Electrical Systems - Comprehensive Study Notes

Key Concepts

Ohm’s Law

Ohm’s Law states that the current flowing through a conductor is directly proportional to the voltage across it, provided temperature remains constant
Formula: V = I × R where:
- V = Voltage (in Volts, V)
- I = Current (in Amperes, A)
- R = Resistance (in Ohms, Ω)
If voltage increases, current increases (assuming resistance stays the same)
If resistance increases, current decreases (assuming voltage stays the same)
This relationship only applies to ohmic conductors (conductors that obey Ohm’s Law)

Series and Parallel Circuits

Series Circuits

Components are connected one after another in a single loop
Current is the same at all points in the circuit: I₁ = I₂ = I₃
Voltage is shared across components: V_total = V₁ + V₂ + V₃
Total resistance increases: R_total = R₁ + R₂ + R₃
If one component fails, the whole circuit stops working
Used in: Christmas lights (older types), some switches

Parallel Circuits

Components are connected across separate branches
Voltage is the same across all branches: V₁ = V₂ = V₃
Current is divided among branches: I_total = I₁ + I₂ + I₃
Total resistance decreases: 1/R_total = 1/R₁ + 1/R₂ + 1/R₃
If one component fails, others continue working
Used in: household electrical circuits, car lighting systems

Resistance — Factors Affecting

Four main factors affect resistance:

Length of conductor
- Longer wire = greater resistance
- Resistance is directly proportional to length
- R ∝ L (R increases as L increases)
Cross-sectional area (thickness)
- Thicker wire = lower resistance
- Resistance is inversely proportional to area
- R ∝ 1/A (R decreases as A increases)
Type of material
- Different materials have different resistivities
- Good conductors (copper, silver, aluminum) = low resistance
- Poor conductors/insulators (rubber, plastic, wood) = high resistance
Temperature
- For most conductors: higher temperature = higher resistance
- As temperature increases, metal ions vibrate more, making it harder for electrons to flow
- For some materials (thermistors), resistance decreases with temperature

Electrical Power and Energy

Electrical Power

Power is the rate of energy transfer or the rate of doing work
Measured in Watts (W) or kilowatts (kW)
1 kW = 1000 W
Formulas for calculating power:
- P = V × I (Power = Voltage × Current)
- P = I² × R (Power = Current² × Resistance)
- P = V²/R (Power = Voltage²/Resistance)

Electrical Energy

Energy is the total amount of electrical work done
Measured in Joules (J) or kilowatt-hours (kWh)
1 kWh = 3,600,000 J = 3.6 × 10⁶ J
Formula: E = P × t where:
- E = Energy (in J or kWh)
- P = Power (in W or kW)
- t = Time (in seconds s, or hours h)
Alternative formula: E = V × I × t

Cost of Electricity

Electricity bills are based on energy consumption in kWh
Cost = Energy used (kWh) × Cost per kWh
Example: If electricity costs $0.28 per kWh, and you use 500 kWh, cost = 500 × 0.28 = $140

Circuit Diagrams and Calculations

Standard Circuit Symbols

Must use correct symbols in circuit diagrams
Lines represent connecting wires (conductors)
Components are represented by standardized symbols

Important Definitions

Voltage (V): The electrical potential difference between two points; the energy transferred per unit charge. Measured in Volts (V).

Current (I): The rate of flow of electric charge through a conductor. Measured in Amperes (A).

Resistance ®: The opposition to the flow of electric current in a conductor. Measured in Ohms (Ω).

Ohm’s Law: The relationship stating that voltage is directly proportional to current in an ohmic conductor at constant temperature, expressed as V = I × R.

Series Circuit: A circuit in which components are connected end-to-end in a single path, so the same current flows through all components.

Parallel Circuit: A circuit in which components are connected across common points, providing multiple paths for current to flow.

Conductor: A material that allows electric current to flow through it easily (low resistance).

Insulator: A material that does not allow electric current to flow through it easily (high resistance).

Ohmic Conductor: A conductor that obeys Ohm’s Law, maintaining a constant resistance as voltage changes.

Power (P): The rate at which electrical energy is transferred or converted. Measured in Watts (W).

Energy (E): The total amount of electrical work done or the capacity to do work. Measured in Joules (J) or kilowatt-hours (kWh).

Ammeter: An instrument used to measure electric current, connected in series with the component.

Voltmeter: An instrument used to measure voltage (potential difference), connected in parallel across the component.

