Current Electricity JEE Notes | EduRev

JEE : Current Electricity JEE Notes | EduRev

 Page 1


J E E - P h y s i c s
E
1
NODE6 (E)\Data\2014\Kota\JEE-Advanced\SMP\Phy\Unit-08\Current electricity\Eng\Theory.p65
CURRENT ELECTRICITY
In previous chapters we deal largely with electrostatics that is, with charges at rest. With this chapter we begin to focus
on electric currents, that is, charges in motion.
ELECTRIC CURRENT
Electric charges in motion constitute an electric current. Any medium having practically free electric charges,
free to migrate is a conductor of electricity. The electric charge flows from higher potential energy state to lower
potential energy state.
e
–
I
Positive charge flows from higher to lower potential and negative charge flows from lower to higher. Metals such
as gold, silver, copper, aluminium etc. are good conductors. When charge flows in a conductor from one place
to the other, then the rate of flow of charge is called electric current (I). When there is a transfer of charge from
one point to other point in a conductor, we say that there is an electric current through the area. If the moving
charges are positive, the current is in the direction of motion of charge. If they are negative the current is
opposite to the direction of motion. If a charge ?Q crosses an area in time ?t then the average electric current
through the area, during this time as
• Average current I
av
 = 
Q
t
?
?
• Instantaneous current 
t 0
Q dQ
I Lim
t dt
? ?
?
? ?
?
GOLDEN KEY POINTS
• Current is a fundamental quantity with dimension  [M
0
L
0
T
0
A¹ ]
• Current is a scalar quantity with its S ? unit ampere.
Ampere : The current through a conductor is said to be one ampere if one coulomb of charge is flowing
per second through a cross–section of wire.
• The conventional direction of current is the direction of flow of positive charge or applied field. It is opposite
to direction of flow of negatively charged electrons.
                         
• The conductor remains uncharged when current flows through it because the charge entering at one end
per second is equal to charge leaving the other end per second.
• For a given conductor current does not change with change in its cross–section because current is simply
rate of flow of charge.
• If n particles each having a charge q pass per second per unit area then current associated with cross–
sectional area A is 
q
nqA
t
?
? ? ?
?
.
• If there are n particles per unit volume each having a charge q and moving with velocity v then current
through cross–sectional area A is 
q
nqvA
t
?
? ? ?
?
• If a charge q is moving in a circle of radius r with speed v then its time period is T = 2 ?r/v. The equivalent
current 
q qv
T 2 r
? ? ?
?
.
JEEMAIN.GURU
Page 2


J E E - P h y s i c s
E
1
NODE6 (E)\Data\2014\Kota\JEE-Advanced\SMP\Phy\Unit-08\Current electricity\Eng\Theory.p65
CURRENT ELECTRICITY
In previous chapters we deal largely with electrostatics that is, with charges at rest. With this chapter we begin to focus
on electric currents, that is, charges in motion.
ELECTRIC CURRENT
Electric charges in motion constitute an electric current. Any medium having practically free electric charges,
free to migrate is a conductor of electricity. The electric charge flows from higher potential energy state to lower
potential energy state.
e
–
I
Positive charge flows from higher to lower potential and negative charge flows from lower to higher. Metals such
as gold, silver, copper, aluminium etc. are good conductors. When charge flows in a conductor from one place
to the other, then the rate of flow of charge is called electric current (I). When there is a transfer of charge from
one point to other point in a conductor, we say that there is an electric current through the area. If the moving
charges are positive, the current is in the direction of motion of charge. If they are negative the current is
opposite to the direction of motion. If a charge ?Q crosses an area in time ?t then the average electric current
through the area, during this time as
• Average current I
av
 = 
Q
t
?
?
• Instantaneous current 
t 0
Q dQ
I Lim
t dt
? ?
?
? ?
?
GOLDEN KEY POINTS
• Current is a fundamental quantity with dimension  [M
0
L
0
T
0
A¹ ]
• Current is a scalar quantity with its S ? unit ampere.
Ampere : The current through a conductor is said to be one ampere if one coulomb of charge is flowing
per second through a cross–section of wire.
• The conventional direction of current is the direction of flow of positive charge or applied field. It is opposite
to direction of flow of negatively charged electrons.
                         
• The conductor remains uncharged when current flows through it because the charge entering at one end
per second is equal to charge leaving the other end per second.
• For a given conductor current does not change with change in its cross–section because current is simply
rate of flow of charge.
• If n particles each having a charge q pass per second per unit area then current associated with cross–
sectional area A is 
q
nqA
t
?
? ? ?
?
.
• If there are n particles per unit volume each having a charge q and moving with velocity v then current
through cross–sectional area A is 
q
nqvA
t
?
? ? ?
?
• If a charge q is moving in a circle of radius r with speed v then its time period is T = 2 ?r/v. The equivalent
current 
q qv
T 2 r
? ? ?
