free electron gas mode. Lorentz theory€¦ · MVJCE Engineering Physics Module II 1 MODULE II...

MVJCE Engineering Physics Module II

1

MODULE II

Electrical Conductivity of Metals notes complied Dr. Ramdas Balan

Classical free electron theory of metal was proposed by Drude and Lorentz to account electrical

conduction in metals.

Drude postulated that the metals consist of positive ion core with the valence electrons

moving freely among the core. These electrons are however, bound to move within the metal due

to electrostatic attraction between the positive ion core and the electron. The behavior of free

electron moving inside the metal is considered to be similar to that of atoms or molecules in the

perfect gas. These free electrons are therefore, also called as free electron gas and the theory is

accordingly named as free electron gas mode.

Based on the Drude considerations that the electron gas behaves as a perfect gas, Lorentz

postulated that the electrons constituting the electron gas obeys Maxwell Boltzmann statistics

under equilibrium conditions. The combined idea of Drude and Lorentz is known as Drude and

Lorentz theory.

CLASSICAL FREE ELECTRON THEORY - ASSUMPTIONS

1. The valence electron of the atom is loosely bound to their respective atom. In metals they

become the free electrons. These electrons are free to move throughout the metal and are

hence termed as free electrons. Since electrical conductivity is due to these free electrons,

these are also termed as conduction electrons.

2. These free electrons move in the metal according to Kinetic theory of gases.

3. Like the molecules of gas in a container, free electrons in these metals move randomly (in

the absence of electric field) with a velocity, which is the average velocity or rms (root

mean square) velocity. In the absence of externally applied potential difference there are

on an average as many electrons wandering through a given cross section of the

conductor in a given direction as there are in the opposite direction. Hence the net current

is zero.

4. However on application of electrical field, the random motion gets slightly affected and

the electrons experience a drift velocity in the direction of the applied field. This drift

velocity is less than the rms velocity by several orders of magnitude.


2

DRIFT VELOCITY, MEAN COLLISION TIME AND MEAN FREE PATH, RELAXATION

TIME

Since, free electrons in the metals behave like molecules/atoms in the gas. So, it is

assumed that classical kinetic theory of gases can be applied to a free electron gas. Thus the

electron can be assigned with mean free path (), mean collision time () and average speed ( c ).

MEAN FREE PATH

The distance travelled by an electron between two successive collisions is called as free

path and its mean distance is called as mean free path.

The mean free path is the average distance travelled by an electron between two successive

collisions with other free electrons.

= c

MEAN COLLISION TIME

The average duration of time that elapses between two successive collisions is called

mean collision time.

= c

DRIFT VELOCITY

Drift velocity is defined as the average velocity acquired by the free electron in a

particular direction during the presence of electric field.

When an electric field is applied on an electron of charge e, then it moves in opposite direction to

field with a velocity vd. This velocity is known as drift velocity.

Lorentz force acting on the electron:

F= - e E (1)

where, E is the electronic field and e is the electronic charge.

The opposing force can be expressed as

F’ = mvd /

When the system in steady state the driving force is equal and opposite to the opposite force

i.e, equation (1) & (2) are equal F = F’

mvd /eE

i,e

m

eEvd


3

EXPRESSION FOR ELECTRICAL CONDUCTIVITY IN METALS

The drift velocity of the electron is

vd = (em) E (1)

From ohm’s law

J = E or J/E (2)

where, J – current density and - electrical conductivity of electrons

But, the current density in terms of drift velocity can be stated as

J= nevd (3)

substituting vd from (1) in (3)

J = ne (em) E or J/E = ne2m (4)

But, J/E so, comparing the (2) and (4)

(5)

Thus eqn. (5) gives the expression for electrical conductivity

EFFECT OF IMPURITY AND TEMPERATURE ON ELECTRICAL RESISTIVITY OF

METALS

The two factors that effects the electrical resistivity (or electrical conductivity) of metals are

1. Temperature

2. Impurity

The resistivity of metals is attributed to the scattering of conduction electrons. The

scattering of electrons takes place because of two reasons: one due collision of conduction

electrons with the vibrating lattice ions and the other is caused by scattering of electrons by the

impurities present in the metal.

The resistivity due to scattering of electrons by the lattice vibrations called phonons is

denoted by . This increases with temperature. It arises even in a pure conductor and hence

called the ideal resistivity. Whereas, the resistivity in the metal caused due to scattering of

electrons by the impurities is denoted by i. This is independent of temperature and present even

at absolute zero of temperature and hence called residual resistivity.

