Understanding the combined resistance of resistors in series is fundamental in the study of electrical circuits, particularly within the context of Kirchhoff’s Laws. For AS & A Level Physics students, mastering this concept is essential for analyzing and designing D.C. circuits effectively. This article delves into the formula for calculating combined resistance in series, its theoretical underpinnings, practical applications, and advanced problem-solving techniques.

Key Concepts

Understanding Resistors in Series

Resistors are electrical components that impede the flow of electric current, measured in Ohms (Ω). When resistors are connected end-to-end in a single path, they are said to be in series. In a series configuration, the same current flows through each resistor, and the total or equivalent resistance is the sum of the individual resistances. This setup is crucial for controlling current in circuits and for dividing voltage among components.

Formula for Combined Resistance in Series

The fundamental formula to calculate the total resistance (R_total) of resistors connected in series is given by:

$$ R_{total} = R_1 + R_2 + R_3 + \dots + R_n $$

Here, R₁, R₂, R₃, ..., R_n represent the individual resistances of the resistors in the series.

Derivation of the Series Resistance Formula

Starting with Ohm’s Law, which states that V = IR, where V is voltage, I is current, and R is resistance. In a series circuit, the current (I) is the same through each resistor. The total voltage (V_total) across the series is the sum of the voltages across each resistor:

$$ V_{total} = V_1 + V_2 + V_3 + \dots + V_n $$

Applying Ohm’s Law to each resistor:

$$ V_1 = I R_1 $$ $$ V_2 = I R_2 $$ $$ \vdots $$ $$ V_n = I R_n $$

Substituting these into the total voltage equation:

$$ V_{total} = I R_1 + I R_2 + \dots + I R_n = I (R_1 + R_2 + \dots + R_n) $$

Rearranging for the total resistance:

$$ R_{total} = R_1 + R_2 + R_3 + \dots + R_n $$>

Practical Example

Consider three resistors connected in series with resistances of 4 Ω, 6 Ω, and 10 Ω. To find the total resistance:

$$ R_{total} = 4\,Ω + 6\,Ω + 10\,Ω = 20\,Ω $$>

If a current of 2 A flows through the circuit, the total voltage supplied by the source is:

$$ V_{total} = I R_{total} = 2\,A \times 20\,Ω = 40\,V $$>

The voltage drop across each resistor can be calculated as:

V₁ = 2 A × 4 Ω = 8 V
V₂ = 2 A × 6 Ω = 12 V
V₃ = 2 A × 10 Ω = 20 V

This example illustrates how the total resistance and voltage are distributed in a series circuit.

Applications of Series Resistance

Series resistance configurations are commonly used in various electrical and electronic applications, including:

Voltage Dividers: Creating specific voltage levels from a higher voltage source.
Current Limiters: Controlling the current flowing through sensitive components.
Battery Packs: Connecting multiple batteries in series to achieve a desired voltage.

Impact of Adding Resistors in Series

Adding more resistors in series increases the total resistance, which in turn affects the current and voltage distribution in the circuit:

Increased Resistance: More resistors mean higher total resistance.
Decreased Current: According to Ohm’s Law, higher resistance leads to lower current if the voltage remains constant.
Voltage Distribution: The total voltage is divided among the resistors based on their individual resistances.

Series vs. Parallel Resistors

Understanding the difference between series and parallel resistor configurations is crucial:

Series: Resistors add up; same current flows through each resistor.
Parallel: Reciprocal of total resistance is the sum of reciprocals of individual resistances; same voltage across each resistor.

Kirchhoff’s Laws and Series Resistance

Kirchhoff’s Current Law (KCL) and Kirchhoff’s Voltage Law (KVL) are essential for analyzing circuits with series resistors:

KCL: States that the total current entering a junction equals the total current leaving. In a series circuit, since there is only one path, the current remains constant.
KVL: States that the sum of all electrical potential differences around a loop is zero. This law is directly applied when summing the voltages across each resistor in a series configuration.

Calculating Power in Series Resistor Networks

The power dissipated by resistors in a series can be calculated using the formula:

$$ P = I^2 R $$>

Where P is power in watts (W), I is current in amperes (A), and R is resistance in ohms (Ω).

