Download Resistance * Learning Outcomes

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.
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)
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 = 𝑅1 + 𝑅2 + 𝑅3 + ⋯
Resistors in Series
 To prove: R T = 𝑅1 + 𝑅2 + 𝑅3
 Let 𝑉1 , 𝑉2 , 𝑉3 be the voltages across each resistor.
 Let 𝐼 be the current through each resistor.
 By formula, 𝑉1 = 𝐼𝑅1 , 𝑉2 = 𝐼𝑅2 , 𝑉3 = 𝐼𝑅3
 But VT = 𝑉1 + 𝑉2 + 𝑉3
 ⇒ 𝑉𝑇 = 𝐼𝑅1 + 𝐼𝑅2 + 𝐼𝑅3
 ⇒ 𝐼𝑅𝑇 = 𝐼𝑅1 + 𝐼𝑅2 + 𝐼𝑅3
 ⇒ 𝐼𝑅𝑇 = 𝐼(𝑅1 + 𝑅2 + 𝑅3 )
 ⇒ 𝑅𝑇 = 𝑅1 + 𝑅2 + 𝑅3
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
𝑅𝑇
=
1
𝑅1
+
1
𝑅2
+
1
𝑅3
+⋯
Resistors in Parallel
 To prove:
1
𝑅𝑇
=
1
𝑅1
+
1
𝑅2
+
1
𝑅3
 Let V be the voltage across each resistor.
 Let 𝐼1 , 𝐼2 , 𝐼3 be the current through each resistor.
 By formula, 𝐼1 =
 But
V
RT
 So
𝑉
𝑅𝑇
 So
1
𝑅𝑇
𝑉
, 𝐼2
𝑅1
=
= 𝐼 = 𝐼1 + 𝐼2 + 𝐼3
=
𝑉
𝑅1
𝑉
+
𝑅2
=
1
𝑅1
+
1
𝑅2
𝑉
+
𝑅3
+
1
𝑅3
𝑉
, 𝐼3
𝑅2
=
𝑉
𝑅3
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?
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.
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
𝑅𝑇
1
𝑅𝑇
=
=
1
𝑅1
+
1
,
𝑅2
2
𝑅1
1
2
 ⇒ 𝑅𝑇 = 𝑅1
 The same is true if the resistors
are in contact.
 ⇒ 𝑅𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 ∝
1
𝐴𝑟𝑒𝑎
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 mm2 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 𝜌𝑐𝑜𝑝𝑝𝑒𝑟 = 1.7 × 10−8 Ω𝑚.
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
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:
𝑅1
𝑙1
 =
𝑅2
𝑙2
 Usually one resistor is unknown
and this formula can be used
to find it.