Electricity & DC Circuits

Introduction to Electricity and DC Circuits

This topic forms the foundation of understanding how electrical circuits function. We explore the fundamental quantities of charge, current, potential difference, and resistance, and then use powerful laws to analyse circuits, from simple series-parallel combinations to more complex networks.

1. Fundamental Concepts

Electric Charge (Q): The fundamental property of matter responsible for electric phenomena. The elementary charge, *e*, is 1.60 × 10⁻¹⁹ C. In metals, the charge carriers are free electrons.

Electric Current (I):

• Definition: **Electric current** is defined as the rate of flow of electric charge through a point.
• Formula: I = ΔQ / Δt
• Unit: The **ampere (A)**, which is one coulomb per second (C s⁻¹).
• Conventional Current vs. Electron Flow: A crucial distinction and a common source of confusion. **Conventional current** is the direction that positive charge would flow, from a point of higher potential to lower potential (i.e., from the positive terminal to the negative terminal). In circuit diagrams and analysis, we *always* use conventional current. In reality, in metallic conductors, the charge carriers are electrons, which flow in the opposite direction (from negative to positive). This convention dates back to before the discovery of the electron.

Potential Difference (V) and Electromotive Force (ε):

Both are measures of energy per unit charge, but they describe different processes.

• Potential Difference (p.d.): The **potential difference** between two points is the energy converted from electrical potential energy to other forms (e.g., heat, light) when one unit of charge passes between those points.
• Formula: V = W / Q
• Unit: The **volt (V)**, which is one joule per coulomb (J C⁻¹).
• Electromotive Force (EMF): The **electromotive force** of a source is the energy converted from other forms (e.g., chemical, mechanical) into electrical potential energy to drive one unit of charge around a complete circuit.
• Formula: ε = W / Q
• Key Idea: EMF is the energy *supplied* by the source (a battery, a generator), while p.d. is the energy *used* by a component.

2. Resistance and Ohm's Law

Resistance (R):

• Definition: **Resistance** is the ratio of the potential difference across a component to the current flowing through it.
• Formula: R = V / I
• Unit: The **ohm (Ω)**.
• Ohm's Law: States that for a metallic conductor at constant temperature, the current is directly proportional to the potential difference across it. Such conductors are called **ohmic conductors**.

I-V Characteristics:

A graph of current (I) against potential difference (V) reveals a component's behaviour.

1. Ohmic Conductor (e.g., a resistor at constant temp): A straight line through the origin. The gradient is 1/R.
2. Filament Lamp: An S-shaped curve passing through the origin. As voltage and current increase, the filament gets hotter, its resistance increases, and the gradient (ΔI/ΔV) decreases.
3. Semiconductor Diode: Allows current to flow easily in one direction (forward bias) but not the other (reverse bias). The graph is essentially flat on the negative V-axis and then rises sharply for positive V after a threshold voltage (~0.6V for silicon).

Resistivity (ρ):

Resistance depends on a material's intrinsic properties and its dimensions.

• Definition: **Resistivity** is a measure of a material's ability to oppose the flow of electric current. It is the resistance of a 1m cube of the material.
• Formula: R = ρL / A
• ρ (rho) = resistivity (Ω m)
• L = length of the conductor (m)
• A = cross-sectional area (m²)
• Application: Electrical wiring in homes in cities like Karachi or Lahore uses copper (low ρ). High-voltage transmission lines across the country use thick aluminium cables (low ρ and lighter than copper) to minimise power loss (P = I²R) over long distances.

3. Circuit Analysis

Kirchhoff's Laws: These are two fundamental laws that allow us to analyse any DC circuit.

1. Kirchhoff's First Law (The Junction Rule): Based on the conservation of charge. It 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).

* Exam Trap: Be careful with the directions of currents. If you assume a direction and your calculated value is negative, it simply means the current flows in the opposite direction.

1. Kirchhoff's Second Law (The Loop Rule): Based on the conservation of energy. It states that around any closed loop in a circuit, the algebraic sum of the EMFs is equal to the algebraic sum of the potential drops (Σε = ΣIR).

* Step-by-step application:

1. Draw the circuit and label all knowns and unknowns.
2. Assign a current direction for each branch (if unknown).
3. Choose a closed loop and a starting point.
4. Traverse the loop. If you cross a source from - to +, it's a positive EMF. If you cross a resistor in the same direction as the current, it's a potential drop (+IR). If you cross against the current, it's a potential gain (-IR).
5. Set the sum of EMFs equal to the sum of p.d.s.
6. Repeat for other loops until you have enough independent equations to solve for your unknowns.

Internal Resistance (r):

Real power sources are not perfect. They have their own internal resistance due to the materials they are made from.

• Concept: When a current (I) is drawn from a source with EMF (ε) and internal resistance (r), some energy is dissipated as heat within the source itself. This results in a p.d. across the internal resistance, often called **'lost volts'**.
• Formula: The total EMF is shared between the external circuit (load resistance R) and the internal resistance (r).

ε = IR + Ir or ε = I(R + r)

• Terminal Potential Difference (V): This is the actual p.d. measured across the terminals of the source when it is in a circuit.

V = IR, so by substitution, V = ε - Ir.

• Practical Implication: The terminal p.d. of a battery is only equal to its EMF when no current is being drawn (I=0). As you draw more current, the terminal p.d. drops. This is why car headlights dim slightly when you start the engine, or lights in a Pakistani home flicker when a heavy load like an AC or water pump is switched on while running on a UPS.

4. The Potential Divider

A potential divider circuit uses two or more resistors in series to provide a fraction of the source voltage.

• Circuit: A source V_in is connected across two series resistors, R₁ and R₂. The output voltage, V_out, is taken across R₂.
• Derivation & Formula:

1. Total resistance, R_total = R₁ + R₂.
2. Current in the circuit, I = V_in / R_total = V_in / (R₁ + R₂).
3. Voltage across R₂, V_out = I × R₂.
4. Substituting I: V_out = (R₂ / (R₁ + R₂)) × V_in

• Applications: This circuit is vital for sensor applications. If R₂ is a variable resistor, like a **Thermistor** (resistance changes with temperature) or a **Light-Dependent Resistor (LDR)** (resistance changes with light intensity), then V_out becomes a signal that varies with physical conditions. This is the basis for automatic streetlights, thermostats, and many other electronic control systems.