Topic 3: Electric Circuits (Pearson Edexcel AS Physics 8PH0)
Welcome to your study notes for Topic 3: Electric Circuits! Electricity powers the modern world, from your smartphone to high-speed trains. In this chapter, we will break down fundamental concepts like charge, current, potential difference, and internal resistance into clear, bite-sized ideas. Don't worry if physics equations sometimes feel intimidating—we'll take each formula step-by-step with real-world analogies and clear exam tips.
1. Electric Current, Charge, and the Transport Equation
What is Electric Charge and Current?
Electric Charge (\(Q\)) is a fundamental property of matter, measured in Coulombs (\(\text{C}\)). When charge moves, it forms an electric current.
Electric Current (\(I\)) is defined as the rate of flow of charge. It is measured in Amperes (\(\text{A}\)).
\(I = \frac{\Delta Q}{\Delta t}\)
From this equation, we can see that \(1\text{ Coulomb} = 1\text{ Ampere-second}\) (\(1\text{ C} = 1\text{ As}\)).
The Transport Equation: What Happens Inside the Wire?
Inside a metal conductor, mobile charge carriers (electrons) drift through a lattice of vibrating positive ions. The current flowing through the wire depends on several physical properties of the conductor:
\(I = nqvA\)
Let's break down each term:
• \(I\) = Electric current in Amperes (\(\text{A}\))
• \(n\) = Number density of charge carriers (number of free electrons per unit volume, measured in \(\text{m}^{-3}\))
• \(q\) = Charge of each individual carrier (measured in \(\text{C}\))
• \(v\) = Drift velocity of the charge carriers (measured in \(\text{m s}^{-1}\))
• \(A\) = Cross-sectional area of the conductor (measured in \(\text{m}^2\))
Did You Know? The Reality of Drift Velocity
Did you know? Even though the electric signal travels through a circuit almost instantaneously, the actual physical electrons move incredibly slowly! The drift velocity (\(v\)) is typically a fraction of a millimeter per second (around \(10^{-4}\text{ m s}^{-1}\)). Think of a crowded hallway: when someone pushes at one end, the person at the far end feels it right away, even though individual people only shuffle forward slowly.
Key Takeaway for Section 1
Key Takeaway: Current is the rate of flow of charge (\(I = \frac{\Delta Q}{\Delta t}\)). Microscopically, current is described by \(I = nqvA\), where individual charge carriers drift surprisingly slowly through the conductor.
2. Potential Difference, Electromotive Force, and Resistance
Potential Difference (P.D.) vs. Electromotive Force (e.m.f.)
Students often mix these two terms up, but the distinction is crucial for your exams:
• Potential Difference (\(V\)): The energy transferred per unit charge from electrical energy into other forms (such as heat or light) when charge passes through a component.
• Electromotive Force (\(\varepsilon\)): The energy transferred by a source to each unit of charge (converting chemical, solar, or mechanical energy into electrical energy).
Both are measured in Volts (\(\text{V}\)), where \(1\text{ Volt} = 1\text{ Joule per Coulomb}\) (\(1\text{ V} = 1\text{ J C}^{-1}\)).
\(V = \frac{W}{Q}\)
Where \(W\) is energy transferred in Joules (\(\text{J}\)) and \(Q\) is charge in Coulombs (\(\text{C}\)).
Resistance and Ohm's Law
Resistance (\(R\)) is defined as the ratio of potential difference across a component to the current flowing through it:
\(R = \frac{V}{I}\)
Resistance is measured in Ohms (\(\Omega\)).
Ohm's Law: For an Ohmic conductor, current is directly proportional to potential difference (\(I \propto V\)), provided the temperature remains constant.
Electrical Power and Energy
The rate at which energy is transferred in an electrical circuit is power (\(P\)), measured in Watts (\(\text{W}\)). Using \(P = \frac{W}{t}\) and substituting our electrical definitions, we get three essential power formulas:
\(P = VI\)
\(P = I^2R\)
\(P = \frac{V^