Resistance

AS UNIT 2 Electricity and light 2.2 Resistance Learners should be able to demonstrate and apply their knowledge and understanding of:
a)	The definition of potential difference
b)	The idea that potential difference is measured in volts (V) where [math]V = J C^{-1}[/math]
c)	The characteristics of [math] I – V[/math] graphs for the filament of a lamp, and a metal wire at constant temperature
d)	Ohm’s law, the equation [math][/math] and the definition of resistance
e)	Resistance being measured in ohms (Ω), where [math]Ω = VA^{-1}[/math]
f)	The application of [math] P = IV = I^2 R = \frac{V^2}{R}[/math]
g)	Collisions between free electrons and ions gives rise to electrical resistance, and electrical resistance increases with temperature
h)	The application of [math]R = \frac{\rho l}{A}[/math] the equation for resistivity
i)	The idea that the resistance of metals varies almost linearly with temperature over a wide range
j)	The idea that ordinarily, collisions between free electrons and ions in metals increase the random vibration energy of the ions, so the temperature of the metal increases
k)	What is meant by superconductivity, and superconducting transition temperature
l)	The fact that most metals show superconductivity, and have transition temperatures a few degrees above absolute zero ([math]-273℃[/math]//0
m)	Certain materials (high temperature superconductors) having transition temperatures above the boiling point of nitrogen ([math]-196℃[/math])
n)	Some uses of superconductors for example, MRI scanners and particle accelerators
Specified Practical Work o Investigation of the [math] I – V[/math] characteristics of the filament of a lamp and a metal wire at constant temperature o Determination of resistivity of a metal o Investigation of the variation of resistance with temperature for a metal wire

a) Definition of Potential Difference
⇒ Potential Difference:
Potential Difference (p.d.) is defined as the amount of energy transferred per unit charge as the charge moves between two points in an electric circuit.
It measures how much electrical energy is converted into other forms of energy (e.g., heat, light, etc.) when a charge flows through a component.
[math]V = \frac{W}{Q}[/math]
OR
[math]V = \frac{E}{Q}[/math]
Where:
– V: Potential difference (measured in volts, V),
– W(E): Work done or energy transferred (measured in joules, J)
– Q: Electric charge (measured in coulombs, C).
Figure 1 Potential Difference
b) Potential Difference Measured in Volts (V)
The volt (V) is the SI unit of potential difference.
⇒ Definition:
One volt is the potential difference between two points in a circuit when one joule of energy is transferred per one coulomb of charge.
– In mathematical terms:
[math]1V = 1J/C[/math]
⇒ Interpretation:
If a potential difference of 5 V exists across a component, this means 5 J of energy is transferred per coulomb of charge passing through it.
c) Characteristics of I-V Graphs
1. Filament Lamp
Shape of Graph:
The I-V graph for a filament lamp is non-linear, resembling an S-shape.
⇒ Explanation:
At low currents, the filament behaves like a simple resistor, and the current (I) increases linearly with voltage (V).
As the current increases, the filament’s temperature rises due to electrical resistance.
The resistance of the filament increases with temperature because the increased vibration of the metal ions impedes the flow of electrons.
This results in a reduced rate of current increase at higher voltages, causing the graph to curve.
Figure 2 I-V graph of filament
2. Metal Wire at Constant Temperature
Shape of Graph:
The I-V graph for a metal wire at constant temperature is a straight line through the origin.
⇒ Explanation:
The resistance of the metal wire remains constant as long as the temperature does not change.
This implies that the current (I) is directly proportional to the voltage (V), consistent with Ohm’s Law.
Slope of the line represents [math]1/R[/math], where R is the resistance.
Figure 3 I-V graph of Metal wire
d) Ohm’s Law
⇒ Definition:
Ohm’s Law states that the current (I) flowing through a conductor is directly proportional to the potential difference (V) across it, provided the temperature and other physical conditions remain constant.
Mathematically:
[math]V = IR[/math]
Where:
– V: Potential difference (volts),
– I: Current (amperes),
– R: Resistance (ohms, Ω).
⇒ Resistance
Definition:
– Resistance (R) is a measure of how much a material opposes the flow of electric current.
– It depends on the material, length, cross-sectional area, and temperature of the conductor.
Unit:
– The unit of resistance is the ohm (Ω), where:
[math]1Ω = \frac{1V}{1A}[/math]
Rearranging Ohm’s Law:
[math]R = \frac{V}{I}[/math]
⇒ Practical Applications:
1. Filament Lamp:
– Used in lighting.
– As resistance increases with temperature, the lamp’s power output stabilizes at higher currents.
2. Metal Wires:
– Metals with constant resistance are ideal for electrical wiring and creating stable circuits.
3. Ohm’s Law:
– Crucial for analyzing circuits.
– Used to calculate unknown quantities (voltage, current, or resistance).
⇒ Example Calculations
1. Calculating Current Using Ohm’s Law
If the resistance of a resistor is R=10 Ω and the voltage across it is V=5 V, the current is:
[math]\begin{gather}
I = \frac{V}{R} \\
I = \frac{5}{10} \\
I = 0.5 \text{ A}
\end{gather}[/math]
2. Resistance from a Filament Lamp
If a filament lamp draws I=0.4 A when connected to a 6 V supply:
[math]\begin{gather}
R = \frac{V}{I} \\
R = \frac{6}{0.4} \\
R = 15 \,\Omega
\end{gather}[/math]
3. Work Done by a Current
If Q=3 C of charge moves through a component with a potential difference of V=2 V, the work done (W) is:
[math]\begin{gather}
W = VQ \\
W = 2 \cdot 3 \\
W = 6 \text{ J}
\end{gather}[/math]
e) Resistance and its Measurement
⇒ Definition of Resistance:
Resistance (R) is a measure of how much a material opposes the flow of electric current.
It depends on the material’s properties, dimensions, and temperature.
⇒ Unit of Resistance:
Resistance is measured in ohms (Ω).
One ohm is defined as the resistance when a potential difference of 1 volt drives a current of 1 ampere through a component.
Mathematically:
[math]1Ω = 1V/A[/math]
f) Electrical Power and its Relationship with Resistance
⇒ Electrical Power (P):
Power is the rate at which electrical energy is transferred or converted in a circuit.
Power depends on the current, voltage, and resistance of the circuit.
Figure 4 Relationship between electric power and resistance
⇒ Equations for Power:
1. From basic definitions:
[math]P = IV[/math]
Where:
– P: Power (in watts, W),
– I: Current (in amperes, A),
– V: Voltage (in volts, V).
2. Using Ohm’s Law ([math]V = IR[/math] ), alternative forms of the power equation are derived:
– Substituting
[math]\begin{gather} V = IR \quad \text{into} \quad P = IV \\
P = I^2 R \end{gather}[/math]
(Useful when current and resistance are known.)
– Substituting
[math]I = \frac{V}{R} [/math]
into
[math]\begin{gather}
P = IV \\
P = \frac{V^2}{R}
\end{gather}
[/math]
(Useful when voltage and resistance are known.)

