The relationship between temperature and resistance in thermistors is a fundamental concept in physics, particularly within the study of electricity and resistivity. Understanding that the resistance of a thermistor decreases as temperature increases is crucial for applications ranging from temperature sensing to circuit protection. This topic is essential for students preparing for AS & A Level Physics (9702), providing a basis for exploring more complex electrical phenomena and their practical implementations.

Key Concepts

What is a Thermistor?

A thermistor is a type of resistor whose resistance varies significantly with temperature. The term "thermistor" is derived from the combination of "thermal" and "resistor." Thermistors are widely used in temperature measurement and control applications due to their high sensitivity to temperature changes.

Types of Thermistors

There are primarily two types of thermistors:

Negative Temperature Coefficient (NTC) Thermistors: Resistance decreases as temperature increases.
Positive Temperature Coefficient (PTC) Thermistors: Resistance increases as temperature increases.

This article focuses on NTC thermistors, where the resistance decreases with an increase in temperature.

Electrical Resistance and Temperature

Electrical resistance is a measure of the opposition to the flow of electric current through a conductor. In thermistors, the resistance varies predictably with temperature, allowing them to function effectively as temperature sensors.

The general relationship between resistance (R) and temperature (T) for an NTC thermistor can be described using the Steinhart-Hart equation:

$$ \frac{1}{T} = A + B \ln(R) + C (\ln(R))^3 $$

Where:

T: Absolute temperature in Kelvin.
R: Resistance in ohms.
A, B, C: Steinhart-Hart coefficients specific to the thermistor.

Material Composition of Thermistors

Thermistors are typically made from metal oxides such as manganese, nickel, cobalt, and copper. These materials are mixed and baked at high temperatures to form a ceramic-like body. The precise composition and manufacturing process determine the thermistor's temperature-resistance characteristics.

Applications of Thermistors

NTC thermistors are used in various applications, including:

Temperature Sensing: In digital thermometers and weather monitoring systems.
Inrush Current Limiting: Protecting circuits from sudden surges when devices are powered on.
Battery Charging: Ensuring safe charging by monitoring temperature changes.
Automotive Sensors: Monitoring engine and cabin temperatures.

Temperature Coefficient of Resistance (TCR)

The Temperature Coefficient of Resistance (TCR) quantifies the change in resistance per degree change in temperature. For NTC thermistors, the TCR is negative, indicating a decrease in resistance with increasing temperature.

The formula for TCR is:

$$ \text{TCR} = \frac{\Delta R/R_0}{\Delta T} $$

Where:

ΔR: Change in resistance.
R₀: Original resistance at reference temperature.
ΔT: Change in temperature.

Non-Linear Resistance-Temperature Relationship

Unlike metallic resistors, whose resistance increases linearly with temperature, NTC thermistors exhibit a non-linear relationship. This non-linearity is advantageous in applications requiring high sensitivity at specific temperature ranges.

To accurately model this behavior, empirical equations like the Steinhart-Hart equation or the Beta parameter equation are used.

The Beta Parameter Equation

A simplified equation often used for NTC thermistors is the Beta parameter equation:

$$ \ln\left(\frac{R_1}{R_2}\right) = \beta \left(\frac{1}{T_1} - \frac{1}{T_2}\right) $$

Where:

R₁, R₂: Resistances at temperatures T₁ and T₂ respectively.
β: Beta constant specific to the thermistor.
T₁, T₂: Absolute temperatures in Kelvin.

Practical Example: Calculating Resistance Change

Consider an NTC thermistor with a resistance of 10,000 Ω at 25°C (298 K) and a Beta value of 3435 K. To find its resistance at 35°C (308 K):

Convert temperatures to Kelvin:

25°C = 298 K
35°C = 308 K

Apply the Beta parameter equation:

$$ \ln\left(\frac{R_{35}}{10,000}\right) = 3435 \left(\frac{1}{298} - \frac{1}{308}\right) $$ $$ \ln\left(\frac{R_{35}}{10,000}\right) = 3435 \times (0.003356 - 0.003247) = 3435 \times 0.000109 = 0.374 $$ $$ \frac{R_{35}}{10,000} = e^{0.374} \approx 1.454 $$ $$ R_{35} = 1.454 \times 10,000 \approx 14,540 \, \Omega $$>

However, since this result contradicts the expected decrease in resistance with an increase in temperature for an NTC thermistor, it indicates a misapplication of the formula. The correct approach involves ensuring the Beta parameter equation accounts for the negative temperature coefficient:

$$ \ln\left(\frac{R_2}{R_1}\right) = \beta \left(\frac{1}{T_2} - \frac{1}{T_1}\right) $$

Applying the correct form:

$$ \ln\left(\frac{R_{35}}{10,000}\right) = 3435 \left(\frac{1}{308} - \frac{1}{298}\right) $$ $$ \ln\left(\frac{R_{35}}{10,000}\right) = 3435 \times (-0.000109) = -0.374 $$ $$ \frac{R_{35}}{10,000} = e^{-0.374} \approx 0.689 $$> $$ R_{35} = 0.689 \times 10,000 \approx 6,890 \, \Omega $$>

Hence, the resistance decreases to approximately 6,890 Ω at 35°C, demonstrating the NTC behavior.

