Grade 11
  1. Mechanics  1.1. dynamics  1.1.1. Motion in one dimension  1.1.2. Motion in two and three dimensions  1.1.3. Displacement and Distance  1.1.4. Speed and Velocity  1.1.5. Acceleration and deceleration  1.1.6. Graphical analysis of motion  1.1.7. Equations of motion (advanced derivations)  1.1.8. Free fall and terminal velocity  1.1.9. Projectile motion  1.1.10. Relative velocity in one and two dimensions  1.1.11. Motion of a car on an inclined plane  1.2. Dynamics  1.2.1. Newton's laws of motion and their applications  1.2.2. Inertia and Newton's first law  1.2.3. Force and types of force  1.2.4. Static and kinetic friction  1.2.5. Circular motion and centripetal acceleration  1.2.6. Non-uniform circular motion  1.2.7. Banking of roads and curves  1.2.8. Momentum and Impulse  1.2.9. Conservation of momentum in one and two dimensions  1.2.10. Applications of Newton's Third Law  1.3. Work, Energy and Power  1.3.1. Work done by a constant and variable force  1.3.2. Work–energy theorem  1.3.3. Kinetic and potential energy  1.3.4. Gravitational and elastic potential energy  1.3.5. Conservation of mechanical energy  1.3.6. Power and instantaneous power  1.3.7. Efficiency of machines  1.3.8. Work done by conservative and non-conservative forces  1.4. Rotational motion  1.4.1. Torque and angular acceleration  1.4.2. Rotational equilibrium  1.4.3. Moment of inertia and its applications  1.4.4. Angular momentum and its conservation  1.4.5. Kinetic energy of rolling motion and rotation  1.4.6. Parallel and Perpendicular Axis Theorem  1.4.7. Dynamics of rotational motion
  2. Gravitational force  2.1. Universal gravitation  2.1.1. Newton's law of universal gravitation  2.1.2. Gravitational field and potential  2.1.3. Change in acceleration due to gravity  2.1.4. Kepler's laws of planetary motion  2.1.5. Orbital mechanics and satellites  2.1.6. Escape velocity and orbital velocity  2.1.7. Weightlessness and free fall  2.1.8. Gravitational potential energy and energy in orbits  2.1.9. Black holes and gravitational lensing
  3. Properties of matter  3.1. Fluid mechanics  3.1.1. Density and pressure in liquids  3.1.2. Pascal's theory and applications  3.1.3. Archimedes' Principle and Buoyancy  3.1.4. Bernoulli's equation and applications  3.1.5. Continuity equation and the Venturi effect  3.1.6. Surface tension and capillarity  3.1.7. Viscosity and Poiseuille's Law  3.1.8. Reynolds number and turbulence  3.2. Elasticity and deformation  3.2.1. Hooke's law and the stress-strain relation  3.2.2. Elastic modulus and applications  3.2.3. Deformation and yield strength of solids
  4. Thermal physics  4.1. Heat and temperature  4.1.1. Thermometric properties and temperature scale  4.1.2. Thermal expansion of solids, liquids and gases  4.1.3. Heat transfer system  4.1.4. Specific heat capacity and calorimetry  4.1.5. Latent heat and phase change  4.2. Kinetic theory of gases  4.2.1. Molecular model of gases  4.2.2. Kinetic explanation of temperature  4.2.3. Ideal Gas Equation and Deviations  4.2.4. Mean free path and Maxwell–Boltzmann distribution  4.3. Laws of Thermodynamics  4.3.1. Zeroth Law and Thermal Equilibrium  4.3.2. First Law and Internal Energy  4.3.3. The Second Law and Entropy  4.3.4. Carnot cycle and heat engines  4.3.5. Refrigerators and heat pumps
  5. Waves and oscillations  5.1. Simple Harmonic Motion  5.1.1. Equations of SHM  5.1.2. Energy in SHM  5.1.3. Damped and forced oscillations  5.1.4. Resonance and applications  5.1.5. Pendulum and spring-mass system  5.2. Wave motion  5.2.1. Types of mechanical waves  5.2.2. Wave motion and energy transport  5.2.3. Superposition Principle and Standing Waves  5.2.4. Reflection, Refraction and Diffraction  5.2.5. Doppler effect in sound and light  5.2.6. Pulsations and interference
  6. Electricity and Magnetism  6.1. Electrostatics  6.1.1. Electric charge and conservation  6.1.2. Coulomb's law and its applications  6.1.3. Electric field and electric flux  6.1.4. Electric potential and potential energy  6.1.5. Capacitance and Energy Stored in a Capacitor  6.2. Current Electricity  6.2.1. Ohm's law and electrical resistance  6.2.2. Resistivity and temperature dependence  6.2.3. Kirchhoff's Laws and Circuit Analysis  6.2.4. Electrical power and efficiency  6.2.5. Combination of resistors  6.3. Magnetism and Electromagnetism  6.3.1. Magnetic fields and their sources  6.3.2. Magnetic force on moving charges  6.3.3. Electromagnetic induction  6.3.4. Faraday's and Lenz's laws  6.3.5. Inductance and Transformer  6.3.6. Alternating current and its applications
  7. Optics  7.1. Reflection and Refraction  7.1.1. laws of reflection and refraction  7.1.2. Mirrors and lenses  7.1.3. Total internal reflection and optical fiber  7.2. Wave optics  7.2.1. Interference and Diffraction  7.2.2. Young's double-slit experiment  7.2.3. Polarization of light
  8. Modern Physics  8.1.1. Photoelectric effect and Einstein's theory  8.1.2. Wave–particle duality  8.1.3. Heisenberg's uncertainty principle  8.1.4. Compton Effect  8.2. Atomic and Nuclear Physics  8.2.1. Atomic model and energy levels  8.2.2. Radioactive decay and half-life  8.2.3. Nuclear Fission and Fusion
  9. Electronics and Communication  9.1. Semiconductors  9.1.1. pn junction and diode  9.1.2. Transistors and Logic Gates  9.1.3. Integrated Circuits and Applications  9.2. Communication Systems  9.2.1. Basics of Analog and Digital Communication  9.2.2. Wireless and Optical Communication  9.2.3. Modulation and Demodulation

