Topic 5.2 Heating ﬀ of Electric Currents · Topic 5.2 Heating ﬀ of Electric Currents Resistance...

Topic 5.2 Heating Effect of Electric Currents


© Kari Eloranta2017

Jyväskylän Lyseon lukioInternational Baccalaureate

February 14, 2017

© Kari Eloranta 2017 Topic 5.2 Heating Effect of Electric Currents



In subtopic 5.2 westudy DC circuits.

DC circuits aremodelled withcircuit diagrams,which areidealisations of real,physical circuits.

Circuit symbols arelisted on page fourin the DataBooklet.



5.2 Most Commonly Used Circuit Symbols

(ideal) wire

resistor

A ammeter, internal resistance Rint ≈ 0Ω

V voltmeter, internal resistance Rint ≈∞ (ideal)

cell

lamp

switch

variable resistor

battery

potentiometer

Figure: Two equivalent circuit diagrams for the study of (I ,V ) properties of a lamp.



5.2 (I ,V ) Characteristics of a Lamp

A

V

V

I

Figure: The circuit diagram for the (I ,V ) properties of a lamp. Voltmeter is in parallel with thelamp, and ammeter is in series with the lamp. Conventional electric current I flows from thepositive terminal into negative terminal (in opposite direction to the electrons).



5.2 (I ,V ) Characteristics of a Resistor (Ohm’s Law)

R

A

V

V

I

Figure: The circuit diagram for the (I ,V ) properties of a resistor. Adjustable voltage sourceprovides changing electric potential difference to the circuit.


Topic 5.2 Heating Effect of Electric CurrentsKirchhoff’s Laws

5.2 Kirchhoff’s First Law

In 1845 German physicist Gustav Kirchhoff (1824-1887) suggested two lawsthat give us a way of determining electric currents and electric potentialdifferences in a DC circuit.

Kirchhoff’s First LawThe sum of electric currents entering a junction is equal to the sum of the currentsleaving the junction.

Kirchhoff’s First Law is a natural consequence of the conservation of electriccharge.

If the currents entering a junction are positive, and the currents leaving thejunction negative, the sum of the currents is zero:

ΣI = 0 (node rule) (1)




I1

I2

I3

I4

I5

Figure: Kirchhoff’s First Law. The sum of the electric currents entering the junction equals thesum of the currents leaving the junction (ΣI = 0 ⇒ I1+ I2 = I3+ I4+ I5). This is a result of theconservation of electric charge.




I1

I2

I3

I4

I5

Figure: Kirchhoff’s First Law. The sum of the electric currents entering the junction equals thesum of the currents leaving the junction (I1+ I2 = I3+ I4+ I5).



5.2 Kirchhoff’s Second Law

Kirchhoff’s Second LawThe sum of electric potential differences along any closed loop in a DC circuit iszero.

Kirchhoff’s Second Law is a consequence of the conservation of energy: anyelectric potential change equals the change in electric potential energy perunit 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 anyclosed loop in a dc circuit is zero:

ΣV = 0 (junction rule) (2)



5.2 Kirchhoff’s Second Law

V

R1 R2


Topic 5.2 Heating Effect of Electric CurrentsResistance

5.2 Resistance

ResistanceThe resistance of an electrical component is

R = V

I(3)

where V is the potential difference across the component, and I the electriccurrent flowing through it.

The SI unit of electrical resistance is

R = [V ]

[I ]= V

A=VA= 1Ω (ohm) (4)

according to German physicist Georg Simon Ohm (1789 – 1854).

Resistance is a function of temperature. For example, as the electric currentthrough a filament increases, the filament warms, and the resistance of thefilament increases.


Topic 5.2 Heating Effect of Electric CurrentsOhm’s Law

Ohm’s Law

Ohm’s LawFor a metallic conductor at constant temperature, the electric potential differenceacross the component is directly proportional to the electric current flowingthrough it.

Ohm’s LawFor a metallic conductor at constant temperature, the electric potential differenceV across the component is

V = RI (5)where R is the resistance of the component, and I the electric current flowingthrough the component.



Ohm’s Law (cont.)

The difference between metallic and non-metallic conductors is thatnon-metallic conductors have much less conduction electrons.

As a result, Ohm’s Law is valid at constant temperature also for nonmetallicconductors such as carbon, in which the electric current is carried by freeelectrons.

