Physics | Electric Circuits

Welcome to Electric Circuits!

Hello future Physicists! Electric Circuits might seem overwhelming with all the wires and components, but don't worry. This chapter is fundamental to understanding how almost all modern technology works—from your mobile phone to power grids. We are going to break down the flow of electricity step-by-step, using simple language and relatable examples.

Our goal is to understand the language of circuits: how charge moves, what makes it move, and what slows it down. Let’s get started!

Section 1: Charge, Current, and Potential Difference (PD)

1.1 Electric Charge (\(Q\))

Electricity fundamentally involves movement of charge. In metals (like wires), the charge carriers are electrons (which are negative).

• Definition: Electric Charge, \(Q\), is a fundamental property of matter.
• Unit: The unit of charge is the Coulomb (\(C\)).
• Did you know? One Coulomb is a huge amount of charge! It takes approximately \(6.24 \times 10^{18}\) electrons to make up just one Coulomb of charge.

1.2 Electric Current (\(I\))

Current is simply the rate of flow of electric charge. Think of it like the flow rate of water in a pipe.

The Current Equation

Current is defined as the amount of charge (\(\Delta Q\)) passing a specific point in a circuit per unit time (\(\Delta t\)).

\(I = \frac{\Delta Q}{\Delta t}\)

• Unit: The unit of current is the Ampere (A), where \(1 \text{ A} = 1 \text{ C s}^{-1}\) (one Coulomb per second).
• Measurement: Current is measured using an Ammeter, which must be connected in series (in the line of current flow).

Conventional vs. Electron Flow: A Common Confusing Point!

When circuits were first studied, scientists didn't know about electrons, so they assumed charge flowed from positive (+) to negative (-). This historical standard is called Conventional Current.

In reality, the electrons (which are negative) flow from the negative terminal to the positive terminal.

Key Rule: Unless specifically asked otherwise, always draw and discuss current using the standard Conventional Current (Positive to Negative).

1.3 Potential Difference (PD) or Voltage (\(V\))

For charge to flow, it needs energy—something to push it along. This "push" is the Potential Difference.

Analogy: The Water Pump

Imagine a circuit is a central heating system. The water is the charge.

• The pump provides the push (Potential Difference).
• The water flowing is the current.
• The radiator (light bulb) uses the energy.

PD is defined as the work done (energy transferred) per unit charge.

\(V = \frac{W}{Q}\)

• Unit: The unit of PD is the Volt (V), where \(1 \text{ V} = 1 \text{ J C}^{-1}\) (one Joule per Coulomb).
• If a component has a PD of 1 Volt across it, it means 1 Joule of energy is transferred to or from every 1 Coulomb of charge passing through it.
• Measurement: PD is measured using a Voltmeter, which must be connected in parallel across the component.

Section 2: Resistance and Ohm’s Law

2.1 Defining Resistance (\(R\))

Resistance is the measure of how much a component opposes the flow of electric current.

Analogy: Road Friction

If the current is traffic flow, resistance is the friction or obstacles (potholes, narrow lanes) that slow the cars down. High resistance means high opposition.

• Unit: The unit of resistance is the Ohm (\(\Omega\)).

2.2 Ohm’s Law

For many common components, there is a simple relationship between voltage and current. This relationship is called Ohm's Law, named after Georg Ohm.

Ohm’s Law states that for an ohmic conductor at a constant temperature, the current (\(I\)) is directly proportional to the potential difference (\(V\)) across it.

\(V \propto I\)

Introducing the constant of proportionality, Resistance (\(R\)):

\(V = IR\)

A Note on Ohmic vs. Non-Ohmic

• A component is Ohmic if its resistance \(R\) remains constant (usually meaning the temperature stays constant). A fixed resistor is usually ohmic.
• A component is Non-Ohmic if its resistance changes, often due to changes in temperature or applied voltage (e.g., a light bulb or diode).

Section 3: I-V Characteristics (Graphing Resistance)

The relationship between current (\(I\)) and potential difference (\(V\)) can be shown on a graph. By plotting \(I\) (y-axis) against \(V\) (x-axis), we can visually determine the resistance of a component, since \(R = V/I\), or \(1/R = I/V\).

3.1 Ohmic Resistor (Constant Resistance)

• Shape: A straight line passing through the origin.
• Observation: Since the graph is linear, the ratio \(V/I\) (the resistance) is constant.

3.2 Filament Lamp

A filament lamp is a non-ohmic conductor.

• Shape: The graph starts linearly but then curves off (gets shallower) as voltage increases.
• Why? As current flows through the thin metal filament, it gets hotter. Increased temperature causes the positive metal ions to vibrate more vigorously, leading to more frequent collisions with the moving electrons. These collisions increase the resistance.
• Common Mistake to Avoid: A shallower slope on an I-V graph means higher resistance.

3.3 Semiconductor Diode

A diode is a component designed to let current flow easily in one direction (the forward bias direction) and block it almost entirely in the reverse direction (the reverse bias direction).

• Forward Bias: Very little current flows until a certain threshold voltage (typically around 0.6 V for silicon). Once this voltage is reached, the resistance drops dramatically, and current increases exponentially.
• Reverse Bias: The resistance is extremely high, and practically zero current flows (unless the voltage is dangerously high, known as the breakdown voltage).