Linearity and Compensation

Due to the non-linear resistance-temperature relationship, applications requiring linear responses often incorporate calibration curves or compensation circuits. Using multiple thermistors in series or parallel configurations can help achieve a more linear overall response.

Temperature Range and Stability

Thermistors are designed to operate within specific temperature ranges. Exceeding these ranges can lead to material degradation, altered resistance characteristics, or permanent damage. Stability over time and repeated temperature cycling is essential for reliable performance.

Manufacturing Considerations

The precise control of composition and sintering processes during manufacturing ensures consistent thermistor behavior. Variations in material purity, particle size, and density can significantly impact resistance-temperature characteristics.

Calibration of Thermistors

Accurate temperature measurement using thermistors requires calibration against known temperature standards. Calibration involves determining the specific Beta constant or fitting the Steinhart-Hart coefficients to empirical data.

Environmental Factors Affecting Thermistors

Factors such as humidity, mechanical stress, and exposure to chemicals can influence thermistor performance. Protective coatings and careful circuit design help mitigate these effects, ensuring long-term reliability.

Advanced Concepts

Mathematical Modeling of Thermistors

Accurate modeling of thermistor behavior is essential for designing precise temperature measurement systems. The Steinhart-Hart equation provides a superior fit over a wide temperature range compared to the Beta parameter equation.

The Steinhart-Hart equation is given by:

$$ \frac{1}{T} = A + B \ln(R) + C (\ln(R))^3 $$

This three-term equation allows for greater accuracy across varying temperatures. Deriving the coefficients A, B, and C involves measuring resistance at three known temperatures and solving the resulting system of equations.

Derivation of the Steinhart-Hart Equation

The derivation begins with the assumption that the reciprocal of the temperature is a linear combination of the natural logarithm of resistance and its higher-order terms. By fitting experimental data to this model, the coefficients A, B, and C can be determined to minimize the deviation between theoretical predictions and actual measurements.

The process involves:

Measuring resistance values at three distinct temperatures.
Taking the natural logarithm of each resistance value.
Substituting these values into the Steinhart-Hart equation to form a system of three equations.
Solving the system to find the coefficients A, B, and C.

Thermistor Sensitivity Analysis

Sensitivity refers to the degree of resistance change relative to temperature change. High sensitivity is desirable for precise measurements. Sensitivity can be enhanced by optimizing material composition and thermistor design.

Mathematically, sensitivity (S) can be expressed as:

$$ S = \frac{\Delta R/R}{\Delta T} $$>

A higher absolute value of S indicates greater sensitivity.

Thermal Time Constant

The thermal time constant determines how quickly a thermistor responds to temperature changes. It is influenced by the thermistor's mass, heat capacity, and thermal conductivity. A smaller time constant implies faster response times, which is critical in applications requiring real-time temperature monitoring.

The thermal time constant (τ) is calculated as:

$$ \tau = \frac{mc}{hA} $$

Where:

m: Mass of the thermistor.
c: Specific heat capacity.
h: Convective heat transfer coefficient.
A: Surface area exposed to the heat transfer medium.

Impact of Self-Heating

Passing current through a thermistor generates heat, which can alter its temperature and affect resistance measurements. This phenomenon, known as self-heating, must be minimized to ensure accurate temperature readings.

Strategies to reduce self-heating include:

Using low excitation currents.
Implementing pulse-based measurement techniques.
Designing thermistors with high thermal conductivity.

Non-Linear Compensation Techniques

To achieve linearity in temperature measurements, various compensation techniques are employed:

Electronic Compensation: Utilizing op-amps and other circuitry to counteract the non-linear response.
Software Calibration: Applying mathematical algorithms to linearize the resistance-temperature relationship.
Series/Parallel Configurations: Combining multiple thermistors to average out non-linearities.

Interdisciplinary Connections

The principles governing thermistors intersect with multiple disciplines:

Engineering: Designing HVAC systems and industrial temperature controls.
Medicine: Developing precise medical devices for patient monitoring.
Environmental Science: Monitoring climate variables and environmental conditions.
Electronics: Creating smart devices with integrated temperature sensing capabilities.

Understanding thermistor behavior enhances the ability to innovate across these fields, promoting advancements in technology and improving the efficiency of systems relying on temperature regulation.

Advanced Problem-Solving: Designing a Temperature-Controlled Circuit

Consider designing a temperature-controlled fan circuit using an NTC thermistor. The fan should turn on when the temperature exceeds 30°C and turn off below 25°C, incorporating hysteresis to prevent rapid switching.

Component Selection: Choose an appropriate NTC thermistor with a Beta value suitable for the desired temperature range.
Hysteresis Implementation: Incorporate a Schmitt trigger to introduce hysteresis, ensuring the fan doesn't oscillate around the threshold temperatures.
Power Supply Considerations: Ensure the circuit operates within the voltage and current specifications of the thermistor and fan.
Calibration: Calibrate the thermistor to accurately correspond resistance values with 25°C and 30°C thresholds.
Testing and Validation: Verify the circuit's response to temperature changes and adjust component values as necessary.

