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By the end of this lesson, you will be able to:
- Use the formula for the combined resistance of two or more resistors in series.
- Derive, using Kirchhoff’s laws, a formula for the combined resistance of two or more resistors in parallel.
- Use the formula for the combined resistance of two or more resistors in parallel.
- Use Kirchhoff’s laws to solve simple circuit problems.
By the end of this lesson, you will be able to:
- Understand and use key vocabulary related to electric circuits, such as , , , , , , and .
- Discuss circuit diagrams and problem solutions using appropriate terminology.
Familiarize yourself with these important terms. Pay attention to their translations.
| English Term | Russian Translation (Русский перевод) | Kazakh Translation (Қазақша аудармасы) |
|---|---|---|
| Electric Current (I) | Электрический ток | Электр тогы |
| Voltage (V) / Potential Difference | Напряжение / Разность потенциалов | Кернеу / Потенциалдар айырымы |
| Resistance (R) | Сопротивление | Кедергі |
| Resistor | Резистор | Резистор |
| Series Circuit | Последовательная цепь | Тізбектей жалғау |
| Parallel Circuit | Параллельная цепь | Параллель жалғау |
| Kirchhoff’s Current Law (KCL) | Первый закон Кирхгофа (закон токов) | Кирхгофтың бірінші заңы (токтар заңы) |
| Kirchhoff’s Voltage Law (KVL) | Второй закон Кирхгофа (закон напряжений) | Кирхгофтың екінші заңы (кернеулер заңы) |
| Junction / Node | Узел | Түйін |
| Loop | Контур | Контур |
| Electromotive Force (EMF, ε) | Электродвижущая сила (ЭДС) | Электр қозғаушы күші (ЭҚК) |
| Power (P) | Мощность | Қуат |
Use these flashcards to review the key concepts and terms. You can find many useful sets on Quizlet by searching for «Resistors in Series and Parallel» or «Kirchhoff’s Laws».
Here is an example set (You might need to adjust the link or find a more specific one):
(Source: Quizlet. If the embed doesn’t work, try this link: Electric Circuits Flashcards on Quizlet)
Circuit: A closed loop path that electric current to flow.
Resistor: An electrical that implements electrical resistance as a circuit element.
Current (I): The rate of flow of electric . Measured in Amperes (A).
Voltage (V): The electric potential difference between two points. Measured in Volts (V).
Resistance (R): A measure of the to current flow in an electrical circuit. Measured in Ohms (Ω).
Kirchhoff’s Current Law (KCL): The sum of currents entering a (or node) is equal to the sum of currents leaving the junction. This is a statement of charge conservation.
Kirchhoff’s Voltage Law (KVL): The sum of the electromotive forces (EMFs) in any closed loop is to the sum of the potential drops in that loop. This is a statement of energy conservation.
This section covers the fundamental principles of combining resistors and Kirchhoff’s laws for circuit analysis.
1. Kirchhoff’s Laws
Gustav Kirchhoff formulated two laws that are fundamental to circuit analysis. These laws are based on the conservation of charge and energy.
a) Kirchhoff’s Current Law (KCL) — The Junction Rule
KCL states that the algebraic sum of currents entering any junction (or node) in a circuit must equal the algebraic sum of currents leaving that junction. Essentially, . This law is a consequence of the conservation of electric charge.
ΣIin = ΣIout
For example, if currents I1 and I2 enter a junction, and current I3 leaves it, then: I1 + I2 = I3.
b) Kirchhoff’s Voltage Law (KVL) — The Loop Rule
KVL states that the algebraic sum of the potential differences (voltages) around any closed loop or path in a circuit must be zero. This law is a consequence of the conservation of energy.
ΣΔV = 0 (around a closed loop)
This means that the sum of electromotive forces (EMFs, e.g., from batteries) is equal to the sum of voltage drops across resistors and other components in the loop.
Σε = ΣIR
2. Resistors in Series
When resistors are connected in series, they are connected end-to-end, providing a single path for the current to flow.
