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1
Energy Bands in Solids
1.1
INTRODUCTION
Electronics (coined from the word ‘electron’) is the branch of science and engineering dealing
with the theory and use of a class of devices in which electrons are transported through a
vacuum, gas or semiconductor. The motion of electrons in such devices, called electron devices, is usually controlled by electric fields. Diodes, triodes, transistors etc. are examples of
electron devices.
Electronics can be broadly classified into two branches: physical electronics and electronic
engineering. Physical electronics treats the motion of electrons in a vacuum, gas or
semiconductor. The design, fabrication, and application of electron devices form the subject
matter of electronic engineering. Electronics has inroads into many fields of science and
technology, and has profoundly influenced the life of modern man. Modern computers and
communication systems are intimately linked with advances in electronics.
We list below some areas of application of electronics; the list is, however, far from complete.
(i) In home, we use a variety of electronic systems: radios, TVs, VCRs, CD players,
telephone answering machines, PCs, microwave ovens, pocket calculators, digital
watches, etc.
(ii) Round-the-globe communication via microwave or fibre-optic links and satellites, and
access to internets have been possible with advances in electronics. This has shaped
the global village.
(iii) Sophisticated electronic instruments are used for researches in various fields of science
and engineering.
(iv) The commercial and industrial sectors rely on electronic communication, information
processing, and control systems.
(v) Electronic radar systems are employed for a safe flight from one airport to another. A
modern aircraft is equipped with electronic sensors and computers.
(vi) Electronic communication and radars play a vital role in meleorology, defence, and
military services.
(vii) Modern medical practice relies on precise diagonistic and monitoring electronic
systems.
For better times, and at times for worse, electronics has shaped our lives. Our living standards have improved, but we are also at the peril of deadly weapons that would not have appeared without the progress in electronics.
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2
1.2
Electron Devices and Circuits
CHARGED PARTICLES
Electric charges, positive or negative, occur in multiples of the electronic charge. The electron
is one of the fundamental particles constituting the atom. The charge of an electron is negative
and is denoted by e. The magnitude of e is 1.6 × 10–19 coulomb.
The mass of an electron changes with its velocity in accordance with the theory of relativity. An electron moving with a velocity v has the mass
m=
m0
1 − v2 / c 2
(1.1)
where c is the velocity of light in free space (c = 3 × 108 m/s). If v << c, m ≈ m0, called the rest
mass of the electron. The rest mass of the electron has the value m0 = 9.11 × 10–31 kg.
Equation (1.1) shows that the mass m increases with the velocity v and approaches infinity as v → c. An electron, starting from rest and accelerated through a potential difference of
6 kV, acquires a velocity of 0.15 c. At this high speed, the mass of the electron increases only by
1 per cent. Therefore, the change of the mass of an electron with velocity can be neglected for
accelerating potentials less than 6 kV.
The radius of an electron is about 10–15 m and that of an atom is 10–10 m. The radii are so
small that electrons and atoms are ordinarily taken as point masses.
Electron Volt: Unit of Energy
For energies involved in electron devices, ‘joule’ is too large a unit. Such small energies are
conveniently measured in electron volt, abbreviated eV. The electron volt is the kinetic energy
gained by an electron, initially at rest, in moving through a potential difference of 1 volt. Since
e = 1.6 × 10–19 coulomb,
1 eV = 1.6 × 10–19 coulomb × 1 volt = 1.6 × 10–19 joule.
1.3
ATOMIC ENERGY LEVELS
An atom of an element is generally made up of electrons, protons, and neutrons. The only
exception is the hydrogen atom which possesses one electron and one proton, but no neutron.
While an electron is negatively charged, the proton is a positively charged particle. The charge
of a proton is numerically equal to that of an electron, but the mass of a proton is 1837 times
that of an electron. A neutron is a neutral particle having a mass nearly equal to the proton
mass. Because the protons and neutrons carry practically the entire mass of the atom, they
remain almost immobile in a region, called the atomic nucleus. The electrons revolve round the
nucleus in definite orbits which are circular or elliptical. The motion is analogous to that of
planets around the sun. The atom is electrically neutral because the number of orbital electrons is equal to the number of protons in the nucleus. The atom of one element differs from
that of another due to the differing numbers of protons, neutrons, and electrons in the atom.
