clip will attract itself to the North Pole or the South Pole of a magnet, depending on which one is
closer to it. Examples of Ferromagnetic materials are Iron, Cobalt, Nickel, Magnetite etc.
In most cases, it is almost impossible to get paramagnetic materials to attract on a magnet as the force of attraction is only
3% to 5% of the weight of the material itself. This is also the reason why we see an iron rod (ferromagnetic) attracting a
magnet while an Aluminium rod (paramagnetic) does not. Even though there is attraction, it is very feeble and cannot be
noticed under ordinary conditions. So in a way, the difference between Paramagnetism and Ferromagnetism is only that of
intensity. A 15kg of Iron will attract to a magnet while even a few hundred grams of Aluminium will not as
Ferromagnetic materials are far more intense than Paramagnetic materials.
Here are a few definitions you need to know for the sake of exam...
(a) Magnetic Induction (B): When a piece of any substance is place in an external Magnetic field, the substance becomes
magnetised and this phenomenon is called Magnetic induction (B). It has a unit Tesla (T)
(b) Intensity of Magnetisation (I): The Intensity, or simple the magnetisation of a magnetised substance represents the
extend to which the substance is magnetised. It is defined as the magnetic Moment per unit volume of the magnetised
substance and is denoted by I. Numerically, 𝑰 = 𝝁/𝑉 where V is the volume and 𝝁 is the Magnetic Moment. It has a
unit of A/m
(c) Magnetic Intensity or Magnetic Field Strength (H): When a substance is place in an external Magnetic field, it
becomes magnetised. The actual magnetic field inside the substance is the sum of the external magnetic field and the
field due to its magnetisation. The capability of the magnetising field to magnetise the substance is expressed by
means of a vector H, called the Magnetic Intensity of the Field. The magnetic intensity is define through the vector
relation
9
qed_amk_gesXII1011
𝐁
−𝐈
μ0
Where 𝜇0 is the permeability of free space. From a practical point of view, H is very important, especially in the case
of ferromagnetic materials, as it can be controlled by adjusting the current, say in coils or solenoids. Its unit is the
same as that of I A/m
𝐇=
(d) Magnetic Permeability (µ): It is the measure of conduction of Magnetic field lines through it. It is defines as the ratio
of Magnetic Induction B inside the magnetised substance to the Magnetic Intensity H of the magnetising field.
𝐁
μ=
𝐇
(e) Relative Permeability (µr or κm): It is the ratio of magnetic permeability of the substance to the permeability of free
space. It is a dimensionless quantity, just a number.
𝜇
𝜇𝑟 =
𝜇0
(f) Magnetic Susceptibility (𝜒𝑚 ): It is a measure of how easily a substance is magnetised in a magnetising field. It may
be define as the ratio of the intensity of magnetisation to the magnetic intensity of the magnetising field that is,
𝑰
𝜒𝑚 =
𝑯
We have seen that for all materials dia, para or ferromagnetic, the field inside the material is different from what the field
would be without external field. In Many cases, the field inside (B according to our definitions above) is proportional to
the External Field (Bo). So we can write that
𝑩 ∝ 𝑩𝟎 ⟹ 𝑩 = μr 𝑩𝟎
So now if we look at the values of μr , we can easily see the difference between dia, para and ferromagnetic materials. We
know that just as in the case of relative permittivity, the value of relative permeability must also be always greater than
one. However, in the case of diamagnetic materials, it is very slightly lesser than one. So if we now plug that value into
the equation 𝑩 = μr 𝑩𝟎, we clearly see that the value of field inside the material (B) is smaller than the value of the
external field (Bo) showing that it opposes the external magnetic field. On the other hand, the value of μr for para
magnetic material is slightly greater that 1 and so we see that in the relation 𝑩 = μr 𝑩𝟎, the value of B is slightly greater
than the value of the external field (B). Finally for ferromagnetic material, the value of μr is very large indicating that the
field inside is much, much greater that the field outside.
Now, since the value of relative permeability for dia and para are very close to 1, it is convenient to express it as
μr = 1 + 𝜒𝑚
And therefore, the value of Magnetic Susceptibility ( 𝜒𝑚 ) also can decides whether a substance is dia, para or
ferromagnetic in nature. The table below will give you a feel for the numbers. This is NOT for you to study!