This problem integrates knowledge of thermistor behavior, electronic circuit design, and control systems, showcasing the application of theoretical concepts in practical scenarios.

Mathematical Derivation: From Beta Equation to Steinhart-Hart

Starting with the Beta parameter equation, we can extend it to derive the more comprehensive Steinhart-Hart equation.

The Beta equation is given by:

$$ \ln\left(\frac{R_2}{R_1}\right) = \beta \left(\frac{1}{T_2} - \frac{1}{T_1}\right) $$>

However, this linearization is limited over narrow temperature ranges. To achieve better accuracy over a broader range, higher-order terms are included:

$$ \frac{1}{T} = A + B \ln(R) + C (\ln(R))^3 $$>

By expanding the Beta equation and incorporating additional terms, the Steinhart-Hart equation provides a cubic relationship between the reciprocal temperature and the natural logarithm of resistance.

Thermistor Networks and Signal Conditioning

Integrating thermistors into electronic systems often requires signal conditioning to interface with digital controllers or analog systems. Common techniques include:

Voltage Dividers: Using thermistors in combination with fixed resistors to create a voltage that varies with temperature.
Bridge Circuits: Employing Wheatstone bridges for precision measurements.
Analog-to-Digital Conversion: Translating the analog resistance changes into digital signals for microcontroller processing.

Proper signal conditioning ensures accurate temperature readings and reliable system performance.

Noise and Interference in Thermistor Measurements

Electrical noise and external interference can affect the accuracy of thermistor-based measurements. Mitigation strategies include:

Shielding and proper grounding of circuit components.
Using low-noise power supplies.
Implementing filtering techniques to remove unwanted frequency components.

Addressing noise and interference is critical for maintaining the integrity of temperature data in sensitive applications.

Thermistor Aging and Reliability

Over time, thermistors can experience drift in resistance values due to factors like material degradation, thermal cycling, and mechanical stress. Understanding aging mechanisms is important for designing systems with predictable long-term behavior.

Reliability testing involves subjecting thermistors to accelerated aging conditions to estimate their lifespan and performance stability.

Emerging Technologies and Future Trends

Advances in nanotechnology and materials science are leading to the development of thermistors with enhanced properties, such as:

Higher sensitivity and faster response times.
Improved thermal stability and resistance to harsh environments.
Integration with flexible and wearable electronics.

These innovations expand the applications of thermistors, enabling their use in cutting-edge technologies like smart textiles and advanced medical devices.

Comparison Table

Aspect	NTC Thermistor	PTC Thermistor
Resistance-Temperature Relationship	Resistance decreases with increasing temperature.	Resistance increases with increasing temperature.
Applications	Temperature sensing, inrush current limiting.	Overcurrent protection, self-regulating heating elements.
Temperature Range	Typically limited to moderate temperature ranges.	Can operate over a wider range, including high temperatures.
Material Composition	Metal oxides like manganese, nickel.	Polymer-based composites or ceramic materials.
Response Time	Fast response to temperature changes.	Slower response due to high resistance changes.

Summary and Key Takeaways

NTC thermistors exhibit a decrease in resistance as temperature rises.
The resistance-temperature relationship is non-linear, described by the Steinhart-Hart and Beta equations.
Thermistors are essential in various applications, including temperature sensing and circuit protection.
Advanced concepts involve mathematical modeling, sensitivity analysis, and signal conditioning.
Understanding thermistor behavior facilitates interdisciplinary innovations across engineering, medicine, and environmental science.

Examiner Tip

Tips

To excel in understanding thermistors, remember the acronym NTC stands for Negative Temperature Coefficient, indicating resistance decreases with temperature. Practice plotting resistance vs. temperature curves to visualize their behavior. Additionally, when dealing with equations, double-check units and signs to ensure accurate calculations. These strategies will enhance your problem-solving skills for exams.

Did You Know

Did you know that thermistors were first developed in the early 20th century and have since become integral components in modern electronics? Additionally, the term "thermistor" is a blend of "thermal" and "resistor," highlighting their temperature-sensitive nature. In everyday life, thermistors play a crucial role in devices like digital thermostats, ensuring our homes maintain comfortable temperatures efficiently.

Common Mistakes

Students often confuse NTC and PTC thermistors, leading to incorrect assumptions about their behavior. For example, assuming an NTC thermistor's resistance increases with temperature can result in calculation errors. Another frequent mistake is misapplying the Beta parameter equation without accounting for its limitations, which can lead to inaccurate temperature predictions. To avoid these pitfalls, always verify the type of thermistor and use the appropriate equations.

FAQ

What is the primary difference between NTC and PTC thermistors?

NTC thermistors decrease in resistance as temperature increases, while PTC thermistors increase in resistance with rising temperature.

How does the Beta parameter affect thermistor performance?

The Beta parameter determines the rate at which a thermistor's resistance changes with temperature, influencing its sensitivity and accuracy in temperature measurements.

Why is calibration important for thermistors?

Calibration ensures that the thermistor's resistance accurately corresponds to specific temperatures, enhancing measurement precision and reliability in applications.

What causes self-heating in thermistors?

Self-heating occurs when electrical current passes through a thermistor, generating heat that can alter its temperature and affect resistance readings.

Can thermistors be used in high-temperature environments?