?
.
JEEMAIN.GURU
J E E - P h y s i c s
2
E
NODE6 (E)\Data\2014\Kota\JEE-Advanced\SMP\Phy\Unit-08\Current electricity\Eng\Theory.p65
CL ASSIFICATION OF MATERIALS ACCORDING TO CONDUCTIVITY
( i ) C o n d u c t o r
In some materials, the outer electrons of each atoms or molecules are only weakly bound to it. These electrons
are almost free to move throughout the body of the material and are called free electrons. They are also known
as conduction electrons. When such a material is placed in an electric field, the free electrons move in a
direction opposite to the field. Such materials are called conductors.
( i i ) I n s u l a t o r
Another class of materials is called insulators in which all the electrons are tightly bound to their respective atoms
or molecules. Effectively, there are no free electrons. When such a material is placed in an electric field, the
electrons may slightly shift opposite to the field but they can’t leave their parent atoms or molecules and hence
can’t move through long distances. Such materials are also called dielectrics.
( i i i ) S e m i c o n d u c t o r
In semiconductors, the behaviour is like an insulator at low levels of temperature. But at higher temperatures,
a small number of electrons are able to free themselves and they respond to the applied electric field. As the
number of free electrons in a semiconductor is much smaller than that in a conductor, its behaviour is in between
a conductor and an insulator and hence, the name semiconductor. A freed electron in a semiconductor leaves a
vacancy in its normal bound position. These vacancies also help in conduction.
Behavior of conductor in absence of applied potential difference :
In absence of applied potential difference electrons have random motion. The average displacement and
average velocity is zero. There is no flow of current due to thermal motion of free electrons in a conductor.
The free electrons present in a conductor gain energy from temperature of surrounding and move randomly
in the conductor.
The speed gained by virtue of temperature is called as thermal speed of an electron 
2
rms
1
mv
2
=
3
2
kT
So thermal speed v
rms
 = 
3kT
m
 where m is mass of electron
At room temperature T = 300 K, v
rms
 = 10
5
 m/s
• Mean free path ? ? : ( ?~10Å)
 , 
total distance travelled
number of collisions
? ?
• Relaxation time : The time taken by an electron between two successive collisions is called as relaxation
time ? ? : ( ?~10
–14
s), Relaxation time : ? ?
total time taken
number of collisions
Behavior of conductor in presence of applied potential difference :
When two ends of a conductors are joined to a battery then one end is at higher potential and another at
lower potential. This produces an electric field inside the conductor from point of higher to lower potential
E = 
V
L
 where V = emf of the battery, L = length of the conductor. .
The field exerts an electric force on free electrons causing acceleration of each electron.
Acceleration of electron 
F eE
a=
m m
?
?
? ?
?
JEEMAIN.GURU
Page 3


J E E - P h y s i c s
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1
NODE6 (E)\Data\2014\Kota\JEE-Advanced\SMP\Phy\Unit-08\Current electricity\Eng\Theory.p65
CURRENT ELECTRICITY
In previous chapters we deal largely with electrostatics that is, with charges at rest. With this chapter we begin to focus
on electric currents, that is, charges in motion.
ELECTRIC CURRENT
Electric charges in motion constitute an electric current. Any medium having practically free electric charges,
free to migrate is a conductor of electricity. The electric charge flows from higher potential energy state to lower
potential energy state.
e
–
I
Positive charge flows from higher to lower potential and negative charge flows from lower to higher. Metals such
as gold, silver, copper, aluminium etc. are good conductors. When charge flows in a conductor from one place
to the other, then the rate of flow of charge is called electric current (I). When there is a transfer of charge from
one point to other point in a conductor, we say that there is an electric current through the area. If the moving
charges are positive, the current is in the direction of motion of charge. If they are negative the current is
opposite to the direction of motion. If a charge ?Q crosses an area in time ?t then the average electric current
through the area, during this time as
• Average current I
av
 = 
Q
t
?
?
• Instantaneous current 
t 0
Q dQ
I Lim
t dt
? ?
?
? ?
?
GOLDEN KEY POINTS
• Current is a fundamental quantity with dimension  [M
0
L
0
T
0
A¹ ]
• Current is a scalar quantity with its S ? unit ampere.
Ampere : The current through a conductor is said to be one ampere if one coulomb of charge is flowing
per second through a cross–section of wire.
• The conventional direction of current is the direction of flow of positive charge or applied field. It is opposite
to direction of flow of negatively charged electrons.
                         
• The conductor remains uncharged when current flows through it because the charge entering at one end
per second is equal to charge leaving the other end per second.
• For a given conductor current does not change with change in its cross–section because current is simply
rate of flow of charge.
• If n particles each having a charge q pass per second per unit area then current associated with cross–
sectional area A is 
q
nqA
t
?