g) Resistance and Temperature
⇒ Origin of Electrical Resistance:
Resistance arises due to collisions between free electrons and ions in a conductor.
In metals:
Free electrons flow when a potential difference is applied.
The fixed ions in the lattice vibrate and scatter the moving electrons, impeding their motion.
This scattering increases with temperature as the ions vibrate more vigorously, increasing resistance.
⇒ Temperature Dependence:
In metals:
Resistance increases with temperature because more collisions occur as the ion vibrations intensify.
In semiconductors:
Resistance decreases with temperature because more charge carriers are generated.
e) Resistivity and the Resistivity Equation
⇒ Resistivity:
Resistivity (ρ) is a material property that quantifies how strongly a material opposes the flow of electric current.
It is a fundamental characteristic of a material, independent of its shape or size.
Equation for Resistance in Terms of Resistivity:
[math]R = ρ \frac{l}{A}[/math]
Where:
– R: Resistance (in ohms, Ω),
– ρ: Resistivity (in ohm-meters, [math][/math]),
– l: Length of the conductor (in meters, m),
– A: Cross-sectional area of the conductor (in square meters, m²).
⇒ Understanding the Resistivity Equation
1. Dependence on Length (l):
Resistance is directly proportional to the length of the conductor:
[math]R ∝ l[/math]
– A longer conductor provides more opportunity for electrons to collide with ions, increasing resistance.
2. Dependence on Cross-sectional Area (AAA):
Resistance is inversely proportional to the cross-sectional area of the conductor:
[math]R ∝ \frac{1}{A}[/math]
– A larger cross-sectional area allows more electrons to flow simultaneously, reducing resistance.
3. Dependence on Material (ρ):
– Materials with lower resistivity (e.g., copper, silver) are better conductors.
– Materials with higher resistivity (e.g., rubber, glass) are insulators.
⇒ Key Applications of the Resistivity Equation
1. Designing Conductors:
Wires are made from materials with low resistivity (e.g., copper) to minimize energy loss.
2. Determining Wire Dimensions:
To reduce resistance in high-power circuits, wires are designed to be short and thick.
3. Material Characterization:
Resistivity helps identify the suitability of materials for specific applications (e.g., metals for conductors, ceramics for insulators).
⇒ Example Calculations
1. Calculating Resistance Using Resistivity:
A copper wire ([math]ρ = 1.7 × 10^{-8} Ωm[/math] ) has a length [math]l = 2m[/math] and cross-sectional area [math]A = 1 × 10^{-6} m^2[/math]. Find its resistance.
[math]\begin{gather}
R = \frac{\rho l}{A} \\
R = \frac{(1.7 \times 10^{-8}) \times 2}{1 \times 10^{-6}} \\
R = 3.4 \times 10^{-2} \,\Omega
\end{gather}[/math]
2. Calculating Power Dissipated:
A resistor with [math]R = 10 Ω[/math] carries a current of [math]l =2m[/math]. Find the power dissipated.
[math]\begin{gather}
P = I^2 R \\
P = (2)^2 \times 10 \\
P = 40 \text{ W}
\end{gather}[/math]
i) Resistance of Metals and Temperature Dependence
⇒ Relationship Between Resistance and Temperature in Metals
The resistance (R) of metals generally increases linearly with temperature over a wide range.
This is because, as temperature rises, the metal ions in the lattice vibrate more intensely, increasing the probability of collisions between free electrons and ions.
⇒ Mathematical Relationship:
The resistance R of a metal at a temperature T can be expressed as:
[math]R_T = R_0 (1 + αΔT)[/math]
Where:
– [math]R_T[/math]: Resistance at temperature T,
– [math]R_o[/math] : Resistance at a reference temperature (usually [math][/math] or room temperature),
– α: Temperature coefficient of resistance (a material-dependent constant),
[math]ΔT = T – T_0[/math]: Change in temperature.
Figure 5 Temperature and resistivity
⇒ Characteristics:
1. At low temperatures, the resistance of most metals approaches a minimum value, as thermal vibrations reduce.
2. At higher temperatures, the relationship between resistance and temperature becomes less linear.
j) Collisions Between Free Electrons and Ions in Metals
⇒ Resistance Increases Temperature
Mechanism:
– In a metal, free electrons flow when a potential difference is applied, creating an electric current
– As these electrons move, they collide with the metal ions in the lattice.
– Each collision transfers energy from the electrons to the ions, causing the ions to vibrate more intensely.
⇒ Energy Transfer:
The increased vibration energy of the ions results in a rise in the temperature of the metal.
This process is responsible for the conversion of electrical energy into heat, which explains why conductors heat up when current flows through them.
⇒ Effects:
1. The greater the current (flow of electrons), the more frequent the collisions, leading to a higher temperature.
2. This is the basis of Joule heating in resistors and electric wires.
k) Superconductivity and Transition Temperature
⇒ Superconductivity:
Superconductivity is a phenomenon in which a material’s electrical resistance drops to exactly zero when it is cooled below a certain critical temperature, known as the superconducting transition temperature (T_C).