Therefore, the total resistivity of a metal can be written as the sum of the two resistivities.

This is called Matthiessen’s rule. Mathematically,

i

Since 2ne

m , we can rewrite the above equation as

m

ne

2


4

pne

m

2 +

ine

m

2

where, p and i are the mean collision times of electron with phonons and impurities

respectively. At lower temperatures tends to zero as the amplitude of lattice vibrations

becomes small which essentially means that all the resistivity will be due to impurities, i.e .,

=i . At higher temperature increases with temperature however the curves of versus T at

room temperature remains linear.

FAILURE OF CLASSICAL FREE ELECTRON THEORY

Though classical free electron theory successfully explained the Electrical and Thermal

conductivity in metal but it failed to explain many other experimental facts among which the

notables are

i. Specific heat

ii. Temperature dependence of electrical conductivity

SPECIFIC HEAT

The molar specific heat of a gas at constant volume is Cv =3/2 R. But, the experimental

value of specific heat of a metal was found to be Cv = 10-4

RT

Thus the value of sp. heat predicted is higher than the experimental value and also the

theory predicted that the sp. heat is temperature independent whereas experimental results are

temperature dependent.

TEMPERATURE DEPENDENCE OF ELECTRICAL CONDUCTIVITY

From experiment results it is well known that conductivity is inversely proportional to

temperature

i.e exp1/T (1)

But according to main assumption of classical free electron theory

2

3 KT = 2

1 mvth2 ;

vthm

KT3 where,

vth - thermal velocity

therefore vth T (2)

and also vth

1 (3)

Comparing (2) and (3) T

1 (4)

but we know m

ne 2 i.e (5)


5

comparing equation (4) and (5)

T

1 (6)

comparing the experimentally derived conductivity equation (1) and the classically derived

equation (6) both are not same. Hence, it is failure.

QUANTUM FREE ELECTRON THEORY

In 1928, Sommerfeld proposed the quantum free electron theory in order to overcome the

failure of classical free electron theory base on quantum concepts.

They are,

In a metal, free electrons are fully responsible for electrical conductivity.

Free electron are free to move anywhere within the metal.

Electron acquires constant velocity known as drift velocity under the influence of electric

field.

Electrons obey quantum concepts.

Electrons are treated as wave like particles.

The velocity and energy of the electron are determined by Fermi- Dirac distribution

function.

Electrons obey Pauli’s exclusion principle.

FERMI ENERGY AND FERMI FACTOR

In a single atom there will be many allowed energy levels whereas in a solid each such

energy level will spread over a range of few eV. If there are N numbers of atom, there will be N

closely spaced energy levels in each energy band of the solid. According to Pauli’s exclusion

principle, each such energy level can accommodate two electrons. At absolute zero temperature,

two electrons with least energy with opposite spins occupy the lowest available energy level. The

next two electrons with opposite spins will occupy next energy level and so on. Thus, the top

most energy level occupied by electrons at absolute zero temperature is called Fermi energy

level. The energy corresponding to that energy level is called Fermi energy.

Fermi energy, Ef , is defined as the higher most energy occupied by an electron at

absolute 0 K. Below which all energy levels are completely occupied and above which all the

energy levels completely empty. Thus Fermi energy represents maximum energy that electrons

can have at absolute zero temperature.

FERMI DISTRIBUTION OR FERMI FACTOR

At absolute zero all energy levels below Fermi energy are completely filled and above it

are completely empty. But at any given temperature, the electrons get thermally excited and

move up to higher energy levels. As a result there will be many vacant energy levels below as


6

well as above Fermi energy level. Under thermal equilibrium, the distribution of electrons

among various energy levels is given by statistical function f(E). The function f(E) is called the

Fermi factor which gives the probability of occupation of a given energy level under thermal

equilibrium.

The distributions of energy in metals are explained by Fermi Dirac statics, since it deals

with the particles having half integral spin like electron.

The probability function f(E) of an electron occupying an energy level E is given by

f(E) = KT

EfE

e)(

1

1

where, Ef - Fermi energy, and K - Boltzmann constant

The probability functions lies between 0 and 1. Hence, there are three probabilities

Case 1: Probability of occupation at T = 0K, when E < Ef

f(E) = e1

1 =

01

1

= 1

Therefore, there is 100% probability that the electron occupy the energy level below Fermi

energy.