![[Изображение: Резисторы соединенные последовательно]](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%201%201'%3E%3C/svg%3E)
Key characteristics of a series circuit:
- Current: The current is the through each resistor: Itotal = I1 = I2 = I3 = …
- Voltage: The total voltage across the combination is the sum of the individual voltage drops across each resistor: Vtotal = V1 + V2 + V3 + …
- Resistance: The total (or equivalent) resistance RT is the sum of the individual resistances:
RT = R1 + R2 + R3 + …
Derivation (using Ohm’s Law V=IR and KVL):
Consider resistors R1, R2, …, Rn in series. The total voltage VT is VT = V1 + V2 + … + Vn (from KVL).
Since V1 = IR1, V2 = IR2, etc. (current I is the same),
VT = IR1 + IR2 + … + IRn = I(R1 + R2 + … + Rn).
If RT is the equivalent resistance, then VT = IRT.
Therefore, IRT = I(R1 + R2 + … + Rn), which simplifies to RT = R1 + R2 + … + Rn.
3. Resistors in Parallel
When resistors are connected in parallel, they are connected across the same two points (junctions), providing multiple paths for the current to flow.
![[Изображение: Резисторы соединенные параллельно]](data:image/svg+xml,%3Csvg%20xmlns=%22http://www.w3.org/2000/svg%22%20viewBox=%220%200%20210%20140%22%3E%3C/svg%3E)
Key characteristics of a parallel circuit:
- Voltage: The voltage is the across each resistor: Vtotal = V1 = V2 = V3 = …
- Current: The total current is the sum of the currents through each branch (from KCL): Itotal = I1 + I2 + I3 + …
- Resistance: The reciprocal of the total (or equivalent) resistance RT is the sum of the reciprocals of the individual resistances:
1/RT = 1/R1 + 1/R2 + 1/R3 + …
Derivation (using Ohm’s Law I=V/R and KCL):
Consider resistors R1, R2, …, Rn in parallel. The total current IT is IT = I1 + I2 + … + In (from KCL).
Since I1 = V/R1, I2 = V/R2, etc. (voltage V is the same across all),
IT = V/R1 + V/R2 + … + V/Rn = V(1/R1 + 1/R2 + … + 1/Rn).
If RT is the equivalent resistance, then IT = V/RT.
Therefore, V/RT = V(1/R1 + 1/R2 + … + 1/Rn), which simplifies to 1/RT = 1/R1 + 1/R2 + … + 1/Rn.
4. Solving Circuit Problems using Kirchhoff’s Laws
For more complex circuits that cannot be simplified to simple series or parallel combinations, Kirchhoff’s laws are essential.
Steps for applying Kirchhoff’s Laws:
- Label currents: Assign a current direction to each branch of the circuit. If your assumed direction is wrong, the calculated current will be negative. Label each current (e.g., I1, I2, I3).
- Apply KCL: Write KCL equations for (N-1) junctions, where N is the number of junctions. This ensures independent equations.
- Apply KVL: Choose independent loops in the circuit. Traverse each loop, writing KVL equations.
- When traversing a resistor in the direction of the assumed current, the potential change is -IR.
- When traversing a resistor opposite to the direction of the assumed current, the potential change is +IR.
- When traversing an EMF source from the negative to the positive terminal, the potential change is +ε.
- When traversing an EMF source from the positive to the negative terminal, the potential change is -ε.
- Solve the system of equations: You will have a system of linear equations. Solve them simultaneously to find the unknown currents (and then voltages or resistances if needed).
Theory Questions:
1. Easy: If three resistors of 10 Ω, 20 Ω, and 30 Ω are connected in series, what is their total resistance?
2. Medium: Two resistors, R1 = 6 Ω and R2 = 3 Ω, are connected in parallel. This combination is then connected in series with a 4 Ω resistor and a 12V battery. What is the total current supplied by the battery?
3. Medium: State Kirchhoff’s Current Law and Kirchhoff’s Voltage Law. Explain the physical principle (conservation law) upon which each law is based.
4. Hard (Critical Thinking): Imagine you have a complex circuit with multiple batteries and resistors. Why might it be impossible to simplify this circuit using only the rules for series and parallel combinations? How do Kirchhoff’s laws provide a more approach to solving such circuits?
