Lecture Sheet-4 Electricity & Magnetism

Current
The rate of flow of charge through any area of cross-section of a conductor is called current
or electric current
Let, an infinitesimal amount of charge dq is passing through any area of a conductor in a very
small interval of time dt, then by the definition of electric current in the conductor is
𝑑𝑞
𝑖=
𝑑𝑡
If the rate of flow of charge is constant, then the current is said to be steady (constant)
current. The steady current in the conductor is denoted by I.
𝑞
𝐼=
𝑡
Where, q is the total charge that flows in time t.

The SI unit of current is Ampere.

1 𝐶𝑜𝑢𝑙𝑜𝑚𝑏
1 𝐴𝑚𝑝𝑒𝑟𝑒 =
1 𝑠𝑒𝑐𝑜𝑛𝑑
Current density
The current density is the electric current per unit area of cross-section. The current density
at a point is the quantity of charge that flows through a unit area of cross-section that point
per unit time. It is denoted by J.

We can write the amount of current through an element as 𝐽⃗. 𝑑𝐴⃗ (Where 𝑑𝐴⃗ is the vector area
of the element). The total current through a surface is then

𝐼 = ∫ 𝐽⃗. 𝑑𝐴⃗

If the current is distributed uniformly across the surface and parallel to dA, then J is uniform
and parallel to dA. Then
𝐼 = ∫ 𝐽⃗. 𝑑𝐴⃗ = ∫ 𝐽𝑑𝐴 = 𝐽 ∫ 𝑑𝐴 = 𝐽𝐴

𝐼
∴ 𝐽=
𝐴
Where, A is the total surface area.
The SI unit of current density is Ampere/m2.

Drift velocity

The drift velocity is the average velocity of electron attains due to an electric field.
If the potential difference between two ends of a conductor is applied to the free electron, it is
moved towards – 𝐸⃗⃗ , that is opposite to the field inside the conductor, with an average velocity
vd is called drift velocity.

Md. Matiur Rahman, Sr. Lecturer, Dept. of Medical Physics, KYAU Page: 1
Lecture Sheet-4 Electricity & Magnetism

Expression for drift velocity:

Fig. 1
If N is the number of conduction electrons per unit volume of a conductor then the number of
conduction electrons n inside the conductor of volume 𝐴𝑙 (Fig. 1.), is given by
𝑛 = 𝑁𝐴𝑙 (1)
Therefore, the total charge q is given by

𝑞 = 𝑁𝐴𝑙𝑒 (2)
Where, e is the electronic charge. If charge q flows through the conductor of volume Al in
time t, then on the average the electrons cover a distance l in time t, so that
𝑙
𝑡=
𝑣𝑑
According to the definition of current
𝑞
𝐼=
𝑡

𝑁𝐴𝑙𝑒
𝐼=
𝑡

(𝑁𝐴𝑙𝑒)𝑣𝑑
𝐼=
𝑙

𝐼 = 𝑁𝐴𝑒𝑣𝑑

𝐼
𝑣𝑑 = (3)
𝑁𝐴𝑒
We know the current density
𝐼
𝐽= (4)
𝐴
Comparing equations (3) and (4) we can write
𝐽
𝑣𝑑 = (5)
𝑁𝑒
Equation (5) is the required expression for the drift velocity of electrons in a conductor
conducting N electrons per unit volume.

Md. Matiur Rahman, Sr. Lecturer, Dept. of Medical Physics, KYAU Page: 2
Lecture Sheet-4 Electricity & Magnetism

Resistance
Resistance is the property of a substance due to which it opposes the flow of electricity
through it. It is denoted by R.
It is measured by Ohm’s law. According to Ohm’s law
𝑉
𝑅=
𝐼
Where, I is the current and V is the potential difference.
The unit of resistance is Ohm (Ω).
Resistivity
Resistivity is the property of the material of a conductor. The electric resistance of a
conductor is directly proportional to the length of the conductor and inversely proportional
to the cross-sectional area of the conductor. This relation is expressed as
𝑙
𝑅∞
𝐴
Where, l is the length of the conductor and A is the cross-sectional area of the conductor. We
can write,
𝑙
𝑅 = 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 ×
𝐴
𝑙
𝑅=𝜌
𝐴
Where, is called the resistivity of the material and is constant for given material.
𝐼𝑓 𝑙 = 1 𝑚 𝑎𝑛𝑑 𝐴 = 1 𝑚2 𝑡ℎ𝑒𝑛 𝑅 = 𝜌. Thus the resistivity is defined as the resistance of a
material of unit cross-section and unit length.
The unit of resistivity is Ohm-meter (Ω-m) or Ohm-centimeter (Ω-cm).
Conductance
Conductance is the reciprocal of resistance where the resistance of a conductor measures the
opposition which it offers to the flow of current, the conductance measures the inducement
which it offers to its flow.
If the conductance of a current be represented by G and its resistance by R, then
1
𝐺=
𝑅
𝐴 1𝐴
𝐺= =
𝜌𝑙 𝜌 𝑙
𝐴
𝐺=𝜎
𝑙
Where,is the conductivity and
1
𝜎=
𝜌
Thus the conductance is directly proportional to the cross-sectional area of the conductor
and inversely proportional to length of the conductor.

Md. Matiur Rahman, Sr. Lecturer, Dept. of Medical Physics, KYAU Page: 3
Lecture Sheet-4 Electricity & Magnetism

Conductivity
Conductivity is the property of a conductor by means of which it allowed the current to flow
through the conductor when the potential difference is applied across the conductor. It is the
reciprocal of resistivity and is represented by 
1
𝜎=
𝜌
The unit of conductivity is Ohm-1-meter-1 or mho/meter.