Using the earlier example with R_total = 20 Ω and I = 2 A:

$$ P = (2\,A)^2 \times 20\,Ω = 4 \times 20 = 80\,W $$>

This total power is distributed among the resistors based on their individual resistances.

Series Resistors in Real-World Circuits

In practical circuits, resistors are rarely connected purely in series or parallel. Many circuits combine both configurations to achieve desired electrical characteristics. Understanding series resistance is the first step towards analyzing more complex resistor networks.

Troubleshooting Series Resistor Circuits

Common issues in series resistor circuits include:

Open Circuits: A broken connection interrupts the current flow entirely.
Incorrect Resistor Values: Using resistors with unintended values can lead to improper voltage distribution.
Heat Dissipation: Higher resistance can lead to excessive heat generation, potentially damaging components.

Practical Measurement of Series Resistance

To measure the total resistance in a series configuration:

Ensure the power supply is disconnected to prevent electric shock.
Use a digital multimeter set to the resistance (Ω) mode.
Connect the multimeter probes across the entire series network.
Record the reading, which should be the sum of all individual resistances.

Limitations of Series Resistor Calculations

While the series resistance formula is straightforward, certain factors can affect accuracy:

Temperature Variations: Resistor values can change with temperature, affecting total resistance.
Tolerance Levels: Resistors come with manufacturing tolerances that can cause deviations from nominal values.
Interconnect Resistance: The resistance of connecting wires, although usually negligible, can influence total resistance in precision circuits.

Using Series Resistance in Circuit Design

Designing circuits with series resistors involves strategic placement to control current and voltage. For instance, LEDs are often connected in series with resistors to ensure each LED receives the appropriate current, preventing damage and ensuring consistent brightness.

Advanced Concepts

Mathematical Derivation of Series Resistance in Complex Circuits

In circuits with multiple looped paths containing resistors in series and parallel, determining the total resistance requires applying Kirchhoff’s Laws systematically. Consider a circuit with several resistors arranged in a combination of series and parallel configurations. By identifying equivalent resistances step-by-step—reducing complex networks to simpler series or parallel groups—one can derive the overall resistance.

For example, consider a circuit with three resistors: R₁ and R₂ in parallel, connected in series with R₃. The total resistance is calculated as:

$$ R_{parallel} = \frac{R_1 R_2}{R_1 + R_2} $$> $$ R_{total} = R_{parallel} + R_3 = \frac{R_1 R_2}{R_1 + R_2} + R_3 $$>

This approach can be extended to circuits with more resistors, using recursive reduction techniques.

Power Distribution in Series Networks

When dealing with multiple resistors in series, understanding how power is distributed is crucial for efficient circuit design. The power consumed by each resistor depends on its resistance and the current flow:

$$ P_i = I^2 R_i $$>

Alternatively, using voltage across each resistor:

$$ P_i = \frac{V_i^2}{R_i} $$>

This dual approach allows for flexibility in analyzing power consumption based on known parameters.

Impact of Temperature on Series Resistance

Resistors exhibit temperature coefficients that dictate how their resistance changes with temperature. In series circuits, as individual resistors heat up, their resistances may increase or decrease, affecting the total resistance and, consequently, the current flow.

The temperature dependence of resistance is given by:

$$ R(T) = R_0 [1 + \alpha (T - T_0)] $$>

Where:

R(T) is the resistance at temperature T.
R₀ is the original resistance at reference temperature T₀.
α is the temperature coefficient of resistance.

Series Resistors in High-Frequency AC Circuits

While the basic principles of series resistance apply to D.C. circuits, alternating current (AC) introduces additional considerations such as inductive and capacitive reactance. In high-frequency AC circuits, resistors in series also exhibit inductive properties, altering the total impedance.

The total impedance (Z) in a series RLC circuit is calculated as:

$$ Z = \sqrt{R^2 + (X_L - X_C)^2} $$>

Where:

X_L: Inductive reactance = $2\pi f L$
X_C: Capacitive reactance = $\frac{1}{2\pi f C}$

Interdisciplinary Connections: Series Resistance in Electrical Engineering

In electrical engineering, series resistance calculations are foundational for designing complex systems such as power distribution networks, electronic devices, and signal processing circuits. Engineers must account for series resistances to ensure signal integrity, minimize power loss, and optimize performance.

