Topic 5.2 Heating Effect of Electric Currents
Topic 5.2 Heating Effect of Electric Currents
© Kari Eloranta
2015
Jyväskylän Lyseon lukio
International Baccalaureate
May 21, 2015
© Kari Eloranta 2015
Topic 5.2 Heating Effect of Electric Currents
Topic 5.2 Heating Effect of Electric Currents
Topic 5.2 Heating Effect of Electric Currents
In subtopic 5.2 we
study DC circuits.
DC circuits are
modelled with circuit
diagrams, which are
idealisations of real,
physical circuits.
Circuit symbols are
listed on page four in
the Data Booklet.
© Kari Eloranta 2015
Topic 5.2 Heating Effect of Electric Currents
Topic 5.2 Heating Effect of Electric Currents
5.2 Most Commonly Used Circuit Symbols
(ideal) wire
switch
resistor
variable resistor
A
ammeter, internal resistance R int ≈ 0 Ω
V
voltmeter, internal resistance R int ≈ ∞ (ideal)
cell
battery
lamp
potentiometer
Figure : Two equivalent circuit diagrams for the study of (I ,V ) properties of a lamp.
© Kari Eloranta 2015
Topic 5.2 Heating Effect of Electric Currents
Topic 5.2 Heating Effect of Electric Currents
5.2 (I ,V ) Characteristics of a Lamp
V
A
V
I
Figure : The circuit diagram for the (I ,V ) properties of a lamp. Voltmeter is in parallel with the
lamp, and ammeter is in series with the lamp. Conventional electric current I flows from the
positive terminal into negative terminal (in opposite direction to the electrons).
© Kari Eloranta 2015
Topic 5.2 Heating Effect of Electric Currents
Topic 5.2 Heating Effect of Electric Currents
5.2 (I ,V ) Characteristics of a Resistor (Ohm’s Law)
V
R
A
V
I
Figure : The circuit diagram for the (I ,V ) properties of a resistor. Adjustable voltage source
provides changing electric potential difference to the circuit.
© Kari Eloranta 2015
Topic 5.2 Heating Effect of Electric Currents
Topic 5.2 Heating Effect of Electric Currents
Kirchhoff’s Laws
5.2 Kirchhoff’s First Law
In 1845 German physicist Gustav Kirchhoff (1824-1887) suggested two laws that
give us a way of determining electric currents and electric potential differences in
a DC circuit.
.
Kirchhoff’s
First Law
.
The sum of electric currents entering a junction is equal to the sum of the currents
leaving
the junction.
.
Kirchhoff’s First Law is a natural consequence of the conservation of electric
charge.
If the currents entering a junction are positive, and the currents leaving the
junction negative, the sum of the currents is zero:
ΣI = 0 (node rule)
© Kari Eloranta 2015
Topic 5.2 Heating Effect of Electric Currents
(1)
Topic 5.2 Heating Effect of Electric Currents
Kirchhoff’s Laws
5.2 Kirchhoff’s First Law
I3
I1
I4
I2
I5
Figure : Kirchhoff’s First Law. The sum of the electric currents entering the junction equals the
sum of the currents leaving the junction (ΣI = 0 ⇒ I 1 + I 2 = I 3 + I 4 + I 5). This is a result of the
conservation of electric charge.
© Kari Eloranta 2015
Topic 5.2 Heating Effect of Electric Currents
Topic 5.2 Heating Effect of Electric Currents
Kirchhoff’s Laws
5.2 Kirchhoff’s First Law
I3
I1
I4
I2
I5
Figure : Kirchhoff’s First Law. The sum of the electric currents entering the junction equals the
sum of the currents leaving the junction ( I 1 + I 2 = I 3 + I 4 + I 5).
© Kari Eloranta 2015
Topic 5.2 Heating Effect of Electric Currents
Topic 5.2 Heating Effect of Electric Currents
Kirchhoff’s Laws
5.2 Kirchhoff’s Second Law
.
Kirchhoff’s
Second Law
.
The sum of electric potential differences along any closed loop in a DC circuit is
zero.
.
Kirchhoff’s Second Law is a consequence of the conservation of energy: any
electric potential change equals the change in electric potential energy per unit
charge.
