Topic 5.2 Heating Effect of Electric Currents


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1 Topic 5.2 Heating Effect of Electric Currents Kari Eloranta 2017 Jyväskylän Lyseon lukio International Baccalaureate February 14, 2017
2 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.
3 5.2 Most Commonly Used Circuit Symbols (ideal) wire resistor switch variable resistor A V ammeter, internal resistance R int 0Ω voltmeter, internal resistance R int (ideal) cell lamp battery potentiometer Figure: Two equivalent circuit diagrams for the study of (I,V ) properties of a lamp.
4 5.2 (I,V ) Characteristics of a Lamp V V A 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).
5 5.2 (I,V ) Characteristics of a Resistor (Ohm s Law) V R V A I Figure: The circuit diagram for the (I,V ) properties of a resistor. Adjustable voltage source provides changing electric potential difference to the circuit.
6 Kirchhoff s Laws 5.2 Kirchhoff s First Law In 1845 German physicist Gustav Kirchhoff ( ) 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) (1)
7 Kirchhoff s Laws 5.2 Kirchhoff s First Law I 3 I 1 I 4 I 2 I 5 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.
8 Kirchhoff s Laws 5.2 Kirchhoff s First Law I 3 I 1 I 4 I 2 I 5 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 ).
9 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) (2)
10 Kirchhoff s Laws 5.2 Kirchhoff s Second Law R 1 R 2 V
11 Resistance 5.2 Resistance Resistance The resistance of an electrical component is R = V 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 R = [V ] [I ] = V = VA = 1Ω (ohm) (4) A according to German physicist Georg Simon Ohm ( ). 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.
12 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.
13 Ohm s Law Ohm s Law (cont.) The difference between metallic and nonmetallic conductors is that nonmetallic 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 (nonohmic behaviour).
14 Ohm s Law 5.2 Resistivity Resistivity ρ of a Wire At constant temperature the resistivity of a metallic wire is ρ = R A L where R is the resistance of the metallic wire of length L, and A is the crosssectional area of the wire. (6) The unit of resistivity is ρ = [R][A] = Ωm (7) [l] The smaller the resistivity, the better the electrical conductivity of the material.
15 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 crosssectional area A R = ρ L 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 of a wire are its crosssectional 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.
16 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 P = W t = V q = V q t 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 equals the power dissipated in the component. As a result, the power dissipation in the resistor is equally P = V I.
17 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 ] = VA = 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.
18 Electric Power 5.2 Electric Power P: Joule s Law For an ohmic component, the potential difference across the component is V = RI. Substituting into the electric power equation gives P = V I = (RI )I = I 2 R or, P = V I = V V R = V 2 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 P = I 2 R = V 2 R where I is the current flowing through the component, and V the electric potential difference across the component.
19 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 (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).
20 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 R total = 1 R R (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.
21 Resistors in Series and in Parallel 5.2 Resistors in Parallel: Derivation R 3 I 3 R 2 I I 2 I R 1 I 1 Figure: By Kirchhoff s First Law the total electric current is I = I 1 + I 2 + I 3. From the Ohm s Law V (V = RI I = V /R) it follows that R total = V 1 R 1 + V 2 R 2 + V 3 R 3. Since the ends of the resistors are at the same potential, the electric potential difference across the system is V = V 2 = V 2 = V 3, The potential differences are cancelled, and 1 R total = 1 R R R 3.
22 Resistors in Series and in Parallel 5.2 Three Resistors in Parallel R 3 = 30Ω I 3 R 2 = 30Ω I I 2 I R 1 = 30Ω I 1 Figure: Three identical 30 Ω resistors are connected in parallel. From the equation 1 R total = 1 R R R 3 = 1 30Ω Ω Ω = 3 we get for the total resistance 30Ω R total = 30Ω = 10Ω, which 3 is one third of the resistance of a resistor.
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