Yes, but it's essential to select thermistors designed for high-temperature stability to prevent material degradation and ensure accurate performance.

1. Capacitance

1.1 Energy Stored in a Capacitor

1.1.1 Determine electric potential energy stored in a capacitor from the area under the potential–charge g

1.1.2 Recall and use W = ½QV = ½CV²

1.2 Discharging a Capacitor

1.2.1 Analyze graphs of potential difference, charge, and current during capacitor discharge

1.2.2 Recall and use τ = RC for the time constant during discharge

1.2.3 Use equations of the form x = x₀e^(-t / RC) for current, charge, or potential during discharge

1.3 Capacitors and Capacitance

1.3.1 Define capacitance for isolated spherical conductors and parallel plate capacitors

1.3.2 Recall and use C = Q / V

1.3.3 Derive formulae for combined capacitance of capacitors in series and parallel

1.3.4 Use capacitance formulae for capacitors in series and parallel

2. Nuclear Physics

2.1 Mass Defect and Nuclear Binding Energy

2.1.1 Sketch the variation of binding energy per nucleon with nucleon number

2.1.2 Explain nuclear fusion and nuclear fission

2.1.3 Calculate the energy released in nuclear reactions using E = c²∆m

2.1.4 Understand the equivalence between energy and mass (E = mc²)

2.1.5 Define and use the terms mass defect and binding energy

2.2 Radioactive Decay

2.2.1 Understand that fluctuations in count rate provide evidence for random decay

2.2.2 Understand that radioactive decay is spontaneous and random

2.2.3 Define activity and decay constant, and recall A = λN

2.2.4 Define half-life

2.2.5 Use λ = 0.693 / t₁/₂ for the half-life relationship

2.2.6 Understand exponential decay and use the relationship x = x₀e^(-λt)

3. Kinematics

3.1 Equations of Motion

3.1.1 Define and use distance, displacement, speed, velocity, acceleration

3.1.2 Use graphical methods to represent motion

3.1.3 Determine displacement from velocity–time graph

3.1.4 Determine velocity using gradient of displacement–time graph

3.1.5 Determine acceleration using gradient of velocity–time graph

3.1.6 Derive equations for uniformly accelerated motion

3.1.7 Solve problems of uniformly accelerated motion

3.1.8 Describe experiment to determine acceleration due to gravity

3.1.9 Explain motion with uniform velocity and perpendicular acceleration

4. Deformation of Solids

4.1 Stress and Strain

4.1.1 Understand and use the terms load, extension, compression, and limit of proportionality

4.1.2 Recall and use Hooke’s law

4.1.3 Recall and use the formula for spring constant k = F / x

4.1.4 Define and use terms stress, strain, and Young's modulus

4.1.5 Describe an experiment to determine Young’s modulus

4.1.6 Understand deformation caused by tensile or compressive forces

4.2 Elastic and Plastic Behaviour

4.2.1 Understand elastic and plastic deformation and elastic limit

4.2.2 Understand area under the force–extension graph as work done

4.2.3 Calculate elastic potential energy using EP = ½kx²

5. Gravitational Fields

5.1 Gravitational Field of a Point Mass

5.1.1 Understand that g is approximately constant for small height changes near Earth's surface

5.1.2 Recall and use g = GM / r²

5.1.3 Derive and use g = GM / r² for gravitational field strength

5.2 Gravitational Potential

5.2.1 Define gravitational potential as work done per unit mass in bringing a test mass from infinity

5.2.2 Use ϕ = -GM / r for gravitational potential due to a point mass

5.2.3 Use EP = -GMm / r for gravitational potential energy between two point masses

5.3 Gravitational Field

5.3.1 Define gravitational field as force per unit mass

5.3.2 Represent a gravitational field using field lines

5.4 Gravitational Force Between Point Masses

5.4.1 For point outside uniform sphere, treat mass as a point mass at its center

5.4.2 Analyze circular orbits in gravitational fields

5.4.3 Recall and use Newton's law of gravitation F = Gm₁m₂ / r²

6. Electricity

6.1 Resistance and Resistivity

6.1.1 Understand that the resistance of a thermistor decreases as temperature increases

6.1.2 Define resistance and recall and use V = IR

6.1.3 Sketch I–V characteristics for metallic conductor, semiconductor diode, and filament lamp

6.1.4 Explain that the resistance of a filament lamp increases as current increases

6.1.5 State Ohm’s law and recall and use R = ρL / A

6.1.6 Understand that the resistance of an LDR decreases as light intensity increases

6.2 Electric Current

6.2.1 Understand that an electric current is a flow of charge carriers

6.2.2 Understand that the charge on charge carriers is quantised

6.2.3 Recall and use Q = It

6.2.4 Use I = Anvq for current-carrying conductors

6.3 Potential Difference and Power

6.3.1 Define potential difference as energy transferred per unit charge

6.3.2 Recall and use V = W / Q

6.3.3 Recall and use P = VI, P = I²R, P = V² / R

7. D.C. Circuits

7.1 Practical Circuits

7.1.1 Recall and use the circuit symbols in the syllabus

7.1.2 Draw and interpret circuit diagrams using circuit symbols

7.1.3 Define electromotive force (e.m.f.) of a source as energy transferred per unit charge