Some components and materials obey Ohm’s Law for a range of temperatureswhile some others do not. For example, the resistance of a ceramic resistorstays relatively constant even when the resistor warms (Ohmic behaviour), butthe resistance of a filament increases as the electric current increases(non-ohmic behaviour).



5.2 Resistivity

Resistivity ρ of a WireAt constant temperature the resistivity of a metallic wire is

ρ = R A

L(6)

where R is the resistance of the metallic wire of length L, and A is thecross-sectional area of the wire.

The unit of resistivity is

ρ = [R][A]

[l ]=Ωm (7)

The smaller the resistivity, the better the electrical conductivity of thematerial.



5.2 Factors That Affect ResistanceResistance and ResistivityAt constant temperature the resistance of a metallic wire R is proportional to thelength of the wire L, and inversely proportional to the cross-sectional area A

R = ρL

A(8)

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 ofa 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 thetemperature.

In a required practical you should investigate one or more of the factors thataffect resistance experimentally. The understanding will be tested in Paper 1and Paper 3 questions.


Topic 5.2 Heating Effect of Electric CurrentsElectric 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 theresistor, 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

P = W

∆t= V ∆q

∆t=V ×∆q

∆t=V I (9)

where I is the electric current through the resistor.

Because the electric current is constant, the power supplied by the field equalsthe power dissipated in the component. As a result, the power dissipation inthe resistor is equally P =V I .



5.2 Electric Power P

Electric PowerThe electric power of an electrical component is

P =V I

where V is the potential difference across the component, and I the currentflowing through it.

The SI unit of electric power is

[P ] = [V ][I ] =VA= 1W (watt). (10)

The electric power gives the amount of energy consumed by the component inthe circuit in one second.

The electric power gives also the power at which a voltage source suppliesenergy to the electric circuit. The supplied power is P =V I =electric potential difference across the source×current in circuit.



5.2 Electric Power P : Joule’s Law

For an ohmic component, the potential difference across the component isV = RI . Substituting into the electric power equation gives

P =V I = (RI )I = I 2R or, P =V I =V ×V

R= V 2

R,

where R is the resistance of the component. The equation is known as Joule’sLaw.

Joule’s LawAn ohmic component of resistance R dissipates energy in a DC circuit at the power

P = I 2R = V 2

Rwhere I is the current flowing through the component, and V the electric potentialdifference across the component.


Topic 5.2 Heating Effect of Electric CurrentsResistors in Series and in Parallel

5.2 Resistors in Series

Resistors in SeriesWhen resistors are connected in series, the total resistance Rtotal is the sum of theindividual resistances

Rtotal = R1+R2+ . . . (11)

In electronics literature the total resistance is often called the equivalentresistance.

Adding resistors in series increases the total resistance. Two resistors in seriesdoubles the total resistance, three threefolds, and so on (Rtotal = R +R = 2R,Rtotal = R +R +R = 3R).



5.2 Resistors in Parallel

Resistors in ParallelWhen resistors are connected in parallel, the relation between the total resistanceRtotal and individual resistances is

1

Rtotal= 1

R1+ 1

R2+ . . . (12)

Adding resistors in parallel decreases the total resistance. Two resistors inparallel reduces the total resistance to one half of the original, three to onethird of the original, and so on.



5.2 Resistors in Parallel: Derivation

I I

R1

I1

R2

I2

R3

I3

Figure: By Kirchhoff’s First Law the total electric current is I = I1+ I2+ I3. From the Ohm’s Law(V = RI ≡ I =V /R) it follows that V

Rtotal= V1

R1+ V2

R2+ V3

R3. Since the ends of the resistors are at the

same potential, the electric potential difference across the system is V =V2 =V2 =V3, Thepotential differences are cancelled, and 1

Rtotal= 1

R1+ 1

R2+ 1

R3.



5.2 Three Resistors in Parallel

I I

R1 = 30Ω

I1

R2 = 30Ω

I2

R3 = 30Ω

I3

Figure: Three identical 30Ω resistors are connected in parallel. From the equation1

Rtotal= 1

R1+ 1

R2+ 1

R3= 1

30Ω+ 130Ω+ 1

30Ω = 330Ω we get for the total resistance Rtotal = 30Ω

3 = 10Ω, whichis one third of the resistance of a resistor.


Topic 5.2 Heating ﬀ of Electric Currents · Topic 5.2 Heating ﬀ of Electric Currents Resistance...

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