For instance, in battery technology, connecting cells in series increases the total voltage available to power devices, necessitating precise resistance calculations to balance load distribution and prevent overloading individual cells.

Advanced Problem-Solving: Series Resistance in Multiple Loops

Consider a circuit with three loops, each containing a combination of series and parallel resistors. To find the total resistance:

Identify and simplify parallel resistor groups.
Sum the simplified resistances in series across loops.
Apply Kirchhoff’s Laws to ensure accurate calculation of complex networks.

This method requires meticulous analysis and often multiple iterations to reduce the network to a single equivalent resistance.

Using Series Resistance in Signal Filtering

In signal processing, series resistors combined with capacitors form low-pass or high-pass filters. The resistor’s value, in series with a capacitor, determines the filter's cutoff frequency (f_c), calculated as:

$$ f_c = \frac{1}{2\pi R C} $$>

Designing such filters involves selecting appropriate resistances to achieve desired frequency responses for applications like audio processing or radio signal tuning.

Energy Efficiency and Series Resistors

Optimizing energy efficiency in circuits with series resistors involves minimizing unnecessary resistance to reduce energy loss. This is particularly important in power transmission where high resistance can lead to significant energy dissipation as heat.

Strategies include:

Using higher conductance materials.
Shortening the length of conductive paths.
Implementing superconducting materials where feasible.

Thermal Management in Series Resistor Networks

Managing heat in series resistor systems is vital to maintain component integrity and prevent thermal runaway. Techniques include:

Heat Sinks: Attaching radiative devices to dissipate excess heat.
Ventilation: Ensuring adequate airflow around resistors.
Material Selection: Using resistors with higher thermal ratings.

Series Resistors in Electronic Timing Circuits

Resistors in series with capacitors are integral to timing circuits, such as those found in oscillators and timers. The time constant (τ) of an RC circuit is given by:

$$ \tau = R \times C $$>

This constant defines how quickly the circuit responds to voltage changes, crucial for applications requiring precise timing control.

Simulation Tools for Analyzing Series Resistance

Modern electrical engineering employs simulation software like SPICE (Simulation Program with Integrated Circuit Emphasis) to model and analyze circuits with series resistors. These tools allow for:

Visualizing current flow and voltage distribution.
Predicting circuit behavior under varying conditions.
Optimizing resistor values for specific applications without physical prototyping.

Advanced Materials and Series Resistance

The development of new materials with unique resistive properties affects how series resistors are utilized in cutting-edge technologies. Materials such as graphene and carbon nanotubes offer low-resistance paths with high durability, enabling more efficient and compact circuit designs.

Series Resistance in Renewable Energy Systems

In renewable energy setups like solar panels, series resistors help manage voltage levels and ensure consistent power output. Proper resistance calculations are essential to maximize energy harvesting and maintain system stability.

Quantum Considerations in Series Resistance

At the nanoscale, quantum effects can influence resistance in series configurations. Phenomena like electron tunneling and quantized conductance become significant, requiring advanced theoretical models to accurately predict resistance behaviors.

Case Study: Designing a Series Resistor Network for LED Lighting

Consider designing a series resistor network for a string of LEDs operating at different forward voltages. Calculations must account for varying resistor values to ensure each LED receives appropriate current without exceeding its maximum ratings.

Steps include:

Determining the desired current for each LED.
Calculating individual resistor values using $R = \frac{V_{source} - V_{LED}}{I}$.
Summing the resistances to find total resistance.
Verifying thermal ratings to prevent overheating.

This case study highlights the practical application of series resistance calculations in developing safe and efficient lighting systems.

Impact of Manufacturing Tolerances on Series Resistance

When assembling series resistor networks, manufacturing tolerances can introduce variability in total resistance. Engineers must account for these tolerances to ensure circuit performance remains within specified limits, often by selecting resistors with tighter tolerance grades or implementing calibration procedures.

Innovations in Variable Series Resistors

Variable resistors (potentiometers) connected in series offer adjustable resistance, enabling dynamic control over circuit parameters. Applications include volume controls in audio equipment, adjustable biasing in amplifiers, and tuning circuits in communication devices.