7.1.4 Distinguish between e.m.f. and potential difference (p.d.)

7.1.5 Understand the effects of internal resistance on terminal potential difference

7.2 Kirchhoff’s Laws

7.2.1 Recall Kirchhoff’s first law: conservation of charge

7.2.2 Recall Kirchhoff’s second law: conservation of energy

7.2.3 Derive the formula for combined resistance of resistors in series using Kirchhoff’s laws

7.2.4 Use the formula for combined resistance of resistors in series

7.2.5 Derive the formula for combined resistance of resistors in parallel using Kirchhoff’s laws

7.2.6 Use the formula for combined resistance of resistors in parallel

7.2.7 Use Kirchhoff’s laws to solve simple circuit problems

7.3 Potential Dividers

7.3.1 Understand the principle of a potential divider circuit

7.3.2 Recall and use the principle of the potentiometer for comparing potential differences

7.3.3 Understand the use of a galvanometer in null methods

7.3.4 Explain the use of thermistors and LDRs in potential dividers to provide a potential dependent on te

8. Thermal Equilibrium

8.1 Thermal Energy Transfer

8.1.1 Understand that thermal energy transfers from higher to lower temperature

8.1.2 Understand that regions of equal temperature are in thermal equilibrium

9. Temperature Scales

9.1 Temperature Measurement

9.1.1 Understand physical properties that vary with temperature (e.g., density, gas volume, resistance)

9.1.2 Convert temperatures between Kelvin and Celsius, and recall T(K) = θ(°C) + 273.15

10. Magnetic Fields

10.1 Concept of a Magnetic Field

10.1.1 Understand that a magnetic field is a field of force produced by moving charges or permanent magnets

10.1.2 Represent a magnetic field using field lines

10.2 Force on a Current-Carrying Conductor

10.2.1 Understand that a force acts on a current-carrying conductor in a magnetic field

10.2.2 Recall and use F = BIL sin θ for the force on a current-carrying conductor in a magnetic field

10.2.3 Define magnetic flux density as the force per unit current per unit length on a wire at right angles

10.3 Force on a Moving Charge

10.3.1 Determine the direction of the force on a charge moving in a magnetic field

10.3.2 Recall and use F = BQv sin θ for the force on a moving charge in a magnetic field

10.3.3 Understand the origin of the Hall voltage and use the expression VH = BI / (ntq) for the Hall voltag

10.3.4 Understand the use of a Hall probe to measure magnetic flux density

10.4 Magnetic Fields Due to Currents

10.4.1 Sketch magnetic field patterns due to currents in a straight wire, flat coil, and solenoid

10.4.2 Understand that magnetic field in a solenoid is increased by a ferrous core

10.4.3 Explain the origin of forces between current-carrying conductors and determine the direction of the

10.5 Electromagnetic Induction

10.5.1 Define magnetic flux as the product of magnetic flux density and area perpendicular to flux directio

10.5.2 Recall and use Φ = BA for magnetic flux

10.5.3 Understand and use the concept of magnetic flux linkage

10.5.4 Explain experiments that show changing magnetic flux induces an e.m.f.

10.5.5 Recall and use Faraday’s and Lenz’s laws of electromagnetic induction

11. Specific Heat Capacity and Latent Heat

11.1 Specific Heat Capacity

11.1.1 Define and use specific heat capacity

11.2 Latent Heat

11.2.1 Define and use specific latent heat; distinguish between fusion and vaporisation

12. Dynamics

12.1 Momentum and Newton’s Laws of Motion

12.1.1 Mass resists change in motion

12.1.2 Recall F = ma and solve related problems

12.1.3 Define and use linear momentum

12.1.4 Force as rate of change of momentum

12.1.5 State and apply Newton’s laws

12.1.6 Describe weight as effect of gravitational field

12.1.7 Weight of an object is mass × gravitational acceleration

12.2 Non-uniform Motion

12.2.1 Understand frictional and drag forces

12.2.2 Describe motion in uniform gravitational field with air resistance

12.2.3 Objects moving against resistive force may reach terminal velocity

12.3 Linear Momentum and its Conservation

12.3.1 State conservation of momentum

12.3.2 Apply conservation of momentum to solve problems

12.3.3 For elastic collision, kinetic energy and relative speed are conserved

12.3.4 Momentum is conserved, but some kinetic energy may change

13. Medical Physics

13.1 Production and Use of Ultrasound

13.1.1 Understand that a piezo-electric crystal changes shape when a p.d. is applied and generates an e.m.f

13.1.2 Understand how ultrasound waves are generated and detected by a piezoelectric transducer

13.1.3 Understand how ultrasound reflection at boundaries provides diagnostic information about internal st

13.1.4 Define specific acoustic impedance as Z = ρc

13.1.5 Use the equation IR / I₀ = (Z₁ – Z₂)² / (Z₁ + Z₂)² for the intensity reflection coefficient

13.2 Production and Use of X-rays

13.2.1 Explain that X-rays are produced by electron bombardment of a metal target and calculate the minimum

13.2.2 Understand the use of X-rays in imaging internal body structures and the term 'contrast' in X-ray im