? ? ?
?
.
• If there are n particles per unit volume each having a charge q and moving with velocity v then current
through cross–sectional area A is 
q
nqvA
t
?
? ? ?
?
• If a charge q is moving in a circle of radius r with speed v then its time period is T = 2 ?r/v. The equivalent
current 
q qv
T 2 r
? ? ?
?
.
JEEMAIN.GURU
J E E - P h y s i c s
2
E
NODE6 (E)\Data\2014\Kota\JEE-Advanced\SMP\Phy\Unit-08\Current electricity\Eng\Theory.p65
CL ASSIFICATION OF MATERIALS ACCORDING TO CONDUCTIVITY
( i ) C o n d u c t o r
In some materials, the outer electrons of each atoms or molecules are only weakly bound to it. These electrons
are almost free to move throughout the body of the material and are called free electrons. They are also known
as conduction electrons. When such a material is placed in an electric field, the free electrons move in a
direction opposite to the field. Such materials are called conductors.
( i i ) I n s u l a t o r
Another class of materials is called insulators in which all the electrons are tightly bound to their respective atoms
or molecules. Effectively, there are no free electrons. When such a material is placed in an electric field, the
electrons may slightly shift opposite to the field but they can’t leave their parent atoms or molecules and hence
can’t move through long distances. Such materials are also called dielectrics.
( i i i ) S e m i c o n d u c t o r
In semiconductors, the behaviour is like an insulator at low levels of temperature. But at higher temperatures,
a small number of electrons are able to free themselves and they respond to the applied electric field. As the
number of free electrons in a semiconductor is much smaller than that in a conductor, its behaviour is in between
a conductor and an insulator and hence, the name semiconductor. A freed electron in a semiconductor leaves a
vacancy in its normal bound position. These vacancies also help in conduction.
Behavior of conductor in absence of applied potential difference :
In absence of applied potential difference electrons have random motion. The average displacement and
average velocity is zero. There is no flow of current due to thermal motion of free electrons in a conductor.
The free electrons present in a conductor gain energy from temperature of surrounding and move randomly
in the conductor.
The speed gained by virtue of temperature is called as thermal speed of an electron 
2
rms
1
mv
2
=
3
2
kT
So thermal speed v
rms
 = 
3kT
m
 where m is mass of electron
At room temperature T = 300 K, v
rms
 = 10
5
 m/s
• Mean free path ? ? : ( ?~10Å)
 , 
total distance travelled
number of collisions
? ?
• Relaxation time : The time taken by an electron between two successive collisions is called as relaxation
time ? ? : ( ?~10
–14
s), Relaxation time : ? ?
total time taken
number of collisions
Behavior of conductor in presence of applied potential difference :
When two ends of a conductors are joined to a battery then one end is at higher potential and another at
lower potential. This produces an electric field inside the conductor from point of higher to lower potential
E = 
V
L
 where V = emf of the battery, L = length of the conductor. .
The field exerts an electric force on free electrons causing acceleration of each electron.
Acceleration of electron 
F eE
a=
m m
?
?
? ?
?
JEEMAIN.GURU
J E E - P h y s i c s
E
3
NODE6 (E)\Data\2014\Kota\JEE-Advanced\SMP\Phy\Unit-08\Current electricity\Eng\Theory.p65
DRIFT VELOCITY
Drift velocity is defined as the velocity with which the free electrons get drifted towards the positive terminal
under the effect of the applied external electric field. In addition to its thermal velocity, due to acceleration given
by applied electric field, the electron acquires a velocity component in a direction opposite to the direction of
the electric field. The gain in velocity due to the applied field is very small and is lost in the next collision.
e
–
I
Under the action of electric field : 
Random motion of an electron
with superimposed drift 
At any given time, an electron has a velocity 
1 1 1
v u a ? ? ?
? ? ?
,  where 
1
u
?
= the thermal velocity and
1
a ?
?
 = the velocity acquired by the electron under the influence of the applied electric field.
?
1
 = the time that has elapsed since the last collision. Similarly, the velocities of the other electrons are
2 2 2 3 3 3 N N N
u a , v u a ,... u a v v ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ?
.
The average velocity of all the free electrons in the conductor is equal to the drift velocity 
d
v
?
of the free electrons
1 2 3 N
d
v v v ...v
N
v ?
? ? ?
? ? ? ?
?
1 1 2 2 N N
(u a ) (u a ) ... (u )
N
a ? ? ? ? ? ? ? ? ?
?
? ? ? ? ?
1 2 N 1 2 N
( ... u ) ...
a
N N
u u ? ? ? ? ? ? ? ? ?
? ?
? ?
? ?
? ?
?
?
? ?
?
1 2 N
... u
0
N
u u ? ? ?
?
? ? ?
 ? 
1 2 N
d
...
N
v a
? ? ? ? ? ?
?
? ?