In the Bohr atomic model, the electrons are assumed to move about the nucleus in certain
discrete circular orbits without radiating any energy. In any orbit, the angular momentum of
the electron is equal to an integral multiple of h/(2π), where h is Planck’s constant (h = 6.62
× 10–34 J.s). The integral number n has values 1, 2, 3 etc. for different orbits. The higher the
value of n, the larger the radius of the orbit.
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The allowable discrete values of n show that all energies are
not permitted for the electrons. The electrons can have only certain
discrete energies corresponding to the different integral values of n.
In other words, the electron energy is quantised. The allowable energies are represented by horizontal lines in a diagram, called the
energy level diagram of the atom. Fig. 1.1 shows such a diagram.
When an electron jumps from a higher energy level Eh to a
lower energy level El , an electromagnetic radiation of frequency ν is
emitted, where
Energy (eV)
Energy Bands in Solids
n
¥
5
4
3
2
1
Fig. 1.1 Energy level
diagram of an atom.
Eh − El
(1.2)
h
h being Planck’s constant. On the contrary, on absorbing a photon of energy hν, an electron
initially at the energy state El can move to the energy state Eh.
An electron normally occupies the lowest energy state, called the ground level, in the
atom. The other higher lying energy levels are called the excited levels of the atom. By absorbing more and more energies, an electron can move into excited states which are farther and
farther away from the nucleus. If the energy is sufficiently high, the electron can overcome the
attraction of the nucleus and gets detached from the atom. The energy required for this to
occur is known as the ionization potential. The energy level corresponding to n = ∞ in Fig. 1.1
represents the ionization level.
As the electrons are electrostatically attracted by the positively charged nucleus, the allowed
energies for the electrons are negative. The ionization level represents the zero level of energy.
The energies become more and more negative with decreasing values of n.
The wavelengths emitted from the atom due to electronic transitions from higher energy
states to lower ones give the spectral lines characterising the atom. To explain the details of the
spectral lines of some elements, improvements over the simple Bohr model have been made.
Quantum mechanical treatments have introduced four quantum numbers, designated by n, l,
ml and ms. The quantum numbers can take the following values:
n = 1, 2, 3, ...
l = 0, 1, 2, ..., (n – 1)
ml = 0, ± 1, ± 2, ..., ± l
ν=
1
2
The quantum number n, called the principal quantum number, primarily determines the
energy of the orbital electrons. The quantum number l measures the angular momentum of
the electron, and is referred to as the orbital angular momentum quantum number. The number
ml gives the splitting of the energies for a given n and l in a magnetic field, and is called the
magnetic quantum number. The quantity ms, known as the spin quantum number, shows that
the spin of the electron about its own axis is quantised.
The state of an electron in the atom is uniquely specified by the four quantum numbers.
This is a result of Pauli’s exclusion principle. The principle states that no two electrons in an
electronic system can occupy the same quantum state described by the same set of four quantum
numbers n, l, ml and ms.
ms = ±
Electronic Shells
The specific value of the principal quantum number n determines an electronic shell. All the
electrons of a given atom having the same value of n belong to the same electronic shell. The
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Electron Devices and Circuits
letters K, L, M, N, ... denote the shells for n = 1, 2, 3, 4, ... , respectively. The different values of
l for a given n define the subshells for the shell. The subshells are represented by s, p, d, f, ...
corresponding to l = 0, 1, 2, 3, ... , respectively.
For n = 1, there are two states corresponding to l = 0, ml = 0, and ms = ± 1/2. These two
states are known as 1s states. For n = 2, we have two 2s states corresponding to l = 0, ml = 0,
and ms = ± 1/2. In addition, there are six states for n = 2, l = 1, ml = –1, 0, + 1, and ms = ± 1/2.
These are 2p states. The total number of electrons in the K shell (n = 1) is thus 2, and that in
the L shell (n = 2) is 2 + 6 = 8. Similarly, it can be shown that a maximum of 10 electrons can be
accommodated in a d subshell, a maximum of 14 electrons in an f subshell, and so on.
The number of protons in the nucleus is the atomic number Z. Since the atom is electrically neutral, the number of orbiting electrons is also Z. For sodium, Z = 11. The electronic
configuration of the sodium atom is 1s2 2s2 2p6 3s1, where the superscripts denote the number of
electrons in a particular subshell. Clearly, the sodium atom has one electron in the outermost
subshell.