13.2.3 Recall and use I = I₀e^(-μx) for the attenuation of X-rays in matter

13.2.4 Understand how computed tomography (CT) produces 3D images by combining multiple 2D X-ray images

13.3 PET Scanning

13.3.1 Understand that a tracer contains radioactive nuclei and is introduced into the body for imaging

13.3.2 Understand that β+ decay is used in positron emission tomography (PET)

13.3.3 Understand that annihilation occurs when a particle interacts with its antiparticle, conserving mass

13.3.4 Explain how gamma-ray photons from annihilation are detected to create an image of tracer concentrat

14. Waves

14.1 Progressive Waves

14.1.1 Describe wave motion in ropes, springs, and ripple tanks

14.1.2 Understand and use the terms displacement, amplitude, phase difference, period, frequency, wavelengt

14.1.3 Use time-base and y-gain on a cathode-ray oscilloscope (CRO) to determine frequency and amplitude

14.1.4 Derive and use v = f λ for the wave equation

14.1.5 Recall and use v = f λ

14.1.6 Understand energy transfer by a progressive wave

14.1.7 Recall and use intensity = power / area, intensity ∝ (amplitude)²

14.2 Transverse and Longitudinal Waves

14.2.1 Compare transverse and longitudinal waves

14.2.2 Analyze and interpret graphical representations of transverse and longitudinal waves

14.3 Doppler Effect for Sound Waves

14.3.1 Understand the change in frequency when a sound source moves relative to a stationary observer

14.3.2 Use the expression f₀ = fₛ(v / (v ± vₛ)) for the observed frequency

14.4 Electromagnetic Spectrum

14.4.1 State all electromagnetic waves are transverse waves traveling at the same speed in free space

14.4.2 Recall the range of wavelengths from radio waves to γ-rays

14.4.3 Recall that wavelengths 400–700 nm are visible to the human eye

14.5 Polarisation

14.5.1 Understand that polarisation is a phenomenon associated with transverse waves

14.5.2 Recall and use Malus's law (I = I₀ cos²θ) to calculate intensity of a plane-polarised wave after pas

15. The Mole

15.1 The Mole

15.1.1 Understand that amount of substance is an SI base quantity with the unit mol

15.1.2 Use molar quantities where one mole contains Avogadro's constant of particles

16. Equation of State

16.1 Ideal Gas Law

16.1.1 Understand that a gas obeying pV ∝ T is an ideal gas

16.1.2 Recall and use the equation pV = nRT for an ideal gas

16.1.3 Recall pV = NkT, where N = number of molecules and k is Boltzmann constant

16.1.4 Use the relationship k = R / NA

17. Kinetic Theory of Gases

17.1 Kinetic Theory

17.1.1 State the basic assumptions of the kinetic theory of gases

17.1.2 Explain pressure exerted by a gas due to molecular movement

17.1.3 Derive and use pV = (3/2)NmkT for pressure of a gas

17.1.4 Compare pV = (3/2)NmkT with pV = NkT to deduce average kinetic energy of a molecule

18. Particle Physics

18.1 Atoms, Nuclei and Radiation

18.1.1 Infer the existence and small size of the nucleus from α-particle scattering

18.1.2 Describe the nuclear atom model: protons, neutrons, and electrons

18.1.3 Distinguish between nucleon number and proton number

18.1.4 Understand isotopes as forms of the same element with different neutrons

18.1.5 Use the notation AZX for nuclides representation

18.1.6 Understand conservation of nucleon number and charge in nuclear processes

18.1.7 Describe the composition, mass, and charge of α-, β-, and γ-radiations

18.1.8 Understand antiparticles and positrons as antiparticles of electrons

18.1.9 State that (electron) antineutrinos are produced during β- decay and neutrinos during β+ decay

18.1.10 Represent α- and β-decay using radioactive decay equations

18.1.11 Use unified atomic mass unit (u) for mass

18.2 Fundamental Particles

18.2.1 Understand quarks as fundamental particles with six flavours

18.2.2 Understand hadrons as baryons (three quarks) or mesons (one quark and one antiquark)

18.2.3 Describe changes in quark composition during β- and β+ decay

18.2.4 Describe protons and neutrons as composed of quarks

18.2.5 Recall and use the charge of each quark and its respective antiquark

18.2.6 Recall electrons and neutrinos as fundamental particles (leptons)

19. Internal Energy

19.1 Internal Energy

19.1.1 Understand that internal energy is the sum of kinetic and potential energies of molecules in a syste

19.1.2 Relate rise in temperature to an increase in internal energy

20. Forces, Density and Pressure

20.1 Turning Effects of Forces

20.1.1 Weight acts at a single point: centre of gravity

20.1.2 Define and apply the moment of a force

20.1.3 Understand the concept of a couple

20.1.4 Define and apply torque of a couple

20.2 Equilibrium of Forces

20.2.1 State and apply the principle of moments

20.2.2 System in equilibrium has no resultant force or torque

20.2.3 Use a vector triangle to represent coplanar forces in equilibrium

20.3 Density and Pressure

20.3.1 Define and use density

20.3.2 Define and use pressure

20.3.3 Derive equation for hydrostatic pressure ∆p = ρg∆h

20.3.4 Use the equation ∆p = ρg∆h

20.3.5 Understand upthrust as the effect of hydrostatic pressure