If the electric potential drops are negative in the direction of electric current,
and increases positive, the sum of the changes in electric potentials along any
closed loop in a dc circuit is zero:
ΣV = 0 (junction rule)
© Kari Eloranta 2015
Topic 5.2 Heating Effect of Electric Currents
(2)
Topic 5.2 Heating Effect of Electric Currents
Kirchhoff’s Laws
5.2 Kirchhoff’s Second Law
V5 V
V1 = R 1 I
4
3
ϵ
2
V2 = R 2 I
1
0
ϵ
R1
I
© Kari Eloranta 2015
R2
Topic 5.2 Heating Effect of Electric Currents
Topic 5.2 Heating Effect of Electric Currents
Resistance
5.2 Resistance
.
Resistance
.
The resistance of an electrical component is
V
R=
I
(3)
where V is the potential difference across the component, and I the electric
current
flowing through it.
.
The SI unit of electrical resistance is
[V ] V
= = VA = 1 Ω (ohm)
R=
[I ] A
according to German physicist Georg Simon Ohm (1789 – 1854).
Resistance is a function of temperature. For example, as the electric current
through a filament increases, the filament warms, and the resistance of the
filament increases.
© Kari Eloranta 2015
Topic 5.2 Heating Effect of Electric Currents
(4)
Topic 5.2 Heating Effect of Electric Currents
Ohm’s Law
Ohm’s Law
.
Ohm’s
Law
.
For a metallic conductor at constant temperature, the electric potential difference
across the component is directly proportional to the electric current flowing
through
it.
.
.
Ohm’s
Law
.
For a metallic conductor at constant temperature, the electric potential difference
V across the component is
V = RI
(5)
where R is the resistance of the component, and I the electric current flowing
through
the component.
.
© Kari Eloranta 2015
Topic 5.2 Heating Effect of Electric Currents
Topic 5.2 Heating Effect of Electric Currents
Ohm’s Law
Ohm’s Law (cont.)
The difference between metallic and non-metallic conductors is that non-metallic
conductors have much less conduction electrons.
As a result, Ohm’s Law is valid at constant temperature also for nonmetallic
conductors such as carbon, in which the electric current is carried by free
electrons.
Some components and materials obey Ohm’s Law for a range of temperatures
while some others do not. For example, the resistance of a ceramic resistor stays
relatively constant even when the resistor warms (Ohmic behaviour), but the
resistance of a filament increases as the electric current increases (non-ohmic
behaviour).
© Kari Eloranta 2015
Topic 5.2 Heating Effect of Electric Currents
Topic 5.2 Heating Effect of Electric Currents
Ohm’s Law
5.2 Resistivity
.
Resistivity
ρ of a Wire
.
At constant temperature the resistivity of a metallic wire is
RA
ρ=
L
(6)
where R is the resistance of the metallic wire of length L , and A is the
cross-sectional
area of the wire.
.
The unit of resistivity is
[R][A]
ρ=
= Ωm
[l ]
(7)
The smaller the resistivity, the better the electrical conductivity of the material.
© Kari Eloranta 2015
Topic 5.2 Heating Effect of Electric Currents
Topic 5.2 Heating Effect of Electric Currents
Ohm’s Law
5.2 Factors That Affect Resistance
.
Resistance
and Resistivity
.
At constant temperature the resistance of a metallic wire R is proportional to the
length of the wire L , and inversely proportional to the cross-sectional area A
L
R =ρ
(8)
A
where
the constant of proportionality ρ is called the resistivity of the material.
.
From the equation above we see that the factors that affect the resistance of a
wire are its cross-sectional area A , length L , and resistivity of the metal ρ . If the
temperature is not constant, the resistance may change with the temperature.
In a required practical you should investigate one or more of the factors that
affect resistance experimentally. The understanding will be tested in Paper 1
and Paper 3 questions.
© Kari Eloranta 2015
Topic 5.2 Heating Effect of Electric Currents
Topic 5.2 Heating Effect of Electric Currents
Electric Power
5.2 Power Dissipation P in a Component
All real components dissipate energy in a DC circuit.
#»
Consider a constant electric field E inside a resistor in a DC circuit. As the
amount of electric charge ∆q moves across a potential difference V in the
resistor, the work done by the electric field on the charge carriers is W = V ∆q .