Energy Storage with Series Resistor Networks

In energy storage systems, such as supercapacitors connected in series, accurate resistance calculations are essential to balance charge distribution and prevent overcharging. By managing series resistances, system efficiency and longevity are optimized.

Environmental Factors Affecting Series Resistance

External factors like humidity, exposure to corrosive environments, and mechanical stress can alter the resistance of series networks. Protective measures, such as encapsulation and use of resistant materials, are necessary to maintain consistent circuit performance.

Future Trends in Series Resistance Applications

Advancements in nanotechnology and materials science are paving the way for more efficient and compact resistor networks. Emerging applications include flexible electronics, wearable devices, and integrated circuit designs that leverage optimized series resistance for enhanced functionality.

Comparison Table

Aspect	Series Resistors	Parallel Resistors
Total Resistance	Sum of individual resistances: $R_{total} = R_1 + R_2 + \dots + R_n$	Reciprocal of sum of reciprocals: $ \frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \dots + \frac{1}{R_n} $
Current Flow	Same current flows through each resistor.	Same voltage across each resistor.
Voltage Distribution	Voltage divides among resistors based on their resistances.	Voltage remains constant across all resistors.
Application Example	Voltage dividers, battery packs in series.	Parallel circuits in household wiring, parallel LED configurations.
Advantages	Simplicity in calculation, easy to add resistances.	Lower total resistance, flexibility in current distribution.
Disadvantages	Total resistance increases with each added resistor.	Total resistance decreases with each added resistor.

Summary and Key Takeaways

Series resistors have a total resistance equal to the sum of individual resistances.
Understanding series resistance is crucial for circuit analysis and design.
Keechhoff’s Laws play a vital role in analyzing series resistor networks.
Advanced applications include power distribution, signal filtering, and renewable energy systems.
Managing temperature, tolerance, and material properties is essential for reliable series resistor performance.

Examiner Tip

Tips

Remember the mnemonic "SARA" to differentiate between Series and Parallel resistors:

Series: Sum the resistances.
Amperage remains the same.
Resistance adds up.
Apply the correct formula.

Additionally, always double-check your calculations and consider using circuit simulation tools like SPICE to validate your results before finalizing your answers.

Did You Know

Did you know that the concept of series resistance dates back to the early studies of electrical circuits by Georg Ohm in the 19th century? Understanding series resistors is not only fundamental in physics but also plays a critical role in designing everyday electronic devices like smartphones and computers. Additionally, in space missions, managing series resistance is crucial to ensure the reliability of electrical systems in extreme conditions.

Common Mistakes

One common mistake students make is incorrectly adding resistances in parallel instead of in series, leading to an overestimation of total resistance. For example, mistakenly using the parallel formula $R_{total} = \frac{R_1 R_2}{R_1 + R_2}$ for series resistors instead of $R_{total} = R_1 + R_2$. Another error is neglecting the impact of resistor tolerance, which can cause significant deviations in expected circuit behavior. Ensuring to apply the correct formula and account for tolerance can prevent these mistakes.

FAQ

What is the total resistance of three resistors in series: 5 Ω, 10 Ω, and 15 Ω?

The total resistance R_total is the sum of all resistances: R_total = 5 Ω + 10 Ω + 15 Ω = 30 Ω.

How does adding more resistors in series affect the total resistance?

Can the series resistance formula be applied to both DC and AC circuits?

Yes, the series resistance formula applies to DC circuits directly. For AC circuits, while the resistive component follows the same principle, impedance must consider inductive and capacitive reactances as well.

What happens to the current in a series circuit if one resistor fails (open circuit)?

If one resistor fails in an open circuit, the entire circuit is interrupted, and the current stops flowing through all resistors.

How do temperature changes affect resistors in series?

Temperature changes can alter the resistance of each resistor due to their temperature coefficients. In series, this affects the total resistance and can lead to variations in current and voltage distribution.

Use the formula for combined resistance of resistors in series in AS & A Level Physics. Master key concepts, solve advanced problems, and optimize your D.C. circuit analysis.

series resistors, combined resistance, Kirchhoff’s Laws, D.C. circuits, AS Level Physics, A Level Physics, resistor networks, electrical circuits, power distribution, circuit analysis

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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Use the Formula for Combined Resistance of Resistors in Series

Introduction