⇒ Properties of Superconductors:
1. Zero Electrical Resistance:
When a material becomes superconducting, it can conduct electricity without any energy loss.
Electrons form Cooper pairs, which move without scattering.
2. Expulsion of Magnetic Fields (Meissner Effect):
Superconductors expel all internal magnetic fields, causing magnetic field lines to bend around the material.
⇒ Superconducting Transition Temperature (T_C):
The critical temperature (T_C) is the temperature below which a material becomes superconducting.
⇒ For many materials:
– T_C is typically very low (e.g., 4.2 K for mercury).
– High-temperature superconductors, like certain ceramic compounds, can have T_C>77K, making them practical for some applications.
⇒ Superconductivity
Superconductivity arises from the formation of Cooper pairs of electrons:
– These pairs experience an attractive interaction mediated by lattice vibrations (phonons).
– The pairs move through the lattice without resistance because they are not scattered by the ions.
⇒ Applications of Superconductivity
1. Magnetic Levitation (Maglev Trains):
Superconductors are used to create strong magnetic fields for levitation and propulsion.
2. MRI Machines:
Superconducting magnets are used in medical imaging to generate powerful, stable magnetic fields.
Figure 6 MRI for Brain
3. Particle Accelerators:
Superconducting magnets guide particles at extremely high speeds.
4. Lossless Power Transmission:
Superconducting cables can transmit electricity without energy loss.
l) Superconductivity in Metals and Materials
⇒ Most Metals and Superconductivity
Many metals exhibit superconductivity, meaning they transition to a state of zero electrical resistance below a specific critical temperature, called the superconducting transition temperature (T_C).
For most pure metals:
– T_C is typically only a few degrees above absolute zero (-273℃).
Example:
– Mercury (T_C =2K),
– Lead (T_C =2K),
– Tin (T_C =7K).
m) High-Temperature Superconductors
High-temperature superconductors are materials that become superconducting at higher temperatures, significantly above the transition temperatures of metals.
Examples include certain ceramic compounds like:
– Yttrium barium copper oxide (YBCO) with T_C >90K
– Bismuth strontium calcium copper oxide (BSCCO).
⇒ Transition Temperature and Nitrogen Boiling Point:
The boiling point of liquid nitrogen is (-196 , (77K)).
High-temperature superconductors with T_C >77K are practical because they can operate using liquid nitrogen as a coolant, which is much cheaper and more accessible than liquid helium (used for lower T_C).
n) Applications of Superconductors
1. Magnetic Resonance Imaging (MRI) Scanners
⇒ Purpose:
MRI scanners use superconducting magnets to produce powerful, stable magnetic fields.
These fields align the protons in the human body, and the signals emitted as they return to their original states are used to create detailed images of tissues.
⇒ Why Superconductors?:
i) Superconducting magnets generate stronger magnetic fields with high efficiency, enabling clearer imaging.
ii) They consume significantly less energy due to the zero-resistance property.
2. Particle Accelerators
Purpose:
– Particle accelerators, such as the Large Hadron Collider (LHC), accelerate particles to near the speed of light for research in high-energy physics.
Role of Superconductors:
– Superconducting magnets guide and focus the particle beams within the accelerator by generating ultra-strong magnetic fields.
– Superconducting materials ensure energy efficiency and reduce operational costs.
3. Maglev (Magnetic Levitation) Trains
Functionality:
Superconducting magnets levitate the train above the tracks, reducing friction and allowing high speeds.
Benefits:
High efficiency, smooth motion, and reduced energy consumption.
Example:
Japan’s maglev trains use superconducting magnets to reach speeds of over 500 km/h.
4. Electric Power Transmission
Superconducting cables can transmit electricity without energy loss due to resistance.
Benefits:
High efficiency in power distribution, particularly in large grids.
5. Other Applications
SQUIDs (Superconducting Quantum Interference Devices):
Extremely sensitive magnetic field detectors used in research and medical diagnostics.
Fusion Reactors:
Superconductors are used to confine plasma in reactors like ITER (International Thermonuclear Experimental Reactor).
⇒ Advantages of High-Temperature Superconductors
1. Cost-Effective Cooling:
Liquid nitrogen (-196℃) is cheaper and easier to handle than liquid helium (-269℃).
2. Wider Range of Applications:
High T_C materials make superconductors more feasible for large-scale, real-world applications.
⇒ Challenges in Superconductivity
1. Material Limitations:
Many high-temperature superconductors are brittle ceramics, difficult to manufacture and shape.
2. Cooling Requirements:
Even high-temperature superconductors require cooling to cryogenic temperatures, limiting their use in everyday applications.
3. Magnetic Field Effects:
Superconductivity can be disrupted by strong external magnetic fields (critical field strength).