? ?
? ?
? ?
? ?
d
a v ? ?
? ?
–
m
eE
? ?
?
Note : Order of drift velocity is 10
–4
 m/s.
Relation between current and drift velocity :
Let n= number density of free electrons and A= area of cross–section of conductor.
Number of free electrons in conductor of length L = nAL, Total charge on these free electrons q neAL ? ?
Time taken by drifting electrons to cross conductor 
d
L
t
v
? ? 
q
current I=
t
?
?
?
 = neAL 
? ?
? ?
? ?
d
v
L
 = neAv
d
Example
Find free electrons per unit volume in a metallic wire of density 10
4
 kg/m
3
, atomic mass number  100 and
number of free electron per atom is one.
S o l uti o n
Number of free charge particle per unit volume (n)   = 
volume total
particle e rg a ch free total
?  No. of free electron per atom means total free electrons = total number  of atoms= 
A
W
N
M
M
?
So 
A
23 4
W A
3
W
N
M
M N 6.023 10 10
n d
V M 100 10
?
?
? ?
? ? ? ?
?
= 6.023 × 10
28
JEEMAIN.GURU
Page 4


J E E - P h y s i c s
E
1
NODE6 (E)\Data\2014\Kota\JEE-Advanced\SMP\Phy\Unit-08\Current electricity\Eng\Theory.p65
CURRENT ELECTRICITY
In previous chapters we deal largely with electrostatics that is, with charges at rest. With this chapter we begin to focus
on electric currents, that is, charges in motion.
ELECTRIC CURRENT
Electric charges in motion constitute an electric current. Any medium having practically free electric charges,
free to migrate is a conductor of electricity. The electric charge flows from higher potential energy state to lower
potential energy state.
e
–
I
Positive charge flows from higher to lower potential and negative charge flows from lower to higher. Metals such
as gold, silver, copper, aluminium etc. are good conductors. When charge flows in a conductor from one place
to the other, then the rate of flow of charge is called electric current (I). When there is a transfer of charge from
one point to other point in a conductor, we say that there is an electric current through the area. If the moving
charges are positive, the current is in the direction of motion of charge. If they are negative the current is
opposite to the direction of motion. If a charge ?Q crosses an area in time ?t then the average electric current
through the area, during this time as
• Average current I
av
 = 
Q
t
?
?
• Instantaneous current 
t 0
Q dQ
I Lim
t dt
? ?
?
? ?
?
GOLDEN KEY POINTS
• Current is a fundamental quantity with dimension  [M
0
L
0
T
0
A¹ ]
• Current is a scalar quantity with its S ? unit ampere.
Ampere : The current through a conductor is said to be one ampere if one coulomb of charge is flowing
per second through a cross–section of wire.
• The conventional direction of current is the direction of flow of positive charge or applied field. It is opposite
to direction of flow of negatively charged electrons.
                         
• The conductor remains uncharged when current flows through it because the charge entering at one end
per second is equal to charge leaving the other end per second.
• For a given conductor current does not change with change in its cross–section because current is simply
rate of flow of charge.
• If n particles each having a charge q pass per second per unit area then current associated with cross–
sectional area A is 
q
nqA
t
?
? ? ?
?
.
• If there are n particles per unit volume each having a charge q and moving with velocity v then current
through cross–sectional area A is 
q
nqvA
t
?
? ? ?
?
• If a charge q is moving in a circle of radius r with speed v then its time period is T = 2 ?r/v. The equivalent
current 
q qv
T 2 r
? ? ?
?
.
JEEMAIN.GURU
J E E - P h y s i c s
2
E
NODE6 (E)\Data\2014\Kota\JEE-Advanced\SMP\Phy\Unit-08\Current electricity\Eng\Theory.p65
CL ASSIFICATION OF MATERIALS ACCORDING TO CONDUCTIVITY
( i ) C o n d u c t o r
In some materials, the outer electrons of each atoms or molecules are only weakly bound to it. These electrons
are almost free to move throughout the body of the material and are called free electrons. They are also known
as conduction electrons. When such a material is placed in an electric field, the free electrons move in a
direction opposite to the field. Such materials are called conductors.
( i i ) I n s u l a t o r
Another class of materials is called insulators in which all the electrons are tightly bound to their respective atoms
or molecules. Effectively, there are no free electrons. When such a material is placed in an electric field, the
electrons may slightly shift opposite to the field but they can’t leave their parent atoms or molecules and hence
can’t move through long distances. Such materials are also called dielectrics.
( i i i ) S e m i c o n d u c t o r
In semiconductors, the behaviour is like an insulator at low levels of temperature. But at higher temperatures,
a small number of electrons are able to free themselves and they respond to the applied electric field. As the
number of free electrons in a semiconductor is much smaller than that in a conductor, its behaviour is in between
a conductor and an insulator and hence, the name semiconductor. A freed electron in a semiconductor leaves a
vacancy in its normal bound position. These vacancies also help in conduction.