Activity 1: Match the Term with its Definition
- Current
- Voltage
- Resistance
- Series Circuit
- Parallel Circuit
Definitions:
- A) A measure of opposition to current flow.
- B) A circuit where components are connected end-to-end, providing a single path for current.
- C) The rate of flow of electric charge.
- D) A circuit where components are connected across the same two points, providing multiple paths for current.
- E) The electric potential difference between two points.
Activity 2: Fill in the Blanks
- Kirchhoff’s Current Law is based on the conservation of ______.
- In a series circuit, the ______ is the same through all components.
- In a parallel circuit, the ______ is the same across all components.
- The unit of resistance is the ______.
- Kirchhoff’s Voltage Law states that the sum of ______ around a closed loop is zero.
Watch this video to get a visual understanding of Kirchhoff’s Laws and how they are applied:
(If the video doesn’t load, you can watch it here: Kirchhoff’s Laws on YouTube by The Organic Chemistry Tutor — This is an example, you might find others more suitable for Cambridge AS/A Level specifically)
Another useful video on Series and Parallel Circuits:
(If the video doesn’t load, you can watch it here: Series and Parallel Circuits by The Organic Chemistry Tutor)
Here are some examples demonstrating how to solve circuit problems.
Problem 1: Series-Parallel Combination
Consider the circuit shown below. Find:
a) The total resistance of the circuit.
b) The total current supplied by the battery.
c) The current through the 6 Ω resistor.
(Imagine a 12V battery connected to a 4 Ω resistor, which is then connected to a parallel combination of a 6 Ω resistor and a 3 Ω resistor.)
[/su_spoiler] [/su_spoiler]Problem 2: Applying Kirchhoff’s Laws
For the circuit shown below, use Kirchhoff’s laws to find the currents I1, I2, and I3.
(Imagine a circuit with two loops. Loop 1: 10V battery, R1=2Ω, R2=3Ω. Loop 2: 5V battery, R2=3Ω, R3=4Ω. R2 is common to both loops. Assume I1 flows from 10V through R1, I2 flows through R2 downwards, I3 flows from 5V through R3. Junction A above R2, Junction B below R2.)
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Explore the PhET Interactive Simulation "Circuit Construction Kit: DC" to build and analyze circuits.
(If the simulation doesn't load, visit: PhET Circuit Construction Kit: DC)
Tasks:
- Build a simple series circuit with a battery and three resistors (e.g., 10Ω, 20Ω, 30Ω). Use the voltmeter to measure the voltage across each resistor and the ammeter to measure the current through them. Verify that RTotal = R1 + R2 + R3 and that current is the same through all.
- Build a simple parallel circuit with a battery and two resistors (e.g., 10Ω, 20Ω). Measure the voltage across each resistor and the current through each branch. Verify that 1/RTotal = 1/R1 + 1/R2 and that voltage is the same across both.
- Try to build the circuit from "Solved Example 1". Measure the currents and voltages and compare them to the calculated values.
Work with a partner or in a small group on the following activity:
Activity: Circuit Challenge on LearningApps.org
Explore circuit building or problem-solving activities on LearningApps.org. Here’s an example category (you may need to search for a specific relevant app):
Physics — Electricity on LearningApps.org
Instructions:
- Go to the link above or search for «series parallel circuits quiz» or «Kirchhoff’s laws quiz» on LearningApps.org or Quizizz.com.
- Choose an activity that involves calculating equivalent resistance, current, or voltage in series and parallel circuits, or applying Kirchhoff’s laws.
- Work together to solve the problems presented in the app. Discuss your approaches and reasoning.
- If you choose a quiz, try to achieve the highest score as a team.
Alternatively, use Quizizz: Series and Parallel Circuits on Quizizz
Answer the following questions. Show all your working.
Question 1: A 12.0 V battery is connected to three resistors (R1=4.0 Ω, R2=8.0 Ω, R3=12.0 Ω) connected in series.
a) Calculate the total resistance of the circuit.
b) Calculate the current flowing from the battery.
c) Calculate the potential difference across each resistor.