1.4
ENERGY BANDS IN CRYSTALS
A crystal is a solid consisting of a regular and
r ¬ + Ze ® r
zero
¥
repetitive arrangement of atoms or molecules
level
Ep(r)
(strictly speaking, ions) in space. If the positions of the atoms in the crystal are represented by points, called lattice points, we get
V(r)
a crystal lattice. The distance between the atoms in a crystal is fixed and is termed the
–¥
lattice constant of the crystal.
r ¬ + Ze ® r
(a)
(b)
To discuss the behaviour of electrons in
a crystal, we consider an isolated atom of the
Fig. 1.2 Variation of (a) Potential in the field of a
crystal. If Z is the atomic number, the atomic
nucleus with distance,
nucleus has a positive charge Ze. At a dis(b) Potential energy of an electron with its distance
from the nucleus.
tance r from the nucleus, the electrostatic
potential due to the nuclear charge is (in SI units)
Ze
V(r) =
,
(1.3)
4 πε 0 r
where ε0 is the permittivity of free space. Since an electron carries a negative charge, the
potential energy of an electron at a distance r from the nucleus is
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Potential energy
zero level
Ze 2
(1.4)
4πε 0 r
V(r) is positive while Ep(r) is negative. Both
V(r) and Ep(r) are zero at an infinite distance
from the nucleus. Figs. 1.2(a) and (b) show the
variation of V(r) and Ep(r), respectively with r.
We now consider two identical atoms
–¥
+ Ze
+ Ze
placed close together. The net potential energy
distance
of an electron is obtained as the sum of the poFig. 1.3 Potential energy variation of an electron
tential energies due to the two individual nuclei.
with distance between two identical nuclei.
In the region between the two nuclei, the net
potential energy is clearly smaller than the potential energy for an isolated nucleus (see Fig. 1.3).
Ep (r) = – eV(r) = –
Energy Bands in Solids
5
Potential energy
A zero level
The potential energy along a line
through a row of equispaced atomic nuclei, as
in a crystal, is diagrammatically shown in
Fig. 1.4. The potential energy between the
nuclei is found to consist of a series of humps.
At the boundary AB of the solid, the potential
+ Ze
+ Ze
+ Ze
+ Ze B
energy increases and approaches zero at infinity, there being no atoms on the other side
–¥
Distance
of the boundary to bring the curve down.
Fig. 1.4 Potential energy of an electron along
The total energy of an electron in an
a row of atoms in a crystal.
atom, kinetic plus potential, is negative and
has discrete values. These discrete energy levels in an isolated atom are shown by horizontal
lines in Fig. 1.5(a). When a number of atoms are brought close together to form a crystal, each
atom will exert an electric force on its neighbours. As a result of this interatomic coupling, the
crystal forms a single electronic system obeying Pauli’s exclusion principle. Therefore, each
energy level of the isolated atom splits into as many energy levels as there are atoms in the
crystal, so that Pauli’s exclusion principle is satisfied. The separation between the split-off
energy levels is very small. This large number of discrete and closely spaced energy levels
form an energy band. Energy bands are represented schematically by shaded regions in
Fig. 1.5(b).
The width of an energy band is determined by the parent energy level of the isolated atom
and the atomic spacing in the crystal. The lower energy levels are not greatly affected by the
interaction among the neighbouring atoms, and hence form narrow bands. The higher energy
levels are greatly affected by the interatomic interactions and produce wide bands. The
interatomic spacing, although fixed for a given crystal, is different for different crystals. The
width of an energy band thus depends on the type of the crystal, and is larger for a crystal with
a small interatomic spacing. The width of a band is independent of the number of atoms in the
Potential energy
Zero level
Potential
energy
Bands
Levels
+ Ze
+ Ze
+ Ze + Ze
+ Ze
Crystal
surface
Distance
(a) Isolated atom
(b) Crystal
Fig. 1.5 Splitting of energy levels of isolated atoms into energy bands as these
atoms are brought close together to produce a crystal.
crystal, but the number of energy levels in a band is equal to the number of atoms in the solid.