Behavior of conductor in absence of applied potential difference :
In absence of applied potential difference electrons have random motion. The average displacement and
average velocity is zero. There is no flow of current due to thermal motion of free electrons in a conductor.
The free electrons present in a conductor gain energy from temperature of surrounding and move randomly
in the conductor.
The speed gained by virtue of temperature is called as thermal speed of an electron 
2
rms
1
mv
2
=
3
2
kT
So thermal speed v
rms
 = 
3kT
m
 where m is mass of electron
At room temperature T = 300 K, v
rms
 = 10
5
 m/s
• Mean free path ? ? : ( ?~10Å)
 , 
total distance travelled
number of collisions
? ?
• Relaxation time : The time taken by an electron between two successive collisions is called as relaxation
time ? ? : ( ?~10
–14
s), Relaxation time : ? ?
total time taken
number of collisions
Behavior of conductor in presence of applied potential difference :
When two ends of a conductors are joined to a battery then one end is at higher potential and another at
lower potential. This produces an electric field inside the conductor from point of higher to lower potential
E = 
V
L
 where V = emf of the battery, L = length of the conductor. .
The field exerts an electric force on free electrons causing acceleration of each electron.
Acceleration of electron 
F eE
a=
m m
?
?
? ?
?
JEEMAIN.GURU
J E E - P h y s i c s
E
3
NODE6 (E)\Data\2014\Kota\JEE-Advanced\SMP\Phy\Unit-08\Current electricity\Eng\Theory.p65
DRIFT VELOCITY
Drift velocity is defined as the velocity with which the free electrons get drifted towards the positive terminal
under the effect of the applied external electric field. In addition to its thermal velocity, due to acceleration given
by applied electric field, the electron acquires a velocity component in a direction opposite to the direction of
the electric field. The gain in velocity due to the applied field is very small and is lost in the next collision.
e
–
I
Under the action of electric field : 
Random motion of an electron
with superimposed drift 
At any given time, an electron has a velocity 
1 1 1
v u a ? ? ?
? ? ?
,  where 
1
u
?
= the thermal velocity and
1
a ?
?
 = the velocity acquired by the electron under the influence of the applied electric field.
?
1
 = the time that has elapsed since the last collision. Similarly, the velocities of the other electrons are
2 2 2 3 3 3 N N N
u a , v u a ,... u a v v ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ?
.
The average velocity of all the free electrons in the conductor is equal to the drift velocity 
d
v
?
of the free electrons
1 2 3 N
d
v v v ...v
N
v ?
? ? ?
? ? ? ?
?
1 1 2 2 N N
(u a ) (u a ) ... (u )
N
a ? ? ? ? ? ? ? ? ?
?
? ? ? ? ?
1 2 N 1 2 N
( ... u ) ...
a
N N
u u ? ? ? ? ? ? ? ? ?
? ?
? ?
? ?
? ?
?
?
? ?
?
1 2 N
... u
0
N
u u ? ? ?
?
? ? ?
 ? 
1 2 N
d
...
N
v a
? ? ? ? ? ?
?
? ?
? ?
? ?
? ?
? ?
d
a v ? ?
? ?
–
m
eE
? ?
?
Note : Order of drift velocity is 10
–4
 m/s.
Relation between current and drift velocity :
Let n= number density of free electrons and A= area of cross–section of conductor.
Number of free electrons in conductor of length L = nAL, Total charge on these free electrons q neAL ? ?
Time taken by drifting electrons to cross conductor 
d
L
t
v
? ? 
q
current I=
t
?
?
?
 = neAL 
? ?
? ?
? ?
d
v
L
 = neAv
d
Example
Find free electrons per unit volume in a metallic wire of density 10
4
 kg/m
3
, atomic mass number  100 and
number of free electron per atom is one.
S o l uti o n
Number of free charge particle per unit volume (n)   = 
volume total
particle e rg a ch free total
?  No. of free electron per atom means total free electrons = total number  of atoms= 
A
W
N
M
M
?
So 
A
23 4
W A
3
W
N
M
M N 6.023 10 10
n d
V M 100 10
?
?
? ?
? ? ? ?
?
= 6.023 × 10
28
JEEMAIN.GURU
J E E - P h y s i c s
4
E
NODE6 (E)\Data\2014\Kota\JEE-Advanced\SMP\Phy\Unit-08\Current electricity\Eng\Theory.p65
CURRENT DENSITY (J)
Current is a macroscopic quantity and deals with the overall rate of flow of charge through a section. To specify the
current with direction in the microscopic level at a point, the term current density is introduced. Current density at
any point inside a conductor is defined as a vector having magnitude equal to current per unit area surrounding that
point. Remember area is normal to the direction of charge flow (or current passes) through that point.
