• Recall and use the formula for the combined resistance of two or more resistors in series (R_T = R₁ + R₂ + …).
• State Kirchhoff’s first law (conservation of charge) and Kirchhoff’s second law (conservation of energy).
• Derive, using Kirchhoff’s laws, a formula for the combined resistance of two or more resistors in parallel (1/R_T = 1/R₁ + 1/R₂ + …).
• Recall and use the formula for the combined resistance of two or more resistors in parallel.
• Apply Kirchhoff’s laws to solve simple circuit problems, including those with multiple loops and sources.
• Understand the concept of a potential divider and solve problems involving potential dividers.

• Define and use key vocabulary related to series and parallel circuits, resistance, and Kirchhoff’s laws (e.g., equivalent resistance, junction, loop, potential difference, current, conservation).
• Explain the derivation of formulas for combined resistance using appropriate scientific language and reference to Kirchhoff’s laws.
• Describe how to apply Kirchhoff’s laws to analyze complex circuits.
• Discuss the principles of potential dividers.

Here are some important terms for this lesson. Pay attention to their meanings and translations.

Review the key terms related to resistors in series/parallel and Kirchhoff’s laws using flashcards. You can create your own set on Quizlet or use a pre-existing one focusing on A-Level Physics DC circuits.

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Resistors in Series: Components connected one after another, forming a single path for the current. The total resistance is the sum of individual resistances.

Resistors in Parallel: Components connected across the same two points in a circuit, providing multiple paths for the current. The reciprocal of the total resistance is the sum of the reciprocals of individual resistances.

Kirchhoff’s First Law (Junction Rule): The sum of currents entering any junction in a circuit is equal to the sum of currents leaving that junction. This is a statement of conservation of charge. (ΣI_in = ΣI_out)

Kirchhoff’s Second Law (Loop Rule): In any closed loop in a circuit, the algebraic sum of the electromotive forces (e.m.f.s) is equal to the algebraic sum of the potential differences (p.d.s) across the components. This is a statement of conservation of energy. (Σε = ΣIR)

When analyzing electrical circuitsэлектр тізбектері / электрические цепи, we often need to simplify combinations of resistorsрезисторлар / резисторы and solve for unknown currents and voltages. This involves understanding how resistors combine in series and parallel, and applying Kirchhoff’s Laws.

Resistors in Series:

When resistors R₁, R₂, R₃, …, are connected in seriesтізбектей / последовательно, they form a single path for the current (I).

[Image of Resistors in Series Diagram: R1, R2, R3 connected end-to-end]

• The currentток / ток (I) is the same through each resistor: I = I₁ = I₂ = I₃ = …
• The total potential differenceпотенциалдар айырымы / разность потенциалов (V_T) across the combination is the sum of the potential differences across each resistor: V_T = V₁ + V₂ + V₃ + …

Using Ohm’s Law (V=IR) for each resistor and the total combination:

IR_T = IR₁ + IR₂ + IR₃ + …

Dividing by I (since it’s constant):

R_T = R₁ + R₂ + R₃ + …

This is the formula for the equivalent resistanceэквиваленттік кедергі / эквивалентное сопротивление of resistors in series. The total resistance is always greater than the largest individual resistance.

Resistors in Parallel:

When resistors R₁, R₂, R₃, …, are connected in parallelпараллель / параллельно, they are connected across the same two points, providing multiple paths for the current.

[Image of Resistors in Parallel Diagram: R1, R2, R3 connected across the same two points]

• The potential difference (V) is the same across each resistor: V = V₁ = V₂ = V₃ = …
• The total current (I_T) from the source is the sum of the currents in each parallel branch (Kirchhoff’s First Law): I_T = I₁ + I₂ + I₃ + …

Using Ohm’s Law (I=V/R) for each resistor and the total combination:

V/R_T = V/R₁ + V/R₂ + V/R₃ + …

Dividing by V (since it’s constant):

1/R_T = 1/R₁ + 1/R₂ + 1/R₃ + …

This is the formula for the equivalent resistance of resistors in parallel. The total resistance is always less than the smallest individual resistance.

Kirchhoff’s Laws:

For more complex circuits that cannot be simplified to simple series/parallel combinations, we use Kirchhoff’s Laws.

1. Kirchhoff’s First Law (Junction Rule or Current Law — KCL):

Based on the conservation of chargeзарядтың сақталуы / сохранение заряда.

It states that the algebraic sum of currents entering any junctionтүйін / узел (or node) in an electrical circuit is zero. Or, more simply:

The total current flowing into a junction equals the total current flowing out of that junction.

ΣI_in = ΣI_out

[Image of a Junction with currents I1, I2 entering and I3, I4 leaving]

2. Kirchhoff’s Second Law (Loop Rule or Voltage Law — KVL):

Based on the conservation of energyэнергияның сақталуы / сохранение энергии.