Section 4: Resistivity, Power, and Energy

4.1 Resistivity (\(\rho\))

While resistance (\(R\)) depends on the material, its length, and its cross-sectional area, resistivity (\(\rho\)) is a property only of the material itself (at a specific temperature).

We know that resistance is:

• Directly proportional to the length of the wire (\(L\)).
• Inversely proportional to the cross-sectional area of the wire (\(A\)).

The Resistivity Formula

This leads to the defining equation for resistivity:

\(R = \frac{\rho L}{A}\)

Rearranging to find resistivity:

\(\rho = \frac{RA}{L}\)

• Unit: The unit of resistivity is Ohm metre (\(\Omega \text{ m}\)).
• High resistivity materials (insulators) resist current strongly. Low resistivity materials (conductors) allow current to flow easily.

4.2 Electrical Power (\(P\))

Power is the rate at which energy is transferred or dissipated (wasted, usually as heat).

Power Equations

Since \(P = \text{Energy} / \text{Time}\) and \(V = \text{Energy} / \text{Charge}\) and \(I = \text{Charge} / \text{Time}\), the fundamental power equation is:

\(P = VI\)

Using Ohm’s Law (\(V = IR\)), we can derive two other useful forms:

\(P = I^2 R \quad \text{ (Substituting } V=IR \text{)}\)

\(P = \frac{V^2}{R} \quad \text{ (Substituting } I=V/R \text{)}\)

• Unit: The unit of power is the Watt (W).

4.3 Electrical Energy (\(E\))

Since power is the rate of energy transfer, the total energy transferred is:

\(E = P \times t\)

Substituting the power equations:

\(E = VIt\)

• Unit: The unit of energy is the Joule (J).

Section 5: Circuit Rules (Kirchhoff’s Laws and Combinations)

To analyze complex circuits, we rely on two conservation laws formulated by Gustav Kirchhoff.

5.1 Kirchhoff’s First Law (The Junction Rule)

This law is based on the conservation of charge. Charge cannot be created or destroyed.

The sum of currents entering a junction must equal the sum of currents leaving the junction.

\(\sum I_{\text{in}} = \sum I_{\text{out}}\)

Analogy: Imagine a river junction. The total amount of water flowing into the junction must equal the total amount flowing out.

5.2 Kirchhoff’s Second Law (The Loop Rule)

This law is based on the conservation of energy. Energy supplied by the cell must equal the total energy used up by the components.

In any closed loop in a circuit, the sum of the electromotive forces (EMFs) is equal to the sum of the potential differences (PDs) (voltage drops).

\(\sum \mathcal{E} = \sum V\)

5.3 Resistors in Series

In a series circuit, components are connected end-to-end, forming a single path.

• Current (\(I\)): The current is the same everywhere: \(I_{\text{total}} = I_1 = I_2 = I_3\)
• Potential Difference (\(V\)): The voltage is shared across the components: \(V_{\text{total}} = V_1 + V_2 + V_3\)
• Resistance (\(R\)): The total resistance is the sum of individual resistances (always increases total R):

  \(R_{\text{total}} = R_1 + R_2 + R_3\)

5.4 Resistors in Parallel

In a parallel circuit, components are connected across the same points, providing multiple paths for the current.

• Current (\(I\)): The current splits, following Kirchhoff's First Law: \(I_{\text{total}} = I_1 + I_2 + I_3\)
• Potential Difference (\(V\)): The voltage across each branch is the same: \(V_{\text{total}} = V_1 = V_2 = V_3\)
• Resistance (\(R\)): The total resistance is calculated using the reciprocal formula (always decreases total R):

  \(\frac{1}{R_{\text{total}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}\)

Section 6: EMF and Internal Resistance

When we talk about a battery or a cell, we often use the terms PD and EMF interchangeably, but they are physically different when current is being drawn.

6.1 Electromotive Force (EMF, \(\mathcal{E}\))

The EMF of a source (like a battery) is the total energy provided per unit charge passing through the source. It is the maximum potential difference the source can supply, measured when no current is flowing (an open circuit).

6.2 Internal Resistance (\(r\))

All real power sources (batteries, generators) have some internal opposition to the flow of charge due to the components and chemicals inside the cell itself. This is called Internal Resistance (\(r\)).

When a cell supplies current, some energy is inevitably wasted overcoming this internal resistance, usually as heat.

• The voltage wasted inside the cell is called the lost volts (\(v\)).
• Using Ohm's Law, the lost volts are calculated as: \(v = Ir\).

6.3 The EMF Equation

The voltage you measure across the terminals of a battery when it is supplying current (Terminal PD, \(V\)) is always less than the EMF because of the lost volts.

\(\text{EMF} = \text{Terminal PD} + \text{Lost Volts}\)

\(\mathcal{E} = V + Ir\)

Rearranging this gives the common working equation for terminal PD:

\(V = \mathcal{E} - Ir\)

This equation is critical. As the current \(I\) drawn from the battery increases, the term \(Ir\) increases, and the measured terminal PD (\(V\)) drops further below the EMF (\(\mathcal{E}\)).

You have covered the core physics of electric circuits! Don't worry if the algebra involving Kirchhoff's laws feels complicated—practice makes perfect. Remember the underlying conservation principles, and the calculations will follow. Keep up the great work!