• Current density at point P is given by 
dI
J n
dA
?
?
?
+
I
dA
n
J I
?
?
dA
J
dA cos ?
–
• If the cross–sectional area is not normal to the current, but makes an angle ? with the direction of current
then 
dI
J
dA cos
?
?
? dI = JdA cos ? = 
J.dA
? ?
 ? I J . dA ?
?
? ? ? ? ?
• Current density 
J
?
 is a vector quantity. It's direction is same as that of E
?
. It's S.I. unit is
ampere/m
2
 and dimension [L
–2
A].
Example
The current density at a point is ? ?
?
? ?
?
4 2
ˆ
J 2 10 j Jm .
Find the rate of charge flow through a cross sectional area ? ?
2
ˆ ˆ
S 2i 3 j cm ? ?
?
Solution
The rate of flow of charge = current = I = J.dS
?
? ?
 ? I = 
J.S
? ?
 = ? ?
4
2 10 ?
 ? ?
4
ˆ ˆ ˆ
j 2i 3 j 10 A 6A
?
? ?
? ? ? ?
? ?
Example
A potential difference applied to the ends of a wire made up of an alloy drives a current through it. The current
density varies as  J = 3 + 2r, where r is the distance of the point from the axis. If R be the radius of the wire, then
the total current through any cross section of the wire.
Solution
Consider a circular strip of radius r and thickness dr
dI = 
J.dS
? ?
= ? ? ? ?
3 2r 2 rdr cos0 ? ? ?
= ? ?
2
2 3r 2r dr ? ?
? ?
R
2
0
I 2 3r 2r dr ? ? ?
?
 = 
R
2
3
0
3r 2
2 r
2 3
? ?
? ?
? ?
? ?
 = 
2 3
3R 2R
2
2 3
? ?
? ?
? ?
? ?
 units
RELATION BETWEEN CURRENT DENSITY, CONDUCTIVITY AND ELECTRIC FIELD
Let the number of free electrons per unit volume in a conductor = n
Total number of electrons in dx distance = n (Adx)
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CURRENT ELECTRICITY
In previous chapters we deal largely with electrostatics that is, with charges at rest. With this chapter we begin to focus
on electric currents, that is, charges in motion.
ELECTRIC CURRENT
Electric charges in motion constitute an electric current. Any medium having practically free electric charges,
free to migrate is a conductor of electricity. The electric charge flows from higher potential energy state to lower
potential energy state.
e
–
I
Positive charge flows from higher to lower potential and negative charge flows from lower to higher. Metals such
as gold, silver, copper, aluminium etc. are good conductors. When charge flows in a conductor from one place
to the other, then the rate of flow of charge is called electric current (I). When there is a transfer of charge from
one point to other point in a conductor, we say that there is an electric current through the area. If the moving
charges are positive, the current is in the direction of motion of charge. If they are negative the current is
opposite to the direction of motion. If a charge ?Q crosses an area in time ?t then the average electric current
through the area, during this time as
• Average current I
av
 = 
Q
t
?
?
• Instantaneous current 
t 0
Q dQ
I Lim
t dt
? ?
?
? ?
?
GOLDEN KEY POINTS
• Current is a fundamental quantity with dimension  [M
0
L
0
T
0
A¹ ]
• Current is a scalar quantity with its S ? unit ampere.
Ampere : The current through a conductor is said to be one ampere if one coulomb of charge is flowing
per second through a cross–section of wire.
• The conventional direction of current is the direction of flow of positive charge or applied field. It is opposite
to direction of flow of negatively charged electrons.
                         
• The conductor remains uncharged when current flows through it because the charge entering at one end
per second is equal to charge leaving the other end per second.
• For a given conductor current does not change with change in its cross–section because current is simply
rate of flow of charge.
• If n particles each having a charge q pass per second per unit area then current associated with cross–
sectional area A is 
q
nqA
t
?
? ? ?
?
.
• If there are n particles per unit volume each having a charge q and moving with velocity v then current
through cross–sectional area A is 
q
nqvA
t
?
? ? ?
?
• If a charge q is moving in a circle of radius r with speed v then its time period is T = 2 ?r/v. The equivalent
current 
q qv
T 2 r
? ? ?
?
.
JEEMAIN.GURU
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NODE6 (E)\Data\2014\Kota\JEE-Advanced\SMP\Phy\Unit-08\Current electricity\Eng\Theory.p65
CL ASSIFICATION OF MATERIALS ACCORDING TO CONDUCTIVITY
( i ) C o n d u c t o r
In some materials, the outer electrons of each atoms or molecules are only weakly bound to it. These electrons
are almost free to move throughout the body of the material and are called free electrons. They are also known
as conduction electrons. When such a material is placed in an electric field, the free electrons move in a
direction opposite to the field. Such materials are called conductors.