Ohm’s law
Statement: The potential difference between the ends of a conductor varies directly as the
current flowing in it, provided the temperature does not change and the physical state of the
conductor remains the same.
Explanation: If the current flowing through the
conductor is I and the potential difference between
the ends of the conductor is 𝑉𝐴 − 𝑉𝐵 , then we can
write

(𝑉𝐴 − 𝑉𝐵 ) ∞ 𝐼

(𝑉𝐴 − 𝑉𝐵 ) = 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 × 𝐼 Fig. 2

(𝑉𝐴 − 𝑉𝐵 )
∴ = 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝐼
The ratio is constant and is known as the resistance of the material.

Therefore we can write

(𝑉𝐴 − 𝑉𝐵 )
=𝑅
𝐼
Denoting 𝑉 = 𝑉𝐴 − 𝑉𝐵 , we can write
𝑉
=𝑅
𝐼
𝑉 = 𝐼𝑅
This is known as Ohm’s law.
From the above equation we can write
𝑉 𝑙
=𝜌
𝐼 𝐴
𝑉 𝐼
=𝜌
𝑙 𝐴

𝐸 = 𝜌𝐽
This is another form of Ohm’s law.

Md. Matiur Rahman, Sr. Lecturer, Dept. of Medical Physics, KYAU Page: 4
Lecture Sheet-4 Electricity & Magnetism

Resistivity-an atomic view

The conduction electrons in the metal are free to move. We also
assume that the electrons collide not with one another but only
with positive charge of the metal.
If an electron of mass m is placed in an electric field of
magnitude E, the electron will experience an acceleration given
by Newton’s second law Fig. 3. An electron moving from A
to B, making six collisions.
𝐹 𝑒𝐸
𝑎= =
𝑚 𝑚
In the average time  between collisions, the average electron will acquire a drift speed
of 𝑣𝑑 = 𝑎𝜏. Thus, at any instant, on average, the electrons will have drift speed
𝑒𝐸𝜏
𝑣𝑑 = 𝑎𝜏 =
𝑚
We know,
𝐽
𝑣𝑑 =
𝑁𝑒
Comparing above two equations
𝐽 𝑒𝐸𝜏
=
𝑁𝑒 𝑚
𝑚
∴ 𝐸=( )𝐽
𝑁𝑒 2 𝜏
We know
𝐸 = 𝜌𝐽
Comparing above two equations
𝑚
𝜌=
𝑁𝑒 2 𝜏
In the above equation N, the number of conduction electrons per volume, is independent of
the field, and m and e are constants. The average time between collisions , is a constant,
because the drift speed 𝑣𝑑 is so much smaller than the effective speed 𝑣𝑒𝑓𝑓 of the electrons.
Thus, for metals their resistivity  is a constant.
Thus we can say that metals obey Ohm’s law.

Md. Matiur Rahman, Sr. Lecturer, Dept. of Medical Physics, KYAU Page: 5
Lecture Sheet-4 Electricity & Magnetism

Mathematical Problems
Problem-1: What is the drift velocity of electrons in a copper conductor having a cross-
sectional area of 5 × 10−6 𝑚2 , if the current is 10 A? Assume that there are 8 ×
1028 electrons/m3 (𝑞𝑒 = 1.6 × 10−19 𝐶).

Solution: We know
Here,
𝐼
𝑣𝑑 = 𝐴 = 5 × 10−6 𝑚2
𝑁𝐴𝑒
𝐼 = 10 𝐴𝑚𝑝
10
𝑂𝑟 𝑣𝑑 = 𝑁 = 8 × 1028 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑠/𝑚3
8 × 1028 × 5 × 10−6 × 1.6 × 10−19
𝑒 = 𝑞𝑒 = 1.6 × 10−19 𝐶
∴ 𝑣𝑑 = 1.56 × 10−4 𝑚𝑠 −1 (𝑨𝒏𝒔) 𝑣𝑑 =?

Problem-2: A silver wire 1 mm in diameter carries a charge of 90 coulomb in 1 hr. Silver

contains 5.8 × 1028 𝑓𝑟𝑒𝑒 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑠/𝑚3 . Calculate (i) the current in the wire (ii) current
density and drift velocity of electrons in the wire.
Solution: We know, Current
Here,
𝑞 90 2
𝐼= = = 0.025 𝑎𝑚𝑝 (𝑨𝒏𝒔) 1 × 10−3
𝑡 3600 2
𝐴 = 𝜋𝑟 = 3.1416 × ( )
Current density 2
𝐼 0.025 = 7.85 × 10−7 𝑚2
𝐽= =
𝐴 7.85 × 10−7 𝑞 = 90 𝐶𝑜𝑢𝑙𝑜𝑚𝑏

= 3.18 × 10−4 𝑎𝑚𝑝/𝑚2 (𝑨𝒏𝒔) 𝑡 = 1 ℎ𝑟 = 3600 𝑠

Drift velocity 𝑁 = 5.8 × 1028 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑠/𝑚3
𝐼 𝑒 = 1.6 × 10−19 𝐶
𝑣𝑑 =
𝑁𝐴𝑒
𝐼 =?
0.025 𝑣𝑑 =?
𝑂𝑟 𝑣𝑑 =
5.8 × 1028 × 7.85 × 10−7 × 1.6 × 10−19

∴ 𝑣𝑑 = 3.43 × 10−6 𝑚𝑠 −1 (𝑨𝒏𝒔)

Md. Matiur Rahman, Sr. Lecturer, Dept. of Medical Physics, KYAU Page: 6