Diamagnetic materials
μr < 1 𝑜𝑟 𝜒𝑚 < 0
Diamagnetic Material
Bismuth
Copper
Water
Nitrogen (1 atm)
Paramagnetic materials
μr > 1 𝑜𝑟 𝜒𝑚 > 0
𝜒𝑚
Paramagnetic Material
𝜒𝑚
Aluminium (300K)
Oxygen (1 atm, 300K)
Liquid Oxygen (90K)
+2.0 × 10−5
+2.0 × 10−6
+3.5 × 10−3
−4
−1.7 × 10
−1.0 × 10−5
−1.0 × 10−5
−7.0 × 10−9
Ferromagnetic materials
μr ≫ 1 or 𝜒𝑚 ≫ 0
μr ≈ 𝜒𝑚 ≈ 105 ‼!
As we have seen that the magnetic property of a substance is Temperature dependant, we have an interesting phenomenon
in nature. There is a very specific temperature for every substance above which the ferromagnetic substance suddenly
turns paramagnetic in nature. This temperature is called the Curie temperature. This phenomenon was discovered by Curie
in 1895. It turns out that the Curie temperature has a dependence on the magnetic susceptibility (χm ) of the substance as
follows...
1
χm ∝
TC
10
qed_amk_gesXII1011
Where TC is the curie temperature. Iron for instance has a curie temperature of only 770 ºC. Hence, it can be shown
experimentally that iron stick to a magnet. But if we were to heat it, then suddenly, it will lose its magnetic force and fall
off. When it cools, it will attract again!
GRATING (comes under the chapter of diffraction)
Those of you who wear spectacles (especially plastic lens)
would have noticed that when they are scratched (because you
rub it too much with rough cloth or with your hands), it is
difficult to see in the night because all sources of light seem to
spread out the light irregularly instead of coming regularly to
your eye. This happens because the scratch forms a small
potion on the lens that is opaque to light and light diffracts
around the edges of it. The idea of a diffraction grating is just
that. Take a nice, fine piece of rectangular glass plate and
scratch vertical lines on it. How many lines? Of the order of
6000 lines in just 1 cm! Now why is this done? Since light
diffract around the opaque edges the light spreads out and here
since all the lines are made in a very orderly manner and
vertical, the spreading of light is very regular (unlike in the
case of your spectacles where the scratches are very disorderly). Now we also know that the spreading is a property of the
wave length of the light used and so this can be used to study various optical phenomena.
If each of these ‘scratches’ (the opaque region) have a thickness of ‘d’ and each transparent region have a thickness ‘e’.
Since they are regularly made, every opaque region will be followed by a transparent region, then an opaque region, then
a transparent region and so on in a very regular repetitive manner.
From the diagram, remembering the derivations for diffraction, we see that the path difference at a certain angular
dispersion 𝜃 is given by the formula 𝑆2 𝐾 = 𝑆1 𝑆2 sin 𝜃 = (𝑒 + 𝑑) sin 𝜃. The two wavelets from S1 and S2 will reinforce
each other when brought to a focus P by a lens, if the path difference equals a whole number multiple of the wave length
(𝜆). Thus for reinforcement we have
(𝑒 + 𝑑) sin 𝜃 = 𝑚𝜆
Here ‘m’ is called the order of the spectrum. What does this mean? If we use mercury vapour lamp and pass the light
through a prism using a spectrometer, you will see the spectrum in the telescope. This you have already seen in the lab. It
turns out that if you use a grating instead of a prism, you will get to see just the same thing in the same way but with one
difference. There will be more than one set of thee spectra and they are much fainter and they will be symmetrically
arranged on either side of the incident ray. The one that comes first on either side is called the ‘first order spectrum’ for
which we use the value 𝑚 = 1. If you turn the spectrometer further away from the incident ray, you will get another set of
the same spectrum which is much fainter. This is seem on either side and will be the ‘second order spectrum’ 𝑚 = 2. So
in short, it turns out that the grating is sort of a substitute for a prism but the principle behind the formation of the spectra
is dispersion in the prism and diffraction in the case of the grating. If there are N number of lines per cm, then 𝑁(𝑒 +
𝑑) = 1𝑐𝑚 or,
1
𝑁=
𝑒+𝑑
Therefore, we have the relation,
𝑚𝜆
sin 𝜃 =
= 𝑁𝑚𝜆
(𝑒 + 𝑑)
sin 𝜃 = 𝑁𝑚𝜆
This is the equation for the grating that is used in the lab. If we use a monochromatic light of wavelength 𝜆 and if the
grating has N lines per cm and if we are looking at the first order grating where 𝑚 = 1, then the angular dispersion that
can be seen through a spectrometer will be 𝜃 = sin−1 (𝑁𝑚𝜆). So calculating the angle at which say red light is seen and
knowing the value of m and N, we can calculate the wavelength of red light!
11
qed_amk_gesXII1011