1. 14. Electric current
Applying an external source of electric field (a battery)
to the metallic conductor, the forces acting on
conduction electrons cause the charges to move
and a net current establishes.
The electric current is defined as a flow of
charge dQ in time dt:
(14.1)
and does not change with the varying cross
section of the wire.
We introduce also the microscopic quantity,
the current per unit area called the current density of magnitude
The current density is a vector and in general relates to electric current as
(14.2)
Free electrons in a conductor (e.g. Cu) move
around in random directions, suffering
collisions with the stationary copper atoms.
Electric field causes the drift which is
superimposed on the random motion of
charges.There is a net charge flow in a
particular direction (reverse to electric field
for electrons).

2. Electric current, cont.
A wire (cylinder) of length L and cross section A contains a charge , where n is the concentration of charge carriers.
This charge in time t passes through the cross section A giving a current density
where is a drift speed.
The drift speeds of conducting electrons in the copper conductor are very small comparing to the average speeds in the random motion: vd ~ vav , (vd)Cu ≈ 10-2 cm/s.
In a semiconductor, where both electrons and holes give rise to the total current we have
(14.3)
- drift velocities of electrons and holes respectively

3. 14.1. Electric resistance
Electrical resistance is defined as a ratio of potential differene between the ends of a conductor to the resulting current
(14.4)
The unit for resistance is 1 ohm = 1Ω = 1 V/A
The resistivity ρ of the material is defined through the electric field E and the current density j (14.5)
For the conductor wire the resistance depends not only on the material but also on its geometric properties
σ – conductivity
Ohm’s law
There exist materials for which the dependence between
the applied voltage and resulting current is linear.
These materials obey Ohm’s law
(14.6)
The term „law” is used for historical reasons.
Eq.(14.6) is not always obeyd.

4. Variation of a conductance with temperature
From the microscopic point of view the drift velocity depends on the electric field.
Dividing the expression for current density by E one obtains
(14.7)
Introducing the quantity called carriers mobility
(14.8)
one can write eq.(14.7) in a form
or more generally
(14.9)
Both the carrier concentration and the mobility can depend on temperature. The behaviour σ(T) is typical of a given class of materials.
Metals
For metals, excluding the low temperature region, the
resistivity increases nearly linearly with temperature.
The linear depedence ρ = ρ(T)
for Pt is used as a temperature
scale.

5. Variation of a conductance with temperature, cont.
Semiconductors
In intrinsic (non-doped) semiconductors the conductance is influenced by the motion of electrons and holes. In this case
The concentration ni depends exponentially on temperature
and because a mobility is weakly temperature dependent, one obtains for conductivity
(14.10)
Calculating the natural logarithm of both sides of eq.(14.10) we obtain a straight line in coordinates lnσ = f(1/T)
(14.11)
From (14.11) the energy gap Eg for a given
semiconductor can be determined. A
Energy gap for selected semiconductors (eV)
germanium
silicon
diamond

6. 14.2. Electromotive force (emf)
An emf device (source), by doing work on charge carriers, maintains a potential difference between a pair of terminals.
The emf of an emf device can be defined as the work per unit charge
that the device does transferring a charge from its low-potential
terminal „-” to its high potential terminal „+”
ε =dW/dq (14.12)
A single - loop circuit
The variations in potential in the closed
circuit are
or (14.13)
1,2 – points at which the
external circuit is connected
The circuit
spread out
in a line
An external resistance R is connected across a battery
with emf ε and internal resistance r A
Vb + ε – Ir – IR = Vb
ε – Ir – IR = 0

7. 14.3. Kirchhoff’s rules ε1 – I1R1 – I2R2 - ε2 + I4R4 - ε3= 0
Eq.(14.13) can be derived from the general rule that holds for any complete loop, called the loop rule or Kirchhoff’s loop rule (Kirchhoff’s second law):
the algebraic sum of all potential changes around a loop vanishes.
Calculating the changes in potential when moving around a loop the following rules
have to be adopted:
rule 1: moving through a resistance in the direction of the current, the change in
potential is negative –iR
rule 2: moving through an emf source in the direction of emf, i.e. from – to +, the change in potential is + ε.
Example:
ε1 – I1R1 – I2R2 - ε2 + I4R4 - ε3= 0
Terminology:
- any closed arrangement of circuit elements is called a
loop (mesh)
- a node is the point where meet three or more leads
- a branch is the section of the network between adjacent
nodes; in this section the current remains constant. A

8. Kirchhoff’s rules, cont.
From the law of conservation of charge it follows that no charge can be stored at the node or the total incoming current equals to the total outgoing current.
Kirchhoff’s node rule (Kirchhoff’s first law)
The algebraic sum of currents meeting at the node is zero.
i1- i2 + i3 – i4 = 0 or
i1+ i3= i2+ i4
In a multiloop circuit we have many branches in which flow different currents.
The use of Kirchhoff’s laws makes it possibble to find the branch currents and unknown voltages if emf sources and resistances are known. .
If we have n nodes and p branches, we write:
n – 1 independent equations from the first law
p – n + 1 equations from the second law.

9. Sample problem 1
Find the magnitudes of currents in a circuit as shown in the figure, where
1. We choose arbitrarily the directions of all unknown
currents.
2. We write the necessary equations using Kirchhoff’s
laws: one from the first (2 nodes) and two from the second (3 branches)
The set of equations is solved by the Cramer method.
Ordering: Determinant from coefficients:
The first column is replaced by free terms:
The first current: , etc. A

10. Sample problem 2
Determine the unknown resistance Rx in a Wheatstone bridge configuration.
When the bridge is balanced VBC= 0.
Resistance R2 is adjusted to obtain VBC = 0, what
means VC = VB.
In this case we have only two currents: in the upper
and lower branch.
One can then write:
Dividing by sides one obtains:
This is a precise method of determination the unknown resistance.
I1 Rx = I2R3
I1 R2 = I2R4