( i i ) I n s u l a t o r
Another class of materials is called insulators in which all the electrons are tightly bound to their respective atoms
or molecules. Effectively, there are no free electrons. When such a material is placed in an electric field, the
electrons may slightly shift opposite to the field but they can’t leave their parent atoms or molecules and hence
can’t move through long distances. Such materials are also called dielectrics.
( i i i ) S e m i c o n d u c t o r
In semiconductors, the behaviour is like an insulator at low levels of temperature. But at higher temperatures,
a small number of electrons are able to free themselves and they respond to the applied electric field. As the
number of free electrons in a semiconductor is much smaller than that in a conductor, its behaviour is in between
a conductor and an insulator and hence, the name semiconductor. A freed electron in a semiconductor leaves a
vacancy in its normal bound position. These vacancies also help in conduction.
Behavior of conductor in absence of applied potential difference :
In absence of applied potential difference electrons have random motion. The average displacement and
average velocity is zero. There is no flow of current due to thermal motion of free electrons in a conductor.
The free electrons present in a conductor gain energy from temperature of surrounding and move randomly
in the conductor.
The speed gained by virtue of temperature is called as thermal speed of an electron 
2
rms
1
mv
2
=
3
2
kT
So thermal speed v
rms
 = 
3kT
m
 where m is mass of electron
At room temperature T = 300 K, v
rms
 = 10
5
 m/s
• Mean free path ? ? : ( ?~10Å)
 , 
total distance travelled
number of collisions
? ?
• Relaxation time : The time taken by an electron between two successive collisions is called as relaxation
time ? ? : ( ?~10
–14
s), Relaxation time : ? ?
total time taken
number of collisions
Behavior of conductor in presence of applied potential difference :
When two ends of a conductors are joined to a battery then one end is at higher potential and another at
lower potential. This produces an electric field inside the conductor from point of higher to lower potential
E = 
V
L
 where V = emf of the battery, L = length of the conductor. .
The field exerts an electric force on free electrons causing acceleration of each electron.
Acceleration of electron 
F eE
a=
m m
?
?
? ?
?
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DRIFT VELOCITY
Drift velocity is defined as the velocity with which the free electrons get drifted towards the positive terminal
under the effect of the applied external electric field. In addition to its thermal velocity, due to acceleration given
by applied electric field, the electron acquires a velocity component in a direction opposite to the direction of
the electric field. The gain in velocity due to the applied field is very small and is lost in the next collision.
e
–
I
Under the action of electric field : 
Random motion of an electron
with superimposed drift 
At any given time, an electron has a velocity 
1 1 1
v u a ? ? ?
? ? ?
,  where 
1
u
?
= the thermal velocity and
1
a ?
?
 = the velocity acquired by the electron under the influence of the applied electric field.
?
1
 = the time that has elapsed since the last collision. Similarly, the velocities of the other electrons are
2 2 2 3 3 3 N N N
u a , v u a ,... u a v v ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ?
.
The average velocity of all the free electrons in the conductor is equal to the drift velocity 
d
v
?
of the free electrons
1 2 3 N
d
v v v ...v
N
v ?
? ? ?
? ? ? ?
?
1 1 2 2 N N
(u a ) (u a ) ... (u )
N
a ? ? ? ? ? ? ? ? ?
?
? ? ? ? ?
1 2 N 1 2 N
( ... u ) ...
a
N N
u u ? ? ? ? ? ? ? ? ?
? ?
? ?
? ?
? ?
?
?
? ?
?
1 2 N
... u
0
N
u u ? ? ?
?
? ? ?
 ? 
1 2 N
d
...
N
v a
? ? ? ? ? ?
?
? ?
? ?
? ?
? ?
? ?
d
a v ? ?
? ?
–
m
eE
? ?
?
Note : Order of drift velocity is 10
–4
 m/s.
Relation between current and drift velocity :
Let n= number density of free electrons and A= area of cross–section of conductor.
Number of free electrons in conductor of length L = nAL, Total charge on these free electrons q neAL ? ?
Time taken by drifting electrons to cross conductor 
d
L
t
v
? ? 
q
current I=
t
?
?
?
 = neAL 
? ?
? ?
? ?
d
v
L
 = neAv
d
Example
Find free electrons per unit volume in a metallic wire of density 10
4
 kg/m
3
, atomic mass number  100 and
number of free electron per atom is one.
S o l uti o n
Number of free charge particle per unit volume (n)   = 
volume total
particle e rg a ch free total
?  No. of free electron per atom means total free electrons = total number  of atoms= 
A
W
N
M
M
?
So 
A
23 4
W A
3
W
N
M
M N 6.023 10 10
n d
V M 100 10
?
?
? ?
? ? ? ?
?