Grade 11 → Electricity and Magnetism → Current Electricity ↓


Combination of resistors


In the study of electricity, especially when dealing with circuits, we often encounter the concept of combining resistors in various configurations to achieve desired electrical characteristics. Understanding how resistors behave when combined is essential in designing and analyzing electrical circuits. In this explanation, we will take a deeper look at the various ways to combine resistors and how these combinations affect the overall resistance of the circuit.

Introduction to resistance

Resistance is an inherent property of materials that opposes the flow of electric current. It is measured in ohms (Ω). The resistance of a material depends on its composition, length, cross-sectional area and temperature. When dealing with resistors in circuits, we use Ohm's law as a fundamental principle. Ohm's law is represented as:

        V=IR

Where V is the voltage across the resistor, I is the current flowing through it, and R is the resistance.

Types of resistor combinations

Resistors can be connected in a circuit in two basic ways: series combination and parallel combination. Understanding these will allow us to effectively determine the overall resistance of a circuit.

Series combination

In a series combination, resistors are connected end-to-end. The current flowing through each resistor is the same, but the voltage across each resistor may vary. The total or equivalent resistance of resistors connected in series is simply the sum of their individual resistances.

        R_total = R_1 + R_2 + R_3 + ... + R_n

Consider the following example of a simple series circuit with three resistors:

R1 R2 R3

If R_1 = 2Ω, R_2 = 3Ω, and R_3 = 5Ω, then the total resistance R_total will be:

        R_total = 2Ω + 3Ω + 5Ω = 10Ω

The advantage of series connection is that the total resistance can be easily calculated and analyzed, but the drawback is that if one resistor fails (becomes open circuit), the whole circuit stops working.

Parallel combination

In parallel combination, resistors are connected in such a way that each resistor is connected to the same two points, and shares the same voltage between them. The total or equivalent resistance of resistors in parallel is less than the resistance of the smallest individual resistor in the combination.

        1/R_total = 1/R_1 + 1/R_2 + 1/R_3 + ... + 1/R_n

Consider this example of three resistors connected in parallel:

R1 R2 R3

If R_1 = 2Ω, R_2 = 3Ω, and R_3 = 6Ω, then we find the total resistance:

        1/R_total = 1/2 + 1/3 + 1/6 
                  = 3/6 + 2/6 + 1/6 
                  = 6/6 
                  = 1Ω

Therefore, R_total = 1Ω. In parallel connections, the total resistance decreases as you add more resistors. Unlike a series circuit, if one resistor fails (opens), current can still flow through other routes.

Applications and practical examples

Electrical engineers often use combinations of resistors to design circuits with the required resistance levels. The balance of series and parallel resistors allows flexibility in circuit design. To see this in practical terms:

Example 1: Adjusting the total resistance

Let's say you have resistors of 3Ω, 6Ω and 9Ω. You need to get a circuit with a total resistance of 4Ω. One way is to use a combination of series and parallel resistors:

  1. Place the 6Ω resistor in parallel with the 9Ω resistor:
    2.             1/R_parallel = 1/6 + 1/9 = 3/18 + 2/18 = 5/18
            R_parallel = 18/5 = 3.6Ω
  2. Connect this combination in series with a 3Ω resistor:
    3.             R_total(series) = 3.6Ω + 3Ω = 6.6Ω
    This configuration achieves a close approximation, but different resistors can achieve different levels.

Example 2: Using variable resistors

Variable resistors, such as potentiometers, allow for fine adjustments in resistance. In a laboratory setting, they are especially useful when you need to calibrate a circuit. Experimenting with different lengths using the variable knob adjusts the resistance in real time.

Conclusion

Understanding series and parallel resistor combinations is crucial in designing effective circuits for a variety of applications. Mastering these concepts opens the door to more advanced electrical engineering discoveries. Combining resistors effectively requires a balance between theoretical knowledge and practical application, always keeping in mind the purpose of the circuit and the constraints it contains.


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Combination of resistors

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