Specified Practical Work
1. Investigation of the I-V Characteristics of the Filament of a Lamp and a Metal Wire at Constant Temperature
⇒ Aim:
To investigate the relationship between current (I) and potential difference (V) for:

A filament lamp.
A metal wire at constant temperature.

Figure 7 Investigation of the I-V graph
⇒ Apparatus:

DC Power Supply.
Ammeter (to measure current, I).
Voltmeter (to measure potential difference, V).
Filament lamp.
Metal wire (e.g., copper wire).
Variable resistor (or rheostat) to vary current.
Connecting wires.

⇒ Procedure:
1. Set up the circuit:
Connect the filament lamp or metal wire in series with the power supply, ammeter, and variable resistor.
Connect the voltmeter across the lamp or wire to measure V.
2. Measure the I-V Characteristics:
Gradually increase the supply voltage using the variable resistor or adjust the power supply.
Record the corresponding values of current (I) and voltage (V) for each setting.
3. For the Metal Wire:
Ensure the temperature of the metal wire remains constant by using small currents to avoid heating. Allow time between measurements for cooling.
4. For the Filament Lamp:
Repeat the process but note the lamp’s resistance increases as it heats up due to the increase in filament temperature.
5. Plot Graphs:
Plot I (current) on the y-axis and V (voltage) on the x-axis.
⇒ Observations and Results:
Metal Wire:
– The graph will be a straight line, showing a linear relationship between I and V, following Ohm’s Law ( [math]V = IR[/math]).
Filament Lamp:
– The graph will initially be straight but will curve as V increases, indicating that resistance increases with temperature due to the filament heating up.
⇒ Conclusion:
Metal wire: Resistance is constant and follows Ohm’s Law.
Filament lamp: Resistance increases as the temperature of the filament increases, causing the I−V graph to curve.
2. Determination of Resistivity of a Metal
⇒ Aim:
To determine the resistivity (ρ) of a given metal wire.
Figure 8 Determination of resistivity of a metal
⇒ Apparatus:

