20. CURRENT ELECTRICITY
1. INTRODUCTION
Transfer of charge across a cross-section of a conducting medium constitutes an electric current. Any conductor in
general offers some resistance to the flow of electric current through it. This means an electric current cannot flow
continuously all by itself in a conductor. An electric source is needed to continuously drive electric current through
a conductor. Some work is done or energy is supplied by the source to drive the current in a conducting medium.
In this chapter we will study the laws and phenomena that govern the flow of electric current in conductors. We
will discuss the factors that affect the electrical properties of conductors, what constitutes an electric circuit and
the laws of division of current in various branches of a complicated network. Electricity has indeed transformed
our lives beyond imagination. Electric energy is used everywhere, right from the lights of our homes, our electronic
appliances, computers, automobiles, heavy machines used in our industries, hospitals, aircrafts etc. We will mainly
focus on direct current circuits and sources in this chapter. The techniques of circuit analysis developed in this
chapter form the backbone of electrical and electronics science and engineering.
2. ELECTRIC CURRENT
In this chapter we will be mainly dealing with current in a conducting medium. Electric current is defined as the
rate of flow of electric charge through a certain cross-section of a conductor. If there is to be an electric current
through a given surface, there must be a net flow of charge through the surface. The free electrons (conduction
electrons)in an isolated conductor are in random chaotic motion in all directions and on an average same number
of electrons passes through each side of any imaginary surface. Thus, the net charge passing through any surface
in any time interval is zero, and thus the current through the conductor is zero. However, if we connect the ends
of the conductor to a battery, an electric field is applied inside the conductor from positive terminal to negative
terminal, and the motion of the electrons is biased opposite to the electric field, with the result that an ordered
motion with a certain average velocity u opposite to the direction of electric field is superimposed on the chaotic
motion of the electrons. Thus there is a net flow of negative charge opposite to the electric field, or equivalently
flow of net positive charge in the direction of electric field. Thus an electric current flows through the conductor in
the direction of electric field.
If charge dq passes through an imaginary surface in time dt, then the current I through that surface is defined
dq
as I = (definition of current). The direction of the current is the direction of flow of positive charge carriers, or
dt
opposite to the flow of electrons.
Also, we can write dq = i dt. The charge that passes though the surface in a time interval extending from 0 to t is
t
given as:=q ∫=
dq ∫ idt
0
(the current i in general varies with time).
The SI unit for current is coulomb per second or the ampere (A), which is an SI base unit:
1 ampere = 1 A = 1 coulomb per second = 1 Cs-1.
,2 0 . 2 | Current Electricity
3. CURRENT DENSITY
In general the electric current is distributed non-uniformly over the surface through which it passes. So to analyse
the current through an elementary surface of infinitesimal area at any point inside the conducting medium, we
introduce a current density vector j .The magnitude of current density vector at any point P, is equal to the ratio
of current dI through an elementary surface perpendicular to the direction of current at P to the area dS ⊥ of this
elementary surface. The direction of j is the same as the notion of dI at that point, or the direction of velocity
vector u of the ordered motion of positive charge carriers.
∆I
If ∆I be the current through the area ∆S ⊥ ,the magnitude of average current density is j = .
∆S ⊥
dI
The magnitude of current density at the point P is j = .
dS ⊥
S cos
Q
P
Q S n
= i S
t
Figure 20.1: Current and current density
If the area dS is not perpendicular to the current dI through it, i.e. the normal to the area makes some angle θ with
the notion of the current, then the current density is given as,
dI
j= or,
= dI jdS cos θ
dS cos θ
If dS be the area vector corresponding to the area dS, we have dI = j.dS
For a finite area, I = ∫ j.dS
An electric current is not a vector quantity. It does not follow the laws of vector addition. The current density is a
vector quantity.
MASTERJEE CONCEPTS
Direction of Current
•• Direction of drift of electrons is in the opposite direction of electric field in conducting wires.
•• It is not always along the length of the wire (direction of cross section). We take the component of the
velocity along the wire.
Yashwanth Sandupatla (JEE 2012, AIR 821)
4. DRIFT SPEED
A conductor contains a large number of loosely bound electrons called free electrons or conduction electrons.
These electrons move randomly in all directions within the entire volume of the conductor, (see Fig. 20.2) and in
this process keep on colliding with the atoms/molecules/ions of the conductor, changing their direction of motion
at each collision.
