Kirchhoff's laws, resistance combinations and potential dividers
AQA A-Level Physics 5
🔗State Kirchhoff's current and voltage laws
➕Calculate total resistance in series: R = R₁ + R₂
🔀Calculate total resistance in parallel: 1/R = 1/R₁ + 1/R₂
📐Use the potential divider formula to find output voltage
🌡️Describe the use of LDRs and thermistors in potential dividers
🔬Investigate resistors in series and parallel circuits
Kirchhoff's Laws
Kirchhoff's First Law (KCL): The sum of currents entering a junction equals the sum of currents leaving it. (Conservation of charge.)
Kirchhoff's Second Law (KVL): The sum of the EMFs in a closed loop equals the sum of the potential differences across the components in that loop. (Conservation of energy.)
KCL → charge is conserved: no charge builds up at junctions. KVL → energy is conserved: energy supplied by the source equals energy dissipated by resistors in any complete loop.
Series Circuits
Components connected end-to-end in a single path. The same current flows through all components.
Current: I is the same through all components
Voltage: V_total = V₁ + V₂ + V₃ + …
Resistance: R_total = R₁ + R₂ + R₃ + …
Voltage across each resistor: V_n = IR_n
Adding more resistors in series always increases the total resistance. If one component breaks (open circuit), the whole circuit stops working — a key disadvantage of series connections for practical use.
The voltage across each resistor in series is proportional to its resistance: V₁/V₂ = R₁/R₂ (since current is the same).
Parallel Circuits
Components connected across the same two points. They share the same potential difference but the current splits between them.
Voltage: V is the same across all branches
Current: I_total = I₁ + I₂ + I₃ + …
Resistance: 1/R_total = 1/R₁ + 1/R₂ + 1/R₃ + …
For two resistors in parallel:
R_total = (R₁ × R₂)/(R₁ + R₂)
Adding more resistors in parallel always decreases the total resistance (more current paths available). The total resistance is always less than the smallest individual resistor. If one component breaks, the others continue to work — mains circuits in homes use parallel wiring for this reason.
The formula 1/R = 1/R₁ + 1/R₂ gives the reciprocal of the total resistance. Remember to take the reciprocal at the end: R_total = 1/(1/R₁ + 1/R₂).
Potential Dividers
Potential divider: Two (or more) resistors in series across a supply voltage, used to produce a specific output voltage from a larger supply.
V_out = V_in × R₂/(R₁ + R₂)
R₁ is the top resistor (connected to +supply)
R₂ is the bottom resistor (output taken across R₂)
Variable resistor / LDR / Thermistor as R₁ or R₂:
LDR (Light Dependent Resistor): Resistance decreases as light intensity increases. In a potential divider: as light increases → R_LDR ↓ → if LDR is R₂, V_out decreases.
Thermistor (NTC): Resistance decreases as temperature increases. In a potential divider: as temperature rises → R_therm ↓ → if thermistor is R₂, V_out decreases.
By choosing which resistor is top or bottom in the divider, and whether to use an LDR or thermistor, you can design circuits that turn on when it gets dark, or that trigger an alarm when temperature rises.
Three resistors of 4 Ω, 6 Ω, and 12 Ω are connected in series to a 12 V supply. Find the total resistance and the voltage across each resistor.
In a potential divider, R₁ = 10 kΩ fixed and R₂ is an LDR. In darkness, the LDR has resistance 50 kΩ; in bright light, 500 Ω. The supply is 6 V. Find V_out in each condition.
4. In a parallel circuit, the supply current is 6 A. One branch carries 2 A and another 2.5 A. What current flows in the third branch?
By KCL: I₃ = I_total − I₁ − I₂ = 6 − 2 − 2.5 = 1.5 A.
5. A circuit has a 5 Ω resistor in series with two 8 Ω resistors in parallel. Connected to 12 V. Calculate the total current from the supply.
1. A circuit has three resistors: R₁ = 10 Ω in series with the parallel combination of R₂ = 15 Ω and R₃ = 30 Ω. The supply EMF is 18 V with negligible internal resistance. Find: (a) the total resistance, (b) the total current, (c) the current through R₂ and R₃.
(a) R₂₃ = (15 × 30)/(15 + 30) = 450/45 = 10 Ω. R_total = 10 + 10 = 20 Ω. (b) I_total = 18/20 = 0.90 A. (c) V across R₂₃ = 0.90 × 10 = 9 V. I₂ = 9/15 = 0.60 A. I₃ = 9/30 = 0.30 A. Check: 0.60 + 0.30 = 0.90 A ✓.
2. A thermistor with resistance 8 kΩ at 20°C and 1 kΩ at 80°C is used as R₁ in a potential divider with a fixed R₂ = 2 kΩ and a 5 V supply. Find V_out at each temperature and explain whether this circuit could be used as a temperature sensor to trigger a cooling fan when temperature exceeds a threshold.
At 20°C: V_out = 5 × 2/(8+2) = 5 × 0.2 = 1.0 V. At 80°C: V_out = 5 × 2/(1+2) = 5 × 0.667 = 3.33 V. As temperature increases, R_thermistor decreases → more voltage dropped across R₂ → V_out increases. This rising V_out could be connected to a comparator circuit: when V_out exceeds a reference voltage (e.g. 2.5 V), the comparator triggers a relay to switch on the cooling fan. This is a simple thermostat design. Alternatively, swap R₁ and R₂ to get a falling V_out with temperature.
3. Two identical cells, each of EMF 1.5 V and internal resistance 0.5 Ω, are connected in parallel to an external resistance of 3 Ω. Calculate the current from each cell and the terminal voltage. (Hint: two identical EMF sources in parallel have EMF ε but halved combined internal resistance.)
Two identical cells in parallel: combined EMF = 1.5 V (same as one cell). Combined internal resistance = 0.5/2 = 0.25 Ω (parallel combination). Total current I = ε/(r_combined + R) = 1.5/(0.25 + 3) = 1.5/3.25 = 0.462 A. Current from each cell = 0.462/2 = 0.231 A. Terminal voltage = IR = 0.462 × 3 = 1.385 V ≈ 1.39 V. (Alternatively: V_terminal = ε − I_cell × r = 1.5 − 0.231 × 0.5 = 1.5 − 0.115 = 1.385 V ✓.) Cells in parallel maintain the same voltage while sharing the current load.
Required Practical Investigating Resistors in Series and Parallel
Objective: Verify the rules for combining resistors in series and parallel by measuring current and voltage in circuits.
Equipment
Ammeter (in series with circuit) and voltmeter (across components)
Power supply (low voltage, e.g. 6 V)
Selection of resistors (e.g. 10 Ω, 22 Ω, 47 Ω)
Connecting leads and breadboard or circuit board
Method
Series: Connect R₁ and R₂ in series with the ammeter. Measure current I and voltage across each resistor. Calculate R_total = V_supply/I. Compare with R₁ + R₂.
Parallel: Connect R₁ and R₂ in parallel. Use ammeters in each branch and the main branch. Verify I_main = I₁ + I₂. Calculate R_total from V/I_main. Compare with (R₁R₂)/(R₁+R₂).
Repeat with different combinations and record results in a table.
Safety and Good Practice
Never connect an ammeter in parallel (it has very low resistance and will draw a large current, potentially damaging it). Always connect the voltmeter in parallel across the component you are measuring.
Keep supply voltage low (≤ 6 V). Avoid leaving the circuit switched on between readings to prevent resistors heating up, which changes their resistance.