20.3.6 Calculate upthrust using F = ρgV (Archimedes' principle)

21. Astronomy and Cosmology

21.1 Standard Candles

21.1.1 Understand luminosity as the total power radiated by a star

21.1.2 Recall and use the inverse square law F = L / (4πd²) for radiant flux intensity

21.1.3 Understand that an object with known luminosity is a standard candle

21.1.4 Use standard candles to determine distances to galaxies

21.2 Stellar Radii

21.2.1 Recall and use Wien’s displacement law λmax ∝ 1 / T to estimate the surface temperature of a star

21.2.2 Use Stefan–Boltzmann law L = 4πσr²T⁴

21.2.3 Use Wien’s law and Stefan–Boltzmann law to estimate the radius of a star

21.3 Hubble’s Law and Big Bang Theory

21.3.1 Understand redshift and the increase in wavelength of light from distant objects

21.3.2 Use ∆λ / λ = ∆f / f = v / c for redshift

21.3.3 Explain why redshift suggests the Universe is expanding

21.3.4 Recall and use Hubble’s law v = H₀d to explain the Big Bang theory

22. First Law of Thermodynamics

22.1 First Law

22.1.1 Recall and use W = p∆V for work done when the volume of a gas changes at constant pressure

22.1.2 Understand the difference between work done by a gas and work done on a gas

22.1.3 Recall and use the first law of thermodynamics: ∆U = q + W

23. Alternating Currents

23.1 Characteristics of Alternating Currents

23.1.1 Understand and use terms period, frequency, and peak value for alternating current or voltage

23.1.2 Use x = x₀ sin(ωt) for sinusoidal alternating current or voltage

23.1.3 Recall that the mean power in a resistive load is half the maximum power for sinusoidal current

23.1.4 Distinguish between root-mean-square (r.m.s.) and peak values and recall I₀r.m.s = I₀ / √2 and V₀r.m

23.2 Rectification and Smoothing

23.2.1 Distinguish graphically between half-wave and full-wave rectification

23.2.2 Explain the use of a single diode for half-wave rectification

23.2.3 Explain the use of four diodes (bridge rectifier) for full-wave rectification

23.2.4 Analyze the effect of a capacitor in smoothing, including the effect of capacitance and load resista

24. Superposition

24.1 Stationary Waves

24.1.1 Explain and use the principle of superposition

24.1.2 Understand experiments demonstrating stationary waves using microwaves, stretched strings, and air c

24.1.3 Explain the formation of stationary waves using graphical methods and identify nodes and antinodes

24.1.4 Determine wavelength from the positions of nodes or antinodes

24.2 Diffraction

24.2.1 Explain diffraction and show understanding of related experiments

24.2.2 Understand the effect of gap width relative to wavelength in diffraction experiments

24.3 Interference

24.3.1 Understand the terms interference and coherence

24.3.2 Demonstrate two-source interference using water waves, sound, light, and microwaves

24.3.3 Understand the conditions required for two-source interference fringes

24.3.4 Recall and use λ = ax / D for double-slit interference with light

24.4 Diffraction Grating

24.4.1 Recall and use d sin θ = nλ for diffraction grating calculations

24.4.2 Describe the use of a diffraction grating to determine the wavelength of light

25. Simple Harmonic Oscillations

25.1 Simple Harmonic Motion

25.1.1 Understand and use terms displacement, amplitude, period, frequency, angular frequency, and phase di

25.1.2 Understand that simple harmonic motion occurs when acceleration is proportional to displacement from

25.1.3 Use a = -ω²x and recall x = x₀ sin(ωt) as the solution

25.1.4 Use v = v₀ cos(ωt) and v = ±ω√(x₀² - x²) for velocity

26. Energy in Simple Harmonic Motion

26.1 Energy in SHM

26.1.1 Describe the interchange between kinetic and potential energy during SHM

26.1.2 Recall and use E = ½mω²x₀² for the total energy of a system undergoing SHM

27. Quantum Physics

27.1 Energy and Momentum of a Photon

27.1.1 Understand that electromagnetic radiation has a particulate nature

27.1.2 Recall and use E = hf for the energy of a photon

27.1.3 Use the electronvolt (eV) as a unit of energy

27.1.4 Understand that a photon has momentum and use p = E / c for photon momentum

27.2 Photoelectric Effect

27.2.1 Understand that photoelectrons are emitted from a metal surface when illuminated by electromagnetic

27.2.2 Understand and use the terms threshold frequency and threshold wavelength

27.2.3 Explain photoelectric emission in terms of photon energy and work function

27.2.4 Recall and use hf = Φ + ½mv² for the maximum kinetic energy of photoelectrons

27.2.5 Explain why the maximum kinetic energy of photoelectrons is independent of intensity, while current

27.3 Wave-Particle Duality

27.3.1 Understand that the photoelectric effect shows the particulate nature of light, while diffraction sh

27.3.2 Describe and interpret electron diffraction as evidence for the wave nature of particles

27.3.3 Understand the de Broglie wavelength as the wavelength associated with a moving particle

27.3.4 Recall and use λ = h / p for de Broglie wavelength

27.4 Energy Levels in Atoms and Line Spectra