, P hysi cs | 20.3
E
x y x y
(a) (b)
Figure 20.2: Random motion of electron inside conductor
When an electric field is applied inside the conductor, each electron experiences a force in the direction opposite
to the field. The chaotic motion of electrons gets biased in favour of this force. At each collision with a molecule,
the electron changes its direction of motion and moves with a random velocity but gains an additional velocity
eE
ve = τô in the direction opposite to the electric field till the next collision happens and the direction of its motion
m
again changes abruptly. As the average time τ between successive collisions is small, the electrons slowly and
steadily drift opposite to the applied field (see Fig. 20.3) with an average drift speed vd.
The distance drifted during successive collisions can be written as
= 1 a( τ)2 = 1 eE ( τ)2
2 2 m
1 eE
The drift speed will be given by the relation: vd= = τ
τ 2 m
E
A vd
Figure 20.3: Calculating drift speed
Current density can be expressed in terms of the drift speed. Consider a cylindrical conductor of cross-sectional
area A in which electric field E exists. Consider a length L= v d ∆t of the conductor. The volume of this portion is
Av d ∆t . If there are n free electrons per unit volume of the conductor, the number of free electrons in this portion
are nA v d Δt. All these electrons cross the area A in time Δt. Thus, the charge crossing this cross-section in time Δt is
∆Q
Q neAv d ∆t or current through the conductor is,=
∆= I = neAv d
∆t
I
Therefore current density is: =j = nev d
A
Illustration 1: If n =8.5 × 1028 m–3,how long does an electron take to drift from one end of 3 m long wire to its other
end? The area of cross section of the wire is 2.0 × 10-6 m2and it is carrying a current of 3.0 Ampere. (JEE MAIN)
Sol: For constant p.d. across conductor the electrons drifts with constant drift velocity across it. If we find the drift
velocity, the time to drift across wire of constant length is easily found out
Given that: (i) Number density n = 8.5 × 1028 m–3 (ii) Cross-sectional area A = 2.0 × 10-6m2
(iii) Current I = 3 A (iv) Charge on electron e = 1.6 × 10–19 C
Current in terms of drift speed is expressed as I = neAvd
I
⇒ vd = .
neA
Now time taken to cross the length of the wire is:
1. INTRODUCTION
Transfer of charge across a cross-section of a conducting medium constitutes an electric current. Any conductor in
general offers some resistance to the flow of electric current through it. This means an electric current cannot flow
continuously all by itself in a conductor. An electric source is needed to continuously drive electric current through
a conductor. Some work is done or energy is supplied by the source to drive the current in a conducting medium.
In this chapter we will study the laws and phenomena that govern the flow of electric current in conductors. We
will discuss the factors that affect the electrical properties of conductors, what constitutes an electric circuit and
the laws of division of current in various branches of a complicated network. Electricity has indeed transformed
our lives beyond imagination. Electric energy is used everywhere, right from the lights of our homes, our electronic
appliances, computers, automobiles, heavy machines used in our industries, hospitals, aircrafts etc. We will mainly
focus on direct current circuits and sources in this chapter. The techniques of circuit analysis developed in this
chapter form the backbone of electrical and electronics science and engineering.
2. ELECTRIC CURRENT
In this chapter we will be mainly dealing with current in a conducting medium. Electric current is defined as the
rate of flow of electric charge through a certain cross-section of a conductor. If there is to be an electric current
through a given surface, there must be a net flow of charge through the surface. The free electrons (conduction
electrons)in an isolated conductor are in random chaotic motion in all directions and on an average same number
of electrons passes through each side of any imaginary surface. Thus, the net charge passing through any surface
in any time interval is zero, and thus the current through the conductor is zero. However, if we connect the ends
of the conductor to a battery, an electric field is applied inside the conductor from positive terminal to negative
terminal, and the motion of the electrons is biased opposite to the electric field, with the result that an ordered
motion with a certain average velocity u opposite to the direction of electric field is superimposed on the chaotic
motion of the electrons. Thus there is a net flow of negative charge opposite to the electric field, or equivalently
flow of net positive charge in the direction of electric field. Thus an electric current flows through the conductor in
the direction of electric field.
If charge dq passes through an imaginary surface in time dt, then the current I through that surface is defined
dq
as I = (definition of current). The direction of the current is the direction of flow of positive charge carriers, or
dt
opposite to the flow of electrons.