Consequently, as the number of atoms in the crystal increases, the separation between the
energy levels in a band decreases. As the crystal contains a large number of atoms (≈ 1029 m–3),
the spacing between the discrete levels in a band is so small that the band can be treated as
continuous.
The lower energy bands are normally completely filled by the electrons since the electrons
always tend to occupy the lowest available energy states. The higher energy bands may be
completely empty or may be partly filled by the electrons. Pauli’s exclusion principle restricts
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Electron Devices and Circuits
the number of electrons that a band can accommodate. A partly filled band appears when a
partly filled energy level produces an energy band or when a totally filled band and a totally
empty band overlap.
As the allowed energy levels of a single atom expand into energy bands in a crystal, the
electrons in a crystal cannot have energies in the region between two successive bands. In
other words, the energy bands are separated by gaps of forbidden energy.
The average energy of the electrons in the highest occupied band is usually much less
than the zero level marked in Fig. 1.5(b). The rise of the potential energy near the surface of
the crystal, as shown in Fig. 1.5(b), serves as a barrier preventing the electrons from escaping
from the crystal. If sufficient energy is imparted to the electrons by external means, they can
overcome the surface potential energy barrier, and come out of the crystal surface.
1.5
METAL, INSULATOR, AND SEMICONDUCTOR
On the basis of the band structure, crystals can be classified into metals, insulators, and semiconductors.
Metal
A crystalline solid is called a metal if the uppermost energy band is partly filled [Fig. 1.6(a)] or
the uppermost filled band and the next unoccupied band overlap in energy. Here, the electrons
in the uppermost band find neighbouring vacant states to move in, and thus behave as free
particles. In the presence of an applied electric field, these electrons gain energy from the field
and produce an electric current, so that a metal is a good conductor of electricity. The partly
filled band is called the conduction band. The electrons in the conduction band are known as
free electrons or conduction electrons.
Insulator
In some crystalline solids, the forbidden energy gap between the uppermost filled band, called
the valence band, and the lowermost empty band, called the conduction band, is very large. In
such solids, at ordinary temperatures only a few electrons can acquire enough thermal energy
to move from the valence band into the conduction band. Such solids are known as insulators.
Since only a few free electrons are available in the conduction band, an insulator is a bad
conductor of electricity. Diamond having a forbidden gap of 6 eV is a good example of an insulator. The energy band structure of an insulator is schematically shown in Fig. 1.6(b).
Semiconductor
A material for which the width of the forbidden energy gap between the valence and the
conduction band is relatively small (~ 1 eV) is referred to as a semiconductor. Germanium and
silicon having forbidden gaps of 0.78 and 1.2 eV, respectively, at 0 K are typical semiconductors.
As the forbidden gap is not very wide, some of the valence electrons acquire enough thermal
energy to go into the conduction band. These electrons then become free and can move about
under the action of an applied electric field. The absence of an electron in the valence band is
referred to as a hole. The holes also serve as carriers of electricity. The electrical conductivity
of a semiconductor is less than that of a metal but greater than that of an insulator. The band
diagram of a semiconductor is given in Fig. 1.6(c).
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Energy Bands in Solids
7
Partly full
(conduction band)
Empty
(conduction band)
Full
valence
band
Full
Full
Forbidden
gap
Nearly empty
(conduction band)
Forbidden gap
Nearly full
(valence band)
Full
(a)
Full
(b)
(c)
Fig. 1.6 Energy band structure of (a) metal, (b) insulator, and (c) semiconductor.
1.6
FERMI-DIRAC DISTRIBUTION FUNCTION
The free electrons in the conduction band of a metal are essentially in an equipotential region.
Only in the regions very close to an ion, there is a variation of potential. Since such regions
constitute a very small portion of the total volume available for the movement of electrons, the
electrons are assumed to move in a field-free or equipotential space. Due to their thermal
energy, the free electrons move about at random just like gas particles. Hence these electrons
are said to form an electron gas. Owing to the large number of free electrons (~ 1023 cm–3) in a
metal, principles of statistical mechanics are employed to determine their average behaviour.