27.4.1 Understand discrete electron energy levels in atoms (e.g., atomic hydrogen)

27.4.2 Understand the appearance and formation of emission and absorption line spectra

27.4.3 Recall and use hf = E₁ – E₂ for the energy difference between electron energy levels

28. Damped and Forced Oscillations, Resonance

28.1 Damped Oscillations

28.1.1 Understand that damping occurs due to resistive forces

28.1.2 Understand the types of damping: light, critical, and heavy damping

28.2 Resonance

28.2.1 Understand that resonance occurs when an oscillating system is forced to oscillate at its natural fr

29. Work, Energy and Power

29.1 Energy Conservation

29.1.1 Understand the concept of work, and recall W = F × s

29.1.2 Apply the principle of conservation of energy

29.1.3 Understand efficiency as useful energy output / total energy input

29.1.4 Use efficiency concept to solve problems

29.1.5 Define power as work done per unit time

29.1.6 Solve problems using P = W / t

29.1.7 Derive P = Fv and solve problems

29.2 Gravitational Potential Energy and Kinetic Energy

29.2.1 Derive ∆EP = mg∆h for gravitational potential energy changes

29.2.2 Recall and use ∆EP = mg∆h for gravitational potential energy changes

29.2.3 Derive EK = ½mv² for kinetic energy

29.2.4 Recall and use EK = ½mv² for kinetic energy

30. Electric Fields

30.1 Electric Fields and Field Lines

30.1.1 Understand that an electric field is a field of force and define it as force per unit positive charg

30.1.2 Represent an electric field using field lines

30.2 Uniform Electric Fields

30.2.1 Recall and use E = ∆V / ∆d to calculate electric field strength between charged parallel plates

30.2.2 Describe the effect of a uniform electric field on the motion of charged particles

30.3 Electric Force Between Point Charges

30.3.1 Understand that, for a point outside a spherical conductor, the charge may be considered a point mas

30.3.2 Recall and use Coulomb’s law F = Q₁Q₂ / (4πε₀r²) for the force between two point charges

30.4 Electric Field of a Point Charge

30.4.1 Recall and use E = Q / (4πε₀r²) for the electric field strength due to a point charge

30.5 Electric Potential

30.5.1 Define electric potential as work done per unit positive charge in bringing a test charge from infin

30.5.2 Recall and use V = Q / (4πε₀r) for electric potential due to a point charge

30.5.3 Understand how electric potential leads to electric potential energy of two point charges and use EP

31. Physical Quantities and Units

31.1 Physical Quantities

31.1.1 Physical quantities consist of magnitude and unit

31.1.2 Estimate physical quantities from the syllabus

31.2 SI Units

31.2.1 SI base units: mass, length, time, current, temperature

31.2.2 Express derived units from SI base units

31.2.3 Check homogeneity of physical equations

31.2.4 Use prefixes (pico, nano, micro, kilo, etc.)

31.3 Errors and Uncertainties

31.3.1 Understand systematic and random errors

31.3.2 Assess uncertainty in derived quantities

31.4 uniform Electric Fields

31.4.1 Distinguish between precision and accuracy

31.5 Scalars and Vectors

31.5.1 Difference between scalar and vector quantities

31.5.2 Add and subtract coplanar vectors

31.5.3 Represent a vector as components

32. Motion in a Circle

32.1 Centripetal Acceleration

32.1.1 Understand centripetal acceleration and its role in circular motion

32.1.2 Recall and use a = rω² and a = v² / r

32.1.3 Recall and use F = mrω² and F = mv² / r

32.1.4 Understand force causing centripetal acceleration is always perpendicular to motion

32.2 Kinematics of Uniform Circular Motion

32.2.1 Recall and use ω = 2π / T and v = rω

32.2.2 Understand and use angular speed

32.2.3 Define and express angular displacement in radians

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Understanding the Decrease in Resistance of a Thermistor with Increasing Temperature

Introduction

Key Concepts

What is a Thermistor?

Types of Thermistors

Electrical Resistance and Temperature

Material Composition of Thermistors

Applications of Thermistors

Temperature Coefficient of Resistance (TCR)

Non-Linear Resistance-Temperature Relationship

The Beta Parameter Equation

Practical Example: Calculating Resistance Change

Linearity and Compensation

Temperature Range and Stability

Manufacturing Considerations

Calibration of Thermistors

Environmental Factors Affecting Thermistors

Advanced Concepts

Mathematical Modeling of Thermistors

Derivation of the Steinhart-Hart Equation

Thermistor Sensitivity Analysis

Thermal Time Constant

Impact of Self-Heating

Non-Linear Compensation Techniques

Interdisciplinary Connections

Advanced Problem-Solving: Designing a Temperature-Controlled Circuit

Mathematical Derivation: From Beta Equation to Steinhart-Hart

Thermistor Networks and Signal Conditioning

Noise and Interference in Thermistor Measurements

Thermistor Aging and Reliability

Emerging Technologies and Future Trends

Comparison Table

Summary and Key Takeaways

Coming Soon!

Tips

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Common Mistakes

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