Also, we can write dq = i dt. The charge that passes though the surface in a time interval extending from 0 to t is
t
given as:=q ∫=
dq ∫ idt
0
(the current i in general varies with time).
The SI unit for current is coulomb per second or the ampere (A), which is an SI base unit:
1 ampere = 1 A = 1 coulomb per second = 1 Cs-1.
,2 0 . 2 | Current Electricity
3. CURRENT DENSITY
In general the electric current is distributed non-uniformly over the surface through which it passes. So to analyse
the current through an elementary surface of infinitesimal area at any point inside the conducting medium, we
introduce a current density vector j .The magnitude of current density vector at any point P, is equal to the ratio
of current dI through an elementary surface perpendicular to the direction of current at P to the area dS ⊥ of this
elementary surface. The direction of j is the same as the notion of dI at that point, or the direction of velocity
vector u of the ordered motion of positive charge carriers.
∆I
If ∆I be the current through the area ∆S ⊥ ,the magnitude of average current density is j = .
∆S ⊥
dI
The magnitude of current density at the point P is j = .
dS ⊥
S cos
Q
P
Q S n
= i S
t
Figure 20.1: Current and current density
If the area dS is not perpendicular to the current dI through it, i.e. the normal to the area makes some angle θ with
the notion of the current, then the current density is given as,
dI
j= or,
= dI jdS cos θ
dS cos θ
If dS be the area vector corresponding to the area dS, we have dI = j.dS
For a finite area, I = ∫ j.dS
An electric current is not a vector quantity. It does not follow the laws of vector addition. The current density is a
vector quantity.
MASTERJEE CONCEPTS
Direction of Current
•• Direction of drift of electrons is in the opposite direction of electric field in conducting wires.
•• It is not always along the length of the wire (direction of cross section). We take the component of the
velocity along the wire.
Yashwanth Sandupatla (JEE 2012, AIR 821)
4. DRIFT SPEED
A conductor contains a large number of loosely bound electrons called free electrons or conduction electrons.
These electrons move randomly in all directions within the entire volume of the conductor, (see Fig. 20.2) and in
this process keep on colliding with the atoms/molecules/ions of the conductor, changing their direction of motion
at each collision.
, P hysi cs | 20.3
E
x y x y
(a) (b)
Figure 20.2: Random motion of electron inside conductor
When an electric field is applied inside the conductor, each electron experiences a force in the direction opposite
to the field. The chaotic motion of electrons gets biased in favour of this force. At each collision with a molecule,
the electron changes its direction of motion and moves with a random velocity but gains an additional velocity
eE
ve = τô in the direction opposite to the electric field till the next collision happens and the direction of its motion
m
again changes abruptly. As the average time τ between successive collisions is small, the electrons slowly and
steadily drift opposite to the applied field (see Fig. 20.3) with an average drift speed vd.
The distance drifted during successive collisions can be written as
= 1 a( τ)2 = 1 eE ( τ)2
2 2 m
1 eE
The drift speed will be given by the relation: vd= = τ
τ 2 m
E
A vd
Figure 20.3: Calculating drift speed
Current density can be expressed in terms of the drift speed. Consider a cylindrical conductor of cross-sectional
area A in which electric field E exists. Consider a length L= v d ∆t of the conductor. The volume of this portion is
Av d ∆t . If there are n free electrons per unit volume of the conductor, the number of free electrons in this portion
are nA v d Δt. All these electrons cross the area A in time Δt. Thus, the charge crossing this cross-section in time Δt is
∆Q
Q neAv d ∆t or current through the conductor is,=
∆= I = neAv d
∆t
I
Therefore current density is: =j = nev d
A
Illustration 1: If n =8.5 × 1028 m–3,how long does an electron take to drift from one end of 3 m long wire to its other
end? The area of cross section of the wire is 2.0 × 10-6 m2and it is carrying a current of 3.0 Ampere. (JEE MAIN)
Sol: For constant p.d. across conductor the electrons drifts with constant drift velocity across it. If we find the drift
velocity, the time to drift across wire of constant length is easily found out
Given that: (i) Number density n = 8.5 × 1028 m–3 (ii) Cross-sectional area A = 2.0 × 10-6m2
(iii) Current I = 3 A (iv) Charge on electron e = 1.6 × 10–19 C
Current in terms of drift speed is expressed as I = neAvd
I
⇒ vd = .
neA
Now time taken to cross the length of the wire is:
