Resistance Learning Outcomes Define resistance and give its unit. Solve problems about resistance. State Ohm s Law. HL: Derive the formulas for resistors in series and parallel. Solve problems about resistors in series and parallel. Give the factors that affect the resistance of a conductor. Use an ohmmeter. Solve problems about resistivity. Resistance Learning Outcomes Discuss light-dependent resistors (LDRs) and thermistors. Demonstrate LDRs and thermistors. HL: Describe wheatstone bridges. HL: Solve problems about wheatstone bridges. HL: Discuss uses of a wheatstone bridges. HL: Use a metre bridge. Resistance The resistance of a conductor is the ratio of the voltage across it to the current flowing through it. Formula: R = V I Resistance is a scalar quantity measured in ohms (Ω). A conductor has a resistance of 1 ohm if the current through it is 1 ampere when the voltage across it is 1 volt. Resistance is measured using an ohmmeter or multimeter set to measure resistance alternatively calculate it by measuring current and voltage, then using the formula. 1
Resistance Resistance e.g. Find the resistance of a conductor if it carries a current of 4 A when the voltage across it is 20 V. e.g. What potential difference will produce a current of 5 A in a 12 Ω resistor? e.g. At a certain temperature, the current through a conductor is 3 A when the voltage across it is 24 V. Find the resistance of the conductor. When the temperature of the conductor is raised, the same voltage causes a current of 2 A to flow through it. Find the increase in its resistance. Ohm s Law Ohm s Law states the current flowing through a conductor is proportional to the voltage across it at constant temperature. Formula: V I Constant temperature is required since resistance varies with temperature more on this later. Some conductors will also vary their resistance with voltage. For conductors which obey Ohm s Law, the constant of proportionality is resistance: Formula: V = IR (the same formula we covered earlier) 2
Types of Resistor Resistors in Series For two or more resistors in series, their total resistance is the sum of their resistances. Formula: R Total = R 1 + R 2 + R 3 + Resistors in Series To prove: R T = R 1 + R 2 + R 3 Let V 1, V 2, V 3 be the voltages across each resistor. Let I be the current through each resistor. By formula, V 1 = IR 1, V 2 = IR 2, V 3 = IR 3 But V T = V 1 + V 2 + V 3 V T = IR 1 + IR 2 + IR 3 IR T = IR 1 + IR 2 + IR 3 IR T = I(R 1 + R 2 + R 3 ) R T = R 1 + R 2 + R 3 3
Resistors in Series e.g. Calculate the total resistance of the following resistors: Resistors in Parallel For two or more resistors in parallel, their total resistance is given by: Formula: 1 = 1 + 1 + 1 + RT R1 R2 R3 Resistors in Parallel To prove: 1 = 1 + 1 + 1 RT R1 R2 R3 Let V be the voltage across each resistor. Let I 1, I 2, I 3 be the current through each resistor. By formula, I 1 = V, I 2 = V, I 3 = V R1 R2 R3 But V RT = I = I 1 + I 2 + I 3 So V = V + V + V RT R1 R2 R3 So 1 = 1 + 1 + 1 RT R1 R2 R3 4
Resistors in Parallel e.g. Calculate the total resistance of the following resistors: Resistance in Circuits e.g. What is the total resistance in this circuit? What is the potential difference across the 9Ω resistor? Resistance in Circuits e.g. What is the total resistance of this circuit? What is the current flowing through the 3Ω resistor? 5
Resistance in Circuits e.g. If the bulb has resistance 4Ω, what is the total resistance of this circuit? What is the current flowing through the bulb? Resistance in Circuits e.g. What is the total resistance of the following resistors? Factors Affecting Resistance Temperature We already know that the resistance of a conductor depends on temperature. Increased temperature has two effects: Heat releases extra electrons from the atoms, decreasing resistance. Heat causes atoms to vibrate more, increasing resistance. For metallic conductors, very few electrons are released, so resistance increases with increasing temperature. For insulators and semiconductors, lots of electrons are released, so resistance decreases with increasing temperature. 6
Factors Affecting Resistance The resistance of a conductor also depends on: Length, l, Cross-sectional area, A, Resistivity of the material, ρ. Factors Affecting Resistance - Length Consider a cuboid resistor: What is the effect on resistance if a second identical resistor is added in series? It doubles Resistance length Factors Cross-Sectional Area What if the second resistor is instead added in parallel? Using 1 = 1 + 1 RT R1 R2, 1 = 2 RT R1 R T = 1 2 R 1 The same is true if the resistors are in contact. Resistance 1 Area 7
Factors - Resistivity Different materials come with a natural level of resistance we normalise this using resistivity. The resistivity of a material is the resistance of a 1m 1m 1m cube of the material. The relationship is: resistance resistivity Factors Affecting Resistance Putting each of these factors into a single formula, we get: Formula: R = ρl A We also get a definition for resistivity from this: Formula: ρ = RA l Factors Affecting Resistance e.g. A uniform wire of length 2 m has a resistance of 12Ω. Find the resistance of a piece of identical wire of length 14 m. e.g. What length of copper wire of cross-sectional area 2 mm 2 is needed to make a resistor of resistance 10 Ω? Resistivity of copper = 1.7 10 8 Ωm. e.g. A coil of copper wire 20 m long has uniform composition and uniform cross-sectional area. The diameter of the wire is 0.055 mm. Calculate the resistance of the coil if ρ copper = 1.7 10 8 Ωm. 8
Light-Dependent Resistor A light-dependent resistor (LDR) is a semiconductor that decreases its resistance when light shines on it. Light hitting the resistor releases electrons from the molecules, allowing them to conduct electricity. Thermistor A thermistor is a semiconductor designed to decrease its resistance as its temperature increases. The heat energy frees electrons from the material, allowing them to be used for conduction. Wheatstone Bridge In a wheatstone bridge, four resistors are arranged around a galvanometer such that no current flows through the galvanometer. This happens when the ratio of the resistors is given by: R1 R2 = R3 R4 9
Wheatstone Bridge Uses Usually, one resistor is variable and set to monitor something, while the galvanometer is replaced by some circuit. e.g. A thermistor can be used to monitor room / oven temperature and the galvanometer can be replaced by a heater. When the thermistor unbalances the circuit due to falling temperature, a current flows and activates the heater. Also used in fail-safe devices. Wheatstone Bridge e.g. If the bridge pictured is balanced, what is the value of R? Metre Bridge A metre bridge replaces one side of a wheatstone bridge with a resistive wire. The galvanometer connection can be made anywhere on the wire and a balance point will exist where: R 1 R 2 = l 1 l 2 Usually one resistor is unknown and this formula can be used to find it. 10