Diagrams and Structures

Standard Circuit Symbols

Cell:
- Draw two parallel lines, one longer than the other
- Longer line = positive terminal (+)
- Shorter line = negative terminal (-)
Battery:
- Two or more cells connected together
- Draw multiple pairs of long and short lines
Switch (open):
- Draw a gap in the circuit with a diagonal line not touching the other end
Switch (closed):
- Draw a straight line completing the circuit
Lamp/Bulb:
- Draw a circle with an X inside it
Resistor:
- Draw a rectangle (longer than it is wide)
Variable Resistor:
- Draw a rectangle with an arrow through it diagonally
Ammeter:
- Draw a circle with letter A inside
Voltmeter:
- Draw a circle with letter V inside
Connecting Wire:
- Draw straight lines (can have corners but no curves)

Series Circuit Diagram

Components arranged in single loop:
- Start with battery on left side
- Draw wire connecting to first component (e.g., lamp)
- Continue wire to second component (e.g., resistor)
- Return wire back to battery, completing the loop
- Ammeter placed anywhere in the series loop
- Voltmeter connected in parallel across any component you want to measure

Parallel Circuit Diagram

Components arranged in separate branches:
- Battery on left side
- Wire splits into two or more branches at a junction
- Each branch contains one or more components
- Branches reconnect at another junction
- Wire returns to battery
- Ammeter placed in main wire or in individual branches
- Voltmeter connected across the battery or across any component

V-I Graph for Ohmic Conductor

- X-axis: Current (I) in Amperes
- Y-axis: Voltage (V) in Volts
- Draw a straight line passing through origin
- Gradient of line = Resistance (R)
- Straight line indicates constant resistance (Ohm's Law obeyed)

V-I Graph for Non-Ohmic Conductor (e.g., Filament Lamp)

- X-axis: Current (I) in Amperes
- Y-axis: Voltage (V) in Volts
- Draw a curve that starts steep then gradually becomes less steep
- Curve indicates resistance increases as current increases
- This happens because filament heats up, increasing resistance

Worked Examples

Example 1: Ohm’s Law Calculation

Question: A lamp has a resistance of 12 Ω and is connected to a 6 V battery. Calculate the current flowing through the lamp.

Solution:

Step 1: Write down what you know
- Voltage, V = 6 V
- Resistance, R = 12 Ω
- Current, I = ?
Step 2: Write the formula
- V = I × R
Step 3: Rearrange to find I
- I = V/R
Step 4: Substitute values
- I = 6/12
- I = 0.5 A

Answer: The current flowing through the lamp is 0.5 A or 500 mA.

Example 2: Series Circuit Calculation

Question: Two resistors of 4 Ω and 6 Ω are connected in series with a 12 V battery. Calculate: a) Total resistance b) Current in the circuit c) Voltage across each resistor

Solution:

Part (a): Total Resistance

Step 1: Use series formula
- R_total = R₁ + R₂
Step 2: Substitute values
- R_total = 4 + 6 = 10 Ω

Part (b): Current

Step 1: Use Ohm’s Law for whole circuit
- V = I × R_total
Step 2: Rearrange
- I = V/R_total
Step 3: Substitute
- I = 12/10 = 1.2 A
Note: This current is the same through both resistors (series circuit property)

Part ©: Voltage across each resistor

For 4 Ω resistor:

V₁ = I × R₁
V₁ = 1.2 × 4 = 4.8 V

For 6 Ω resistor:

V₂ = I × R₂
V₂ = 1.2 × 6 = 7.2 V

Check: V₁ + V₂ = 4.8 + 7.2 = 12 V ✓ (equals battery voltage)

Answers:

a) Total resistance = 10 Ω
b) Current = 1.2 A
c) Voltage across 4 Ω = 4.8 V; Voltage across 6 Ω = 7.2 V

Example 3: Electrical Power and Energy Calculation

Question: A 2000 W kettle is connected to a 240 V supply and used for 5 minutes to boil water. Calculate: a) The current flowing through the kettle b) The energy consumed in kWh c) The cost of electricity if 1 kWh costs $0.28

Solution:

Part (a): Current

Step 1: Use power formula
- P = V × I
Step 2: Rearrange
- I = P/V
Step 3: Substitute
- I = 2000/240
- I = 8.33 A (or 8.3 A to 2 s.f.)

Part (b): Energy in kWh

Step 1: Convert power to kW
- P = 2000 W = 2 kW
Step 2: Convert time to hours
- t = 5 minutes = 5/60 hours = 0.0833 h
Step 3: Calculate energy
- E = P × t
- E = 2 × 0.0833
- E = 0.167 kWh (or 0.17 kWh to 2 s.f.)