A useful concept is the distribution function that gives the probability of occupancy of a given
state by the electrons. The Maxwell-Boltzmann (MB) distribution function is obtained from
classical ideas and does not incorporate Pauli’s exclusion principle. The MB distribution can be
applied when the number of particles in a system is much less than the number of available
quantum states. The free electrons in a metal cannot therefore be treated with the help of the
MB distribution function. This shortcoming of the MB function was removed in the FermiDirac (FD) distribution function which can be used to determine the energy distribution of free
electrons in a metal.
From statistical mechanics*, the FD distribution function is found to be
f (E) =
1
1 + exp [( E − EF ) / kB T ]
(1.5)
where f (E) is the probability of occupancy of the state with energy E, EF is a characteristic
energy for a particular solid and is referred to as the Fermi level, T is the absolute temperature
and kB is Boltzmann’s constant (kB = 1.38 × 10–23 J/K).
At the absolute zero of temperature, i.e. at T = 0 K,
1
Eq. (1.5) shows that f (E) = 1 for E < EF and f (E) = 0 for
T=0K
T = 300 K
E > EF . Thus all the energy states below EF are occupied f (E) −
T = 2000 K
by the electrons and all the energy states above EF are com1/2
pletely empty. Hence the Fermi energy EF denotes the maximum energy that can be occupied by the electrons at 0 K.
At temperatures greater than the absolute zero,
f (E) > 0 for E > EF, as shown in Fig. 1.7. This means that at
a finite temperature, some of the electrons in the quantum
states below EF acquire thermal energy to move into states
above EF.
0
1 E/EF
Fig. 1.7 Plot of f(E) against E/EF
for T = 0.300 and 2000 K.
*See, for example, ‘‘Quantum Mechanics, Statistical Mechanics, and Solid State Physics: An
Introduction’’ by D. Chattopadhyay and P.C. Rakshit (S. Chand and Co., New Delhi).
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8
Electron Devices and Circuits
When E = EF, Eq. (1.5) shows that f (E) = 1/2 for T > 0. Thus the Fermi level is the energy
level for which the probability of occupancy is 1/2 for a finite non-zero temperature. For most
metals, EF is less than 10 eV.
For a pure semiconductor and an insulator, the Fermi level lies near the middle of the
forbidden energy gap, whereas for a metal, the Fermi level lies within the conduction band
(Fig. 1.8).
Conduction
band
Conduction
band
Fermi
level
Valence
band
Valence
band
Conduction
band
Fig. 1.8 Schematic diagram showing the position of the Fermi level in (a) an insulator,
(b) a semiconductor, and (c) a metal.
1.7
SOLVED PROBLEMS
1. An electron at rest is accelerated through a potential difference of 100 V. Calculate its final
kinetic energy in J and eV. What is its final velocity?
Ans. The final kinetic energy of the electron is
EK = e × 100 J = 1.6 × 10–19 × 100 J = 1.6 × 10–17 J = 100 eV.
If v is the final velocity of the electron, we have
1
EK = m0 v2 where m0 is the rest mass of the electron. Thus
2
2 × 1.6 × 10 −17
= 5.927 × 106 m/s
9.11 × 10 −31
Since v << c, the relativistic correction is not necessary.
v=
2 EK
=
m0
2. An ion has a positive charge numerically equal to twice the electronic charge. The mass of the
ion is 7360 times that of an electron. The ion is initially stationary and accelerated through a
potential difference of 2 kV. Calculate the velocity and the kinetic energy acquired by the ion.
Ans. Let m and q be the mass and the charge of the ion, respectively. If v is the velocity of the
ion after moving through a potential difference of V volt, we have
Given,
1
m v2
2
m = 7360 m0 = 7360 × 9.11 × 10–31 kg,
q = 2 | e | = 2 × 1.6 × 10–19 C,
V = 2000 V.
Hence
v=
qV=
and
2qV
=
m
2 × 2 × 1.6 × 10 −19 × 2000
= 4.369 × 105 m/s
7360 × 9.11 × 10 −31
The kinetic energy of the ion is
1
mv2 = qV = 2 × 1.6 × 10–19 × 2000
2
= 6.4 × 10–16 J = 4000 eV
Since v << c, the relativistic correction is not necessary here.
EK =
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Energy Bands in Solids
9
3. A system of particles obeys Fermi-Dirac distribution function. Show that the probability of
vacancy of an energy level ∆E above the Fermi level EF is the same as the probability of occupancy
of an energy level ∆E below EF.