Metal wire of known material.
DC Power Supply.
Micrometer Screw Gauge (to measure the diameter of the wire).
Ruler (to measure the length of the wire).
Ammeter.
Voltmeter.
Variable resistor.
Connecting wires.

⇒ Procedure:
1. Measure the Dimensions of the Wire:
Use the micrometer screw gauge to measure the diameter of the wire (d) at several points and calculate the average.
Calculate the cross-sectional area:
[math]\begin{gather}
A = \frac{\pi d^2}{4}
\end{gather}[/math]
Measure the length (L) of the wire using a ruler.
2. Set Up the Circuit:
Connect the metal wire in series with the power supply, ammeter, and variable resistor.
Connect the voltmeter across the wire.
3. Measure Current and Voltage:
Gradually increase the voltage using the variable resistor.
Record the corresponding current (I) and voltage (V) for multiple settings.
4. Calculate Resistance (R):
For each pair of I and V, calculate resistance using:
[math]\begin{gather}
R = \frac{V}{I}
\end{gather}[/math]
5. Determine Resistivity (ρ):
Using the formula:
[math]\begin{gather}
\rho = \frac{R A}{L}
\end{gather}[/math]
Substitute the values of R, A, and L to calculate resistivity.
⇒ Precautions:

Use low currents to avoid heating the wire, which would change its resistance.
Take multiple readings of d, L, I, and V to minimize errors.

3. Investigation of the Variation of Resistance with Temperature for a Metal Wire
⇒ Aim:
To investigate how the resistance of a metal wire changes with temperature.
Figure 9 Investigation of the variation of Resistance with temperature
⇒ Apparatus:

Metal wire (e.g., nichrome or copper).
DC power supply.
Ammeter and Voltmeter.
Rheostat.
Water bath or oil bath (to heat the wire).
Thermometer.
Beaker and heater.
Connecting wires.

⇒ Procedure:
1. Set Up the Wire in a Water Bath:
Submerge the metal wire in a water bath or oil bath, ensuring good thermal contact.
2. Measure Resistance at Different Temperatures:
Gradually heat the bath using a heater.
Measure the temperature (T) using the thermometer.
For each temperature, measure the current (I) and voltage (V) and calculate resistance using:
[math]\begin{gather}
R = \frac{V}{I}
\end{gather}[/math]
3. Record Data:
Record R at several temperature points as the bath heats up.
4. Plot a Graph:
Plot R (resistance) on the y-axis and T (temperature) on the x-axis.
⇒ Observations:
Metal Wire:
– Resistance increases almost linearly with temperature.
– This is due to increased collisions between free electrons and vibrating metal ions.
⇒ Conclusion:

Resistance of a metal wire increases with temperature due to increased lattice vibrations.
The graph can be used to calculate the temperature coefficient of resistance (α) using the relation:

[math]\begin{gather}
R_T = R_0 (1 + \alpha \Delta T)
\end{gather}[/math]

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Resistance

AS UNIT 2

Electricity and light

2.2 Resistance

Learners should be able to demonstrate and apply their knowledge and understanding of:

a) Definition of Potential Difference

b) Potential Difference Measured in Volts (V)

c) Characteristics of I-V Graphs

d) Ohm’s Law

e) Resistance and its Measurement

f) Electrical Power and its Relationship with Resistance

g) Resistance and Temperature

e) Resistivity and the Resistivity Equation

i) Resistance of Metals and Temperature Dependence

j) Collisions Between Free Electrons and Ions in Metals

k) Superconductivity and Transition Temperature

l) Superconductivity in Metals and Materials

m) High-Temperature Superconductors

n) Applications of Superconductors

Specified Practical Work

1. Investigation of the I-V Characteristics of the Filament of a Lamp and a Metal Wire at Constant Temperature

2. Determination of Resistivity of a Metal

3. Investigation of the Variation of Resistance with Temperature for a Metal Wire

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