Part ©: Cost

Cost = Energy × Cost per kWh
Cost = 0.167 × $0.28
Cost = $0.047 or approximately $0.05 (5 cents)

Answers:

a) Current = 8.3 A
b) Energy = 0.17 kWh
c) Cost = $0.05

Common Mistakes to Avoid

Calculation Errors

Forgetting to rearrange formulas correctly: When finding I from V = I × R, students often multiply instead of dividing
Using wrong units: Mixing W and kW, or seconds and hours without converting
Not converting minutes to hours: When calculating energy in kWh, time must be in hours (divide minutes by 60)
Rounding too early: Keep at least 3 significant figures during calculations, only round the final answer

Series vs Parallel Confusion

Mixing up rules: Saying current is same in parallel or voltage is same in series
Wrong resistance formulas: Using R_total = R₁ + R₂ for parallel circuits
Forgetting about current split in parallel: Not recognizing that current divides at junctions in parallel circuits

Circuit Diagram Mistakes

Incorrect symbols: Drawing a battery as a single cell, or using non-standard symbols
Wrong meter connections:
- Connecting ammeter in parallel (it should be in series)
- Connecting voltmeter in series (it should be in parallel)
Missing labels: Not labeling values or directions

Ohm’s Law Application

Applying to non-ohmic conductors: Assuming filament lamps follow Ohm’s Law strictly (they don’t because they heat up)
Forgetting constant temperature condition: Not stating that Ohm’s Law requires constant temperature

Factors Affecting Resistance

Confusing length and thickness: Thinking thicker wires have higher resistance (opposite is true)
Not explaining temperature effect properly: Just saying “temperature affects resistance” without explaining the mechanism

Power and Energy

Confusing power and energy: Using them interchangeably or mixing up their units
Wrong formula selection: Using P = V × I when you don’t have V and I, but could use P = I² × R instead

Exam Tips

For Calculations

Always write the formula first before substituting numbers (shows your working even if answer is wrong)
Include units in every step and in final answer (can lose marks for missing units)
Show clear working: Write out: formula → rearrangement → substitution → answer
Check your answer makes sense: A household current of 1000 A is clearly wrong!

For Circuit Diagrams

Use a ruler for straight lines (shows care and gets you marks)
Draw symbols at appropriate size (not too small or too large)
When connecting meters:
- State “ammeter in series” to measure current
- State “voltmeter in parallel” to measure voltage
Label all values clearly including units

Keywords and Mark-Earning Phrases

For Ohm’s Law questions:

“Voltage is directly proportional to current”
“At constant temperature”
“For an ohmic conductor”

For series circuits:

“Current is the same throughout the circuit”
“Voltage is shared across components”
“Total resistance equals sum of individual resistances”

For parallel circuits:

“Voltage is the same across all branches”
“Current is divided among branches”
“Each component has its own complete path”

For resistance factors:

“Resistance is directly proportional to length”
“Resistance is inversely proportional to cross-sectional area”
“As temperature increases, metal ions vibrate more, impeding electron flow”

For power and energy:

“Power is the rate of energy transfer”
“Energy is power multiplied by time”
When calculating cost: “Energy consumed × cost per unit”

Describing Graphs

Always mention: “As [x-variable] increases, [y-variable] increases/decreases”
For straight-line V-I graphs: “The graph is a straight line through the origin, showing Ohm’s Law is obeyed”
For curved graphs: “The gradient decreases, showing resistance increases”

Practical Skills (if applicable)

“To measure current, connect ammeter in series”
“To measure voltage, connect voltmeter in parallel”
“Ensure all connections are tight to avoid high resistance”
“Switch off circuit when not taking readings to prevent overheating”

Quick Summary

✓ Ohm’s Law: V = I × R; voltage is directly proportional to current at constant temperature for ohmic conductors

✓ Series circuits: Same current throughout (I₁ = I₂); voltage shared (V_total = V₁ + V₂); resistance adds (R_total = R₁ + R₂)

✓ Parallel circuits: Same voltage across branches (V₁ = V₂); current divides (I_total = I₁ + I₂); resistance decreases (1/R_total = 1/R₁ + 1/R₂)

✓ Factors affecting resistance: Increases with length and temperature; decreases with greater cross-sectional area; depends on material

✓ Power formulas: P = V × I; P = I² × R; P = V²/R (all measured in Watts)

✓ Energy formula: E = P × t (measured in Joules or kWh, where 1 kWh = 3.6 × 10⁶ J)

✓ Cost of electricity: Cost = Energy used (kWh) × Cost per kWh

✓ Circuit symbols: Know all standard symbols (cell, battery, lamp, resistor, ammeter, voltmeter, switch)

✓ Ammeter: Connected in series to measure current flowing through component

✓ Voltmeter: Connected in parallel to measure voltage across component

✓ V-I graphs: Straight line through origin = ohmic conductor; curved line = non-ohmic conductor (resistance changes)

✓ Always show working: Write formula → rearrange → substitute → answer with units for maximum marks