Question 2: The same three resistors (R1=4.0 Ω, R2=8.0 Ω, R3=12.0 Ω) are now connected in parallel to the 12.0 V battery.
a) Calculate the total resistance of the circuit.
b) Calculate the total current flowing from the battery.
c) Calculate the current flowing through each resistor.
Question 3 (Analysis): A circuit consists of a 24V battery. Resistor A (6Ω) and Resistor B (12Ω) are connected in parallel. This parallel combination is then connected in series with Resistor C (2Ω).
a) Draw the circuit diagram.
b) Calculate the total resistance of the circuit.
c) Calculate the current flowing through Resistor C.
d) Calculate the current flowing through Resistor A and Resistor B.
e) Calculate the power dissipated by Resistor C.
a) Circuit Diagram: [Student should draw a diagram showing a 24V battery, then Resistor C (2Ω) in series with a parallel branch containing Resistor A (6Ω) and Resistor B (12Ω)].
b) Resistance of parallel A & B (RAB):
1/RAB = 1/6 + 1/12 = 2/12 + 1/12 = 3/12 = 1/4. So, RAB = 4Ω.
Total Resistance RTotal = RC + RAB = 2Ω + 4Ω = 6Ω.
c) Current through Resistor C (IC) is the total current ITotal:
ITotal = Vbattery / RTotal = 24V / 6Ω = 4A. So, IC = 4A.
d) Voltage across the parallel combination (VAB):
VAB = ITotal * RAB = 4A * 4Ω = 16V.
Current through Resistor A (IA): IA = VAB / RA = 16V / 6Ω = 8/3 A ≈ 2.67A.
Current through Resistor B (IB): IB = VAB / RB = 16V / 12Ω = 4/3 A ≈ 1.33A.
(Check: IA + IB = 8/3 + 4/3 = 12/3 = 4A = ITotal)
e) Power dissipated by Resistor C (PC):
PC = IC2 * RC = (4A)2 * 2Ω = 16 * 2 = 32W.
(Alternatively, VC = ICRC = 4A * 2Ω = 8V. PC = VCIC = 8V * 4A = 32W)
Question 4 (Synthesis — Kirchhoff’s Laws):
Consider the circuit below. Use Kirchhoff’s laws to determine the magnitude and direction of the current in the 5 Ω resistor.
(Example setup: Left loop: 20V battery, 2Ω resistor. Right loop: 10V battery (opposing 20V if in series aiding), 3Ω resistor. Middle branch connecting the two loops: 5Ω resistor.)
Question 5 (Critical Analysis & Design): You are given three identical resistors, each with resistance R.
a) How many different values of total resistance can you obtain by connecting these three resistors in various combinations (using all three each time)?
b) Draw the circuit diagram for each combination and calculate its total resistance in terms of R.
c) If R = 10Ω, and you need a total resistance of 15Ω using these three resistors, which combination would you use? Explain if it’s possible.
For more detailed explanations, examples, and practice problems, check out these excellent resources:
- Save My Exams (Cambridge AS/A Level Physics): Save My Exams — CIE Physics (Navigate to the relevant DC Circuits / Kirchhoff’s Laws section)
- PhysicsAndMathsTutor.com:
- Notes and Questions: CIE A-Level Physics Revision (Look for DC Circuits or Electricity sections)
- Past Papers: Excellent for practice.
- OpenStax College Physics:
- Chapter on DC Circuits: Chapter 21: DC Circuits (or similar chapter in the non-AP version)
- YouTube Channels:
- The Organic Chemistry Tutor (Has excellent physics explanations, including circuits)
- Flipping Physics (A-Level focused physics videos)
- Science Shorts (Clear and concise physics explanations)
Take a few moments to reflect on your learning:
- What was the most concept for you in this lesson? What steps can you take to understand it better?
- Which part of the lesson did you find most interesting or helpful? Why?
- How confident do you feel about using Kirchhoff’s laws to solve circuit problems now? (Scale of 1-5, where 5 is very confident)
- Can you explain, in your own words, why the total resistance decreases when resistors are added in parallel?
- What is one question you still have about series circuits, parallel circuits, or Kirchhoff’s laws?