The power at which the electric field does work is thus
W V ∆q
∆q
P=
=
=V ×
=V I
∆t
∆t
∆t
(9)
where I is the electric current through the resistor.
Because the electric current is constant, the power supplied by the field equals
the power dissipated in the component. As a result, the power dissipation in the
resistor is equally P = V I .
© Kari Eloranta 2015
Topic 5.2 Heating Effect of Electric Currents
Topic 5.2 Heating Effect of Electric Currents
Electric Power
5.2 Electric Power P
.
Electric
Power
.
The electric power of an electrical component is
P =V I
where V is the potential difference across the component, and I the current
flowing
through it.
.
The SI unit of electric power is
[P ] = [V ][I ] = V A = 1W (watt).
(10)
The electric power gives the amount of energy consumed by the component in
the circuit in one second.
The electric power gives also the power at which a voltage source supplies
energy to the electric circuit. The supplied power is
P = V I = electric potential difference across the source × current in circuit.
© Kari Eloranta 2015
Topic 5.2 Heating Effect of Electric Currents
Topic 5.2 Heating Effect of Electric Currents
Electric Power
5.2 Electric Power P : Joule’s Law
For an ohmic component, the potential difference across the component is
V = R I . Substituting into the electric power equation gives
2
V
V
P = V I = (R I )I = I 2R or, P = V I = V × = ,
R
R
where R is the resistance of the component. The equation is known as Joule’s
Law.
.
Joule’s
Law
.
An ohmic component of resistance R dissipates energy in a DC circuit at the power
2
V
P = I 2R =
R
where I is the current flowing through the component, and V the electric potential
.difference across the component.
© Kari Eloranta 2015
Topic 5.2 Heating Effect of Electric Currents
Topic 5.2 Heating Effect of Electric Currents
Resistors in Series and in Parallel
5.2 Resistors in Series
.
Resistors
in Series
.
When resistors are connected in series, the total resistance R total is the sum of the
individual resistances
R total = R 1 + R 2 + . . .
(11)
.
In electronics literature the total resistance is often called the equivalent
resistance.
Adding resistors in series increases the total resistance. Two resistors in series
doubles the total resistance, three threefolds, and so on (R total = R + R = 2R ,
R total = R + R + R = 3R ).
© Kari Eloranta 2015
Topic 5.2 Heating Effect of Electric Currents
Topic 5.2 Heating Effect of Electric Currents
Resistors in Series and in Parallel
5.2 Resistors in Parallel
.
Resistors
in Parallel
.
When resistors are connected in parallel, the relation between the total resistance
R total and individual resistances is
1
.
1
1
= + +...
R total R 1 R 2
(12)
Adding resistors in parallel decreases the total resistance. Two resistors in
parallel reduces the total resistance to one half of the original, three to one third
of the original, and so on.
© Kari Eloranta 2015
Topic 5.2 Heating Effect of Electric Currents
Topic 5.2 Heating Effect of Electric Currents
Resistors in Series and in Parallel
5.2 Resistors in Parallel: Derivation
R3
I3
R2
I
I2
I
R1
I1
Figure : By Kirchhoff’s First Law the total electric current is I = I 1 + I 2 + I 3. From the Ohm’s Law
(V = R I ≡ I = V /R ) it follows that RVtotal = RV11 + RV22 + RV33 . Since the ends of the resistors are at the
same potential, the electric potential difference across the system is V = V2 = V2 = V3, The
1
potential differences are cancelled, and Rtotal
= R11 + R12 + R13 .
© Kari Eloranta 2015
Topic 5.2 Heating Effect of Electric Currents
Topic 5.2 Heating Effect of Electric Currents
Resistors in Series and in Parallel
5.2 Three Resistors in Parallel
R 3 = 30 Ω
I3
R 2 = 30 Ω
I
I2
I
R 1 = 30 Ω
I1
Figure : Three identical 30 Ω resistors are connected in parallel. From the equation
1
1
1
1
1
1
1
3
30 Ω
=
+
+
=
+
+
=
we
get
for
the
total
resistance
R
=
= 10 Ω, which
total
R total
R1
R2
R3
30 Ω 30 Ω 30 Ω
30 Ω
3
is one third of the resistance of a resistor.
© Kari Eloranta 2015
Topic 5.2 Heating Effect of Electric Currents