Ans. Let the energy above the Fermi energy EF be E1. Then ∆E = E1 –EF, and the
probability of occupancy f(E1) of the level E1 is given by the FD distribution function, i.e.,
f(E1) =
1
1
=
1 + exp ( E1 − E F ) kB T
1 + exp ( ∆E kBT )
The probability of vacancy of the level E1 is
1 – f(E1) = 1 –
1
exp ( ∆E kB T )
=
1 + exp (∆E kB T ) 1 + exp (∆E kB T )
The probability of occupancy of an energy level E2 below EF
where EF – E2 = ∆E, is
f(E2) =
=
1
1
=
1 + exp ( E2 − E F ) kBT
1 + exp ( − ∆E kBT )
exp (∆E kBT )
= 1 – f(E1),
1 + exp (∆E kBT )
which proves the desired result.
REVIEW QUESTIONS
1. What do you mean by the rest mass of an electron? Does the mass of an electron vary with its
velocity? If so, how?
2. What is ‘electron volt’? What is its relation with joule?
3. State and explain Pauli’s exclusion principle. What are electronic shells?
4. Define a crystal. Draw a diagram showing the variation of the potential energy of an electron
along a line of atomic nuclei in a crystal.
5. ‘The energy levels of an atom produce energy bands in a solid’. Explain.
6. What do you mean by the potential energy barrier at the surface of a crystal?
7. What are the factors determining (a) the width of an energy band in a crystal and (b) the number
of energy levels in the band?
8. Explain clearly the differences in the band structure of a metal, an insulator, and a semiconductor.
9. Write the Fermi-Dirac distribution function and explain its importance. What is Fermi level?
OBJECTIVE-TYPE QUESTIONS
1. Choose the correct answer:
(i) The electronic charge is (a) 1 C, (b) 1.6 µC, (c) 1.6 × 10–19 C.
(ii) The rest mass of an electron is (a) 10–20 kg, (b) 9.1 × 10–31 kg, (c) 9.1 × 10–28 kg.
(iii) The radius of an electron is (a) 10–15 m, (b) 10–10 m, (c) 10–9 m.
(iv) The relativistic correction for the electronic mass is necessary for accelerating potentials in
the range (a) 0 to 100 V, (b) 100 V to 6 kV, (c) 6 kV to 20 kV.
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Electron Devices and Circuits
(v) A quantum state specified by the four quantum numbers can be occupied by (a) one electron,
(b) two electrons, (c) any number of electrons.
(vi) The forbidden gap of silicon at 0 K is (a) 0.78 eV, (b) 1.2 eV, (c) 1.5 eV.
(vii) The Fermi level in a metal is usually less than (a) 1 eV, (b) 5 eV, (c) 10 eV.
(viii) The Fermi level in a pure semiconductor lies (a) near the middle of the forbidden gap, (b) in
the valence band, (c) in the conduction band.
Answers to Objective-Type Questions
1. (i) (c), (ii) (b), (iii) (a), (iv) (c), (v) (a), (vi) (b), (vii) (c), (viii) (a).
PROBLEMS
1. An electron, initially at rest, gains a speed of 107 m/s after being accelerated through a potential
difference of V volt. Determine V. What is the final kinetic energy of the electron in J and eV?
(Ans. 284.7 volt, 4.555 × 10–17 J, 284.7 eV)
2. A particle carries a positive charge numerically equal to the electronic charge. It acquires a velocity of 200 km/s after moving through a potential difference of 210 V. Determine the mass of the
particle relative to the electronic rest mass.
(Ans. 1844 m0)
3. A particle of charge 1.2 × 10–8C and mass 5 g travels a distance of 3 m under the action of a
potential difference of 500 V. Calculate the final velocity and the acceleration of the particle if it
starts from rest.
(Ans. 4.898 × 10–2m/s, 4 × 10–4m/s2)
4. A potential difference of 400 volt is applied between two parallel metal plates 4 cm apart. An
electron starts from rest from the negative plate. Obtain (i) the kinetic energy of the electron
when it reaches the positive plate and (ii) the time required by the electron to reach the positive
plate (Mass of an electron = 9.1 × 10–31 kg, electronic charge = 1.6 × 10–19 C).
(Ans. 400 eV, 6.745 × 10–9 s)
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