Case 2: Probability of occupation at T= 0K, when E > Ef

f(E) = e1

1 =

1

1 = 0

the energy levels above Fermi energy level Ef are unoccupied at 0K. Therefore, there is 0 %

probability for the electron to occupy the energy level above the Fermi level

Case 3: Probability of occupation at T = 0 and E = Ef

f(E) = 01

1

e =

11

1

=

2

1

This shows at T = 0K, there is a 50% probability for the electron to occupy the Fermi

energy.

DENSITY OF STATES

Fermi level divides the occupied states from the unoccupied states i.e it is the highest

energy state for the electrons to occupy at absolute zero temperature. To know the actual number

of electrons with a given energy one must know the no. of states in the system, multiplying the

no. of states by the probability of occupation we get the actual no. of electron in the system.

The energy distribution of electron in a metal is determined by Fermi-Dirac statistics. The

ability of a metal to conduct electricity depends upon the no. of quantum states and the

availability energy levels for the electrons. Therefore determination of energy states for the


7

electron is essential. The number of states at each energy level that are available to be

occupied by the electrons, we introduce the concept of the density of states of a system.

The density of states, g(E), is defined as the number of energy levels available per unit volume

per unit energy centered at E. The number of states per unit volume between the energy level E

and E+dE is denoted by g (E)dE.

Density of states = No. of quantum states present between E and E+dE

Volume of the specimen

Density of states expression

EXPRESSION FOR ELECTRICAL RESISTIVITY BASED ON QUANTUM FREE

ELECTRON

Using the concepts of density of states and Fermi-Dirac statistics Sommerfeld arrived at

the following expression for electrical conductivity in metals,

fvm

ne*

2

where, is the mean free path; m* is the effective mass and vf is called the Fermi velocity. The

Fermi velocity can be found out by equating the Fermi energy to the kinetic energy of the

electrons in a metal.

That is , 2

2

1fmv = Ef vf =

m

E f2

The resistivity of the metal is given by

1

therefore

MERITS OF QUANTUM FREE ELECTRON THEORY

Temperature dependence resistivity of metal

The experimental results of electrical conductivity is inversely proportional to temperature.

The expression for electrical conductivity is

m

ne 2

Here as per quantum free electron theory

g(E)dE =3

23

)(28

h

m 21E dE

2

*

ne

vm f


8

fv

, therefore fmv

ne 2

i.e (1)

As the conduction electron transverse in the metal, they are subjected to scattering by the

vibrating ions of the lattice. The vibration occurs such that the displacement of ions takes place

equally in all directions. If the r is the amplitude of the vibrations, then the ions can be

considered to present effectively a circular cross section of area, r2 that blocks the path of the

electrons irrespective of the direction of approach. Since the vibrations of larger area of cross

section should scatter more effectively, it results in a reduction in the value of mean free path of

the electrons.

therefore2

1

r (2)

The amplitude of vibration varies with the temperature i.e the radius of increase with increase in

temperature

hence r2 T (3)

therefore, from (2) and (3) T

1 (4)

comparing eqn (1) and (4 ) we have , T

1

Thus, quantum free electron theory derived temperature dependent conductivity and

experimental values are same.

SPECIFIC HEAT CAPACITY

The quantum theory of free electrons solves the flaws of the classical theory which is discussed

below. Specific heat of free electrons: From quantum theory of free electrons, the specific heat of

free electrons is given by

Cv = fE

k2 RT

For a typical value of Ef = 5eV, we get

Cv = 10-4

RT

which is agreement with the experimental results.

Dependence of electrical conductivity on electron concentration: The electrical conductivity in

metals is given by

fvm

ne*

2

It is clear from this above equation that the electrical conductivity depends both the electron

concentration (n).


9

SEMICONDUCTORS

Materials are classified based on their electrical conductivity as conductor, semiconductor and

Insulators. Using band gap (which is also called as forbidden gap i,e. a gap between conduction

band and valence band) the classification of the materials can be explained. If materials posses

band gap of the order of 1 eV then it is called Semiconductors.

Figure 1: Band gap of semiconductor

Semiconductors are broadly classified in to two types i) Intrinsic semiconductors and

ii) Extrinsic semiconductors.