= 6.023 × 10
28
JEEMAIN.GURU
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NODE6 (E)\Data\2014\Kota\JEE-Advanced\SMP\Phy\Unit-08\Current electricity\Eng\Theory.p65
CURRENT DENSITY (J)
Current is a macroscopic quantity and deals with the overall rate of flow of charge through a section. To specify the
current with direction in the microscopic level at a point, the term current density is introduced. Current density at
any point inside a conductor is defined as a vector having magnitude equal to current per unit area surrounding that
point. Remember area is normal to the direction of charge flow (or current passes) through that point.
• Current density at point P is given by 
dI
J n
dA
?
?
?
+
I
dA
n
J I
?
?
dA
J
dA cos ?
–
• If the cross–sectional area is not normal to the current, but makes an angle ? with the direction of current
then 
dI
J
dA cos
?
?
? dI = JdA cos ? = 
J.dA
? ?
 ? I J . dA ?
?
? ? ? ? ?
• Current density 
J
?
 is a vector quantity. It's direction is same as that of E
?
. It's S.I. unit is
ampere/m
2
 and dimension [L
–2
A].
Example
The current density at a point is ? ?
?
? ?
?
4 2
ˆ
J 2 10 j Jm .
Find the rate of charge flow through a cross sectional area ? ?
2
ˆ ˆ
S 2i 3 j cm ? ?
?
Solution
The rate of flow of charge = current = I = J.dS
?
? ?
 ? I = 
J.S
? ?
 = ? ?
4
2 10 ?
 ? ?
4
ˆ ˆ ˆ
j 2i 3 j 10 A 6A
?
? ?
? ? ? ?
? ?
Example
A potential difference applied to the ends of a wire made up of an alloy drives a current through it. The current
density varies as  J = 3 + 2r, where r is the distance of the point from the axis. If R be the radius of the wire, then
the total current through any cross section of the wire.
Solution
Consider a circular strip of radius r and thickness dr
dI = 
J.dS
? ?
= ? ? ? ?
3 2r 2 rdr cos0 ? ? ?
= ? ?
2
2 3r 2r dr ? ?
? ?
R
2
0
I 2 3r 2r dr ? ? ?
?
 = 
R
2
3
0
3r 2
2 r
2 3
? ?
? ?
? ?
? ?
 = 
2 3
3R 2R
2
2 3
? ?
? ?
? ?
? ?
 units
RELATION BETWEEN CURRENT DENSITY, CONDUCTIVITY AND ELECTRIC FIELD
Let the number of free electrons per unit volume in a conductor = n
Total number of electrons in dx distance = n (Adx)
JEEMAIN.GURU
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NODE6 (E)\Data\2014\Kota\JEE-Advanced\SMP\Phy\Unit-08\Current electricity\Eng\Theory.p65
Total charge dQ = n (Adx)e
Current 
dQ dx
I nAe
dt dt
? ?
 = neAv
d
, Current density 
I
J
A
? = nev
d
= 
eE
ne
m
?
? ?
? ?
? ?
 ? ? ? ?
d
eE
v
m
? ?
? ?
? ?
? ?
 ?
2
ne
J E
m
? ? ?
?
? ?
? ?
? J = ?E, where conductivity  
2
ne
m
?
? ?
? depends only on the material of the conductor and its temperature.
In vector form 
J E ? ?
? ?
 Ohm's law (at microscopic level)
RELATION BETWEEN POTENTIAL DIFFERENCE AND CURRENT (Ohm's Law)
If the physical conditions of the conductor (length, temperature, mechanical strain etc.) remains same, then the
current flowing through the conductor is directly proportional to the potential difference across it's two ends i.e.
I ? V ? V = IR  where R is a proportionality constant, known as electric resistance. Ohm's law (at macroscopic level)
• Ohm's law is not a universal law. The substances, which obey ohm's law are known as ohmic.
• Graph between V and I for a metallic conductor is a straight line as shown.        
?
? ?
V ?
Slope of the line ? ? ? ?
V
tan R
I
GOLDEN KEY POINTS
 • 1 ampere of current means the flow of 6.25 ×  10
18
 electrons per second through any cross section of
conductor.
 • Current is a scalar quantity but current density is a vector quantity.
 • Order of free electron density in conductors = 10
28
 electrons/m
3
 •
v
T
? ? v
d
• If a steady current flows in a metallic conductor of non uniform cross section.
(i) Along the wire I is same.
(ii) Current density and drift velocity depends on area
A
1
A
2
 I
1
 = I
2
 , A
1
 < A
2  
? J
1
 > J
2
 , E
1
 > E
2
 , 
1 2
d d
v v ?
• If the temperature of the conductor increases, the amplitude of the vibrations of the positive ions in the
conductor also increase. Due to this, the free electrons collide more frequently with the vibrating ions and as
a result, the average relaxation time decreases.
• At different temperatures V–I curves are different.
Here tan ?
1
 > tan ?
2  
   So R
1
 > R
2
i.e. T
1
 > T
2
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