DC Circuits and Resistance

**1. Charge, Current, and Potential Difference — The Fundamentals Revisited**

Let's build a deeper understanding of electrical circuits beyond the basics.

Electric charge (Q) is measured in coulombs (C). Current is the rate of flow of charge:

`I = Q/t` → Current (A) = Charge (C) ÷ Time (s)

1 ampere = 1 coulomb of charge flowing per second.

Potential difference (V) is the energy transferred per unit charge:

`V = W/Q` → Voltage (V) = Energy (J) ÷ Charge (C)

When you plug in your phone charger in Karachi (220 V supply), each coulomb of charge transfers 220 joules of energy to the circuit.

Electromotive force (EMF) is the total energy supplied per unit charge by a power source (battery, generator). In a circuit, EMF = sum of all potential differences around the circuit (Kirchhoff's Second Law).

**2. Resistance in Detail**

Resistance (R) opposes current flow. Ohm's Law: `V = IR`

Ohmic conductors (like metal wires at constant temperature) have a linear V-I graph — resistance stays constant. Non-ohmic components have curved V-I graphs — resistance changes.

Factors affecting resistance of a wire:

1. Length (L): resistance ∝ length — longer wire = more resistance (like a longer pipe resists water flow more)
2. Cross-sectional area (A): resistance ∝ 1/area — thicker wire = less resistance
3. Material: different materials have different resistivities
4. Temperature: for metals, resistance increases with temperature (atoms vibrate more, obstruct electron flow)

Resistivity equation: `R = ρL/A` where ρ (rho) is the material's resistivity.

Practical investigation: Use a metre ruler, resistance wire, ammeter, voltmeter, and crocodile clips to investigate how length affects resistance. Plot R vs L — should be a straight line through the origin.

**3. Special Components**

Light-Dependent Resistor (LDR):

• Resistance decreases as light intensity increases.
• In darkness: very high resistance (~1 MΩ). In bright light: low resistance (~100 Ω).
• Used in: automatic streetlights across Karachi's DHA and Clifton.

Thermistor (NTC type):

• Resistance decreases as temperature increases.
• In cold: high resistance. In hot: low resistance.
• Used in: temperature sensors, fire alarms, air conditioning thermostats.

Diode:

• Allows current in one direction only (forward bias).
• In reverse bias, resistance is extremely high — effectively blocks current.
• LED (Light Emitting Diode): emits light when forward biased. Used in the LED bulbs that are replacing old incandescent bulbs across Pakistan to save energy.

**4. Series and Parallel — Advanced Analysis**

Series circuits:

• Same current through all components: `I = I₁ = I₂ = I₃`
• Voltages add up: `V_total = V₁ + V₂ + V₃`
• Total resistance: `R_total = R₁ + R₂ + R₃`

Parallel circuits:

• Same voltage across branches: `V = V₁ = V₂ = V₃`
• Currents add up: `I_total = I₁ + I₂ + I₃`
• Total resistance: `1/R_total = 1/R₁ + 1/R₂ + 1/R₃`

Potential divider circuit:

Two resistors in series can divide the supply voltage into smaller parts:

`V_out = V_in × R₂/(R₁ + R₂)`

This is extremely useful. Replace one resistor with an LDR or thermistor, and V_out changes with light or temperature — the basis for sensor circuits.

Example: An LDR (R₁) in series with a fixed resistor (R₂ = 10 kΩ) connected to 5 V. In darkness, LDR resistance is high → most voltage drops across LDR → V_out across R₂ is low. In bright light, LDR resistance drops → more voltage across R₂ → V_out increases. Connect V_out to a transistor switch and you've built an automatic light sensor.

**5. Electrical Safety and Power**

Power equations:

• `P = IV` (Power = Current × Voltage)
• `P = I²R` (useful when you know current and resistance)
• `P = V²/R` (useful when you know voltage and resistance)

Energy transferred: `E = Pt = IVt` measured in **joules (J)** or **kilowatt-hours (kWh)** for electricity bills.

1 kWh = 1000 W × 3600 s = 3,600,000 J

Pakistan's electricity is billed in kWh — a typical household in Lahore uses 300-500 kWh/month. `Cost = kWh × price per unit`.

Fuses and circuit breakers: protect against excessive current. A fuse melts (breaks circuit) when current exceeds its rating. Always choose a fuse slightly above the normal operating current.

Earthing: The earth wire connects metal casings to ground — if a live wire touches the casing, current flows to earth, blowing the fuse and preventing electrocution.

**Exam Strategy**

• For circuit calculations: identify series or parallel first, then apply the correct rules.
• V-I graphs: know the shapes for ohmic conductor (straight line), filament lamp (curve), and diode (forward/reverse).
• Potential divider: write the formula, identify which resistor changes, predict V_out direction.
• LDR and thermistor: remember — both DECREASE resistance when their stimulus INCREASES (light/temperature).
• Energy calculations: convert to correct units first (W not kW, seconds not minutes).
• Fuse rating question: calculate normal current (I = P/V), choose next fuse up (e.g., 3 A, 5 A, or 13 A).