Semiconductor which does not have any kind of impurities, behaves as an Insulator at 0 K and

behaves as a conductor at higher temperature is known as Intrinsic Semiconductor or Pure

Semiconductors. e.g Silicon (Si), Germanium (Ge).

Figure 2: Electronic structures of fourth group element - Si, Ge (example for Intrinsic

semiconductor)

Extrinsic semiconductors are those in which presence of impurities of large quantity present.

Usually, the impurities can be either 3rd

group elements or 5th

group elements.

Based on the impurities present in the extrinsic semiconductors which introduce either majorly

holes or electrons, upon these they are classified into,

i) n-type semiconductors - electron are majority charge carriers

ii) p-type semiconductors - holes are majority charge carriers


10

Figure 3 : a) n-type semiconductors b) p-type semiconductors

CARRIER CONCENTRATION IN INTRINSIC SEMICONDUCTORS

In semiconductor, two types of charge carrier that is electron and hole can contribute to a current.

Since the current in a semiconductors is determined largely by the number of electrons in the

conduction band and number of holes in the valence band. Under thermal equilibrium condition

number electron in the conduction(ne) is equal to number of holes in the valence band (np).

The distribution of electrons in the conduction band is given by the density of states, g(E) times

the probability that a state is occupied by the electrons, f(E).

Let ‘ne’ be the number of electrons available between energy interval ‘E and E+ dE’ in the

conduction band, then,

(1)

where, g(E) is density of states in the conduction band,

f(E) is the fermi dirac statistics

density of states, dEEmh

dEEg e2

3

3)(

28)(

(2)

Probability of occupancy

Hence,

(3)

band theof top

)()(

cE

e dEEfEgn

)exp(

1)(

res temperatupossible allFor

)exp(1

1)(

kT

EEEf

kTEE

kT

EEEf

f

F

f

)exp()(

)(exp)(

kT

EEEf

kT

EEEf

F

F


11

Substituting equation (2 ) and (3 ) in (1)

(4)

or where ,

Let ‘np’ be the number of holes or vacancies in the energy interval ‘E and E + dE’ in the valence

band.

(5)

density of state , (6)

Probability of un-occupancy

(7)

substituting equation (6) and (7) in (5)

or

where

FERMI LEVEL IN INTRINSIC SEMICONDUCTORS

For Intrinsic semiconductor, number of electrons in the conduction band (ne) is equal to number

of holes in the valence band (np). i,e ne = np

)exp()(24

2

3

3 kT

EEkTm

hn cF

ee

Ev

p dEEFEgnband theof bottom

)}(1){(

)exp(kT

EENn cF

ce

2

3

3)(

24kTm

hN ec

kTEE

kT

EEEf

kT

EEEf

F

f

f

res temperatupossible allFor

)exp(1

1)(1

)exp(1

11)(1

)exp()(24

)exp()(24

2

3

32

3

3 kT

EEkTm

hkT

EEkTm

h

Fve

cFe

)exp(

1)(1

kT

EEEf

f

dEEmh

dEEg e2

3

3)(

28)(

2

3

3)(

24kTm

hN hh

)exp()(24

2

3

3 kT

EEkTm

hn Fv

hp

)exp(

kT

EENn Fv

hp


12

i,e

Taking natural logarithm on both side on above equation

where, Ev +Ec = Eg

Thus, fermi level is in the middle of the band gap for intrinsic semiconductors.

EXPRESSION FOR CONDUCITIVITY IN INTRINSIC SEMICONDUCTORS

Consider an intrinsic semiconductor of area of cross section A, in which a current 'I' flows. let 'v'

be the velocity of electrons, Ne is the number of electron per unit volume and 'e' is the magnitude

of electric charge on the electron.

Then, current (charge flow per second ) I =Ne eAv (1)

Current density J = I/A = Ne e (2)

Mobility of electron , e = /E i,e = e E (3)

substituting (3) in (2) (4)

As per Ohms law J = eE (5)

From equation (4) and (5)