It states that in any closed loopконтур / контур within an electrical circuit, the algebraic sum of the electromotive forces (e.m.f.s, ε) is equal to the algebraic sum of the potential differences (p.d.s, IR drops) across the components in that loop.

Σε = Σ(IR)

When applying KVL:

• Choose a direction (clockwise or counter-clockwise) around the loop.
• An e.m.f. is positive if you traverse it from the negative to the positive terminal, and negative if traversed from positive to negative.
• An IR drop is positive if you traverse the resistor in the same direction as the assumed current, and negative if against the assumed current direction.

[Image of a simple loop with a cell and resistors, showing a chosen loop direction]

Derivation of Parallel Resistance Formula using Kirchhoff’s Laws:
Consider two resistors R₁ and R₂ in parallel, connected to a voltage source V.

Current I_T leaves the source. At the junction, it splits into I₁ (through R₁) and I₂ (through R₂).

By KCL: I_T = I₁ + I₂ (Equation 1)

The voltage across R₁ is V, and across R₂ is V (they are in parallel).

So, I₁ = V/R₁ and I₂ = V/R₂.

Let R_T be the equivalent resistance of the parallel combination. Then I_T = V/R_T.

Substitute these into Equation 1:

V/R_T = V/R₁ + V/R₂

Divide by V (assuming V ≠ 0):

1/R_T = 1/R₁ + 1/R₂. This can be extended for more resistors.

Questions on Theory:

1. (Easy) Three resistors of 2Ω, 3Ω, and 6Ω are connected in series. What is their total resistance?
   
   Answer

   For resistors in series, R_T = R₁ + R₂ + R₃.
   
   R_T = 2Ω + 3Ω + 6Ω = 11Ω.
   
2. (Medium) State Kirchhoff’s First Law and explain which fundamental conservation principle it relates to.
   
   Answer

   Kirchhoff’s First Law (Junction Rule) states that the sum of currents entering any junction in a circuit is equal to the sum of currents leaving that junction (ΣI_in = ΣI_out). It relates to the principle of conservation of charge, as charge cannot accumulate at a junction; the rate at which charge enters must equal the rate at which it leaves.
3. (Medium) Two resistors, 10Ω and 20Ω, are connected in parallel to a 6V battery. Calculate the total current supplied by the battery.
   
   Answer

   First, find the total resistance (R_T) of the parallel combination:
   
   1/R_T = 1/R₁ + 1/R₂ = 1/10Ω + 1/20Ω
   
   1/R_T = 2/20Ω + 1/20Ω = 3/20Ω
   
   R_T = 20/3 Ω ≈ 6.67Ω.
   
   Now, use Ohm’s Law to find the total current (I_T) from the battery:
   
   I_T = V / R_T = 6V / (20/3 Ω) = 6V * (3/20Ω) = 18/20 A = 0.9A.
   
4. (Hard — Critical Thinking) Consider a circuit loop containing a 12V cell, an internal resistance of 1Ω for the cell, and an external resistor of 5Ω. Use Kirchhoff’s Second Law to write an equation for this loop and then calculate the current flowing in the circuit.
   
   Answer

   Let the current flowing in the circuit be I.
   
   Choose a loop direction (e.g., clockwise, starting from the negative terminal of the cell and going through it).
   
   According to Kirchhoff’s Second Law (Σε = ΣIR):
   
   The e.m.f. (ε) of the cell is +12V (traversing from — to +).
   
   The potential drop across the internal resistance (r=1Ω) is Ir = I * 1Ω. This is a ‘drop’ so it’s on the ΣIR side.
   
   The potential drop across the external resistor (R=5Ω) is IR = I * 5Ω. This is also on the ΣIR side.
   
   So, the equation is: 12V = I(1Ω) + I(5Ω)
   
   12V = I(1Ω + 5Ω)
   
   12V = I(6Ω)
   
   Current I = 12V / 6Ω = 2A.
   

Exercise 1: Match the Law/Concept with its Description

1. Resistors in Series Total Resistance
2. Kirchhoff’s First Law
3. Resistors in Parallel Total Resistance
4. Kirchhoff’s Second Law

A. ΣI_in = ΣI_out at a junction.

B. 1/R_T = 1/R₁ + 1/R₂ + …

C. Σε = Σ(IR) in a closed loop.

D. R_T = R₁ + R₂ + …

Exercise 2: Fill in the Blanks

1. For resistors connected in parallel, the __________ across each resistor is the same.
2. Kirchhoff’s Loop Rule is a consequence of the conservation of __________.
3. The total resistance of a series combination is always __________ than the largest individual resistance.
4. At any junction in a circuit, the sum of currents __________ the junction equals the sum of currents __________ the junction.
5. The formula for combined resistance of three resistors R1, R2, and R3 in parallel is 1/R_T = __________.