Similarly, contribution of hole in electrical conductivity,

2

3

)(

)exp(

)exp(

e

h

Fv

cF

m

m

kT

EEkT

EE

2

3

)()exp(

e

hFvCF

m

m

kT

EEEE

2

3

)()2

exp(

e

hgF

m

m

kT

EE

EeJ eeN

ee e eN

hh e hN

)exp()()exp()( 2

3

2

3

kT

EEm

kT

EEm Fv

hcF

e

)2

(

that know tor wesemiconduc intrinsicIn

)2

()log(4

3

)()log(2

32

2

3

cvF

he

cv

e

hF

cv

e

hF

EEE

mm

EE

m

mkTE

kT

EE

m

m

kT

E

2

g

F

EE


13

Total conductivity of semiconductors is given by sum of e and h

i.e

For Intrinsic semiconductors, Nh = Ne = ni therefore,

HALL EFFECT

When magnetic field is applied perpendicular to a current carrying conductor or semiconductor,

voltage is developed across the specimen in a direction perpendicular to both the current and the

magnetic field. This phenomenon is called the Hall effect and voltage developed is called the

Hall voltage (VH).

Consider a rectangular bar of n-type semiconductor material in which a current 'I' flows in the

positive 'x' direction. It means that electron move in negative 'x' direction. Let a magnetic field

'B' be applied along the negative 'z' direction. Under the influence of magnetic field, the electrons

experience the Lorentz force, )( BveF d , (1)

where B - applied magnetic field,

e - magnitude on charge of electron and d - drift velocity of the electron.

Figure 4 : Schematic diagram of Hall effect

As a result, the density of the electrons increases in the lower end of the material, due to which

its bottom edge becomes negatively charged. Hence, top edge of the material become positively

charged. This result with potential called Hall voltage (VH). And electric field, EH is established

between upper and lower surface of the material. This field exert an upward forces (F') on the

electron.

F' = e EH (2)

eh ee eh NN

)(n i ehe

Left hand rule : Hall effect

fore finger - represent the

direction of current

middle finger - represent the

direction of applied magnetic

field

thumb finger - direction of the

drift electron


14

When an equilibrium is reached, the magnetic deflecting force on the electrons are balanced by

the electric forces due to electric field.

Hd eEBve )( i,e )( BvE dH (3)

The relation between current density and drift velocity is J = ne dv , we know that I = ne dv A

where A is area i,e d x w i,e )(dwne

Ivd

therefore equation (3) becomes, (4)

If 'd' is the distance between the upper and lower surfaces of the slab, then d

VE H

H (5)

where, charge density = n e From equation (4) and (5)

Hence,

For p- type semiconductors, the positively charged holes will be deflected to the bottom so that

electron will be maintained at the top of the semiconductor, when similar direction of current and

magnetic field is applied to that above. Hence, negative Hall voltage is maintained.

Based on the polarity of the VH, type of semiconductors can be identified.

Hall co-efficient

For a given semiconductor , the Hall field EH depends upon the current density J and applied

field B,

i,e EH J B or EH =RH J B where RH is called Hall co-efficient

therefore

where from eqution (3) that is EH =B Vd and J =neVd substituting with above equation

)((

dwne

BIEH

new

BIVH

w

BIVH

JB

ER H

H

neBnev

BvR

d

dH

1


15

Superconductivity

Metals show positive temperature co-efficient (PTC) as resistivity

increases with increase in temperature. They are directly

proportional to each other, T . As per Matthensien rule,

resistivity in metal are mainly due to two main factors

i) temperature and ii) impurity. Lowering the temperature,

resistivity decreases for metals as shown in the figure 1.

Figure 1 Temperature Vs Resistivity for metals

In 1908, Dutch physicist Kamerlingh Onnes liquefied Helium at the

standard pressure to obtain 4.2K (i.e -268.6 oC). He has studied

number of metals' electrical resistivity lowering the temperature up to

liquefied helium temperature. Onnes observed abrupt drop in

resistivity to zero at 4.3K for mercury (Hg), as shown in figure 2.

Figure 2 Temperature Vs Electrical resistivity plot of mercury

Electrical resistivity completely vanishes at low temperature is called Superconductivity. And the

materials which shows superconductivity are called Superconductors.

CRITICAL TEMPERATURE (Tc)

The temperature at which the material shows transition from normal state to superconductivity

state (i,e material shows zero resistivity) is called critical temperature. Above the critical

temperature material will be in normal state.

MEISSNER EFFECT

When a superconductor in superconducting state (maintained below

critical temperature) kept in the magnetic field, it is observed to

expel the magnetic flux and the phenomenon is called Meissner

effect. It is similar characteristics like diamagnetism. Hence

superconductors exhibit diamagnetic property.

Figure 3 Meissner effect

Temperature in K

Res

isti

vit

y

()

normal conductor Superconductor


16

CRITICAL FIELD (Hc)

It is studied that apply of high magnetic fields destroy the superconductivity and restores the

normal conducting state. The magnetic field at which the superconducting state is destroyed is

called Critical field.

TYPE SUPERCONDUCTORS

Depending on the characteristics of transition from superconducting state to normal state when it

is exposed to external magnetic field, superconductors are classified in to two types 1) type I and

type II superconductors.

TYPE I SUPERCONDUCTORS

Superconducors that undergo abrupt transition from superconducting state to normal state at

critical magnetic field are known as type Type I superconductors .

Type I superconductors exhibit complete Meissner effect i,e completely expels the magnetic flux

(B) when it is superconducting state (figure 4b). When the applied magnetic field (H) is greater

than the critical field Hc, the entire material becomes normal by losing it superconducting

property completely and the magnetic flux (B) penetrates through the body.

In the presence of an external magnetic field H < Hc, the materials in superconducting state is a

perfect diamagnetic. Since it is a diamagnetic, it possesses negative magnetic moment ( - 4M),

shown in figure 4a.

Figure 4 Type I superconductors

The critical field value for Type I superconductors are found to be very low.

a

b

superconducting state

superconducting

state

normal state


17

TYPE II SUPERCONDUCTORS

Type II superconductor has two critical magnetic fields Hc1 and Hc2. When the applied magnetic

field (H) is lesser than the Hc1, superconductor completely expels the magnetic field and

behaves as a perfect diamagnetic. For the applied field between Hc1 and Hc2 , magnetic flux

partially penetrates the superconductor (shown in figure 5b), where Meissner effect is

incomplete.

Figure 5 Type II superconductors

This partial penetration is in the form of a regular array of

normal conducting regions. These normal regions allow the

penetration of the magnetic field in the form of thin

filament, called vortex. The material surrounding this

normal can have zero resistance. Vortex region are

essentially filaments of normal conductor that run through the

material when an external magnetic field increases , the number of vortex increases. The number

of vortex increase until the field reaches the upper critical field Hc2, the vortex crowd and join up

so that the entire materials become normal.

BCS THEORY

According to classical physics , part of the resistance of a metal is due to collisions between free

electrons and lattice vibration known as phonons. And part of resistance is due to scattering of

electrons from impurities. In 1957 John Bardeen , Cooper and Schrieffer developed a theory of

superconductivity called BCS theory which explains the loss of electrical resistance due to

electron pairing called Copper pair.

In normal metal, the electrical current are carried by electrons which are scattered giving rise to

resistance. since electrons each carry negative electric charge, they repel each other. In a

superconductor, there is an attractive force between electrons of opposite momentum and

a b

super conducting

state, no magnetic

flux penetrate Mixed state,

partially magnetic

flux penetrate

normal conducting

region , which allow

the magnetic flux to

penetrate through


18

opposite spin that overcomes this repulsion enabling them to form a pairs. These pairs are able to

move through the material effectively without being scattered.

An electron passes through the lattice and positive ions are attracted to it, causing a distortion in

their nominal positions. The second electron come along attracted by the displaced ions since

having high positive charge. Hence the electron forms a pair through the phonon interaction.

HIGH TEMPERATURE SUPERCONDUCTOR

High temperature superconductors are which shows transition at higher temperature greater than

the boiling temperature of liquid nitrogen (77K). Until 1986, transition temperature for

superconductors was recorded with 23K. Bednorz and Muller discovered new class of material,

La2CuO4 which has transition at 30K. Soon after other superconducting cuprates materials were

discovered with even high transition temperature. YBa2Cu3O7 which shown superconductivity at

critical temperature of 135K. These temperature could be achieved with liquid nitrogen (77K) to

cool them.

APPLICATION OF SUPERCONDUCTOR

MAGLEV VEHICLES

Magnetically levitated vehicles are called Maglev vehicles.

The phenomenon on which the magnetic levitation is based on

Meissner effect. The Levitating vehicle consist of powerful

electromagnet made from superconductors built on its base.

Normal electromagnets on a guideway repel the superconducting electromagnets to levitate the

vehicle. During motion of the vehicle , the absence of contact between the moving and

stationary systems, the friction is eliminated. Hence great speed could be achieved with very low

energy consumption.

free electron gas mode. Lorentz theory€¦ · MVJCE Engineering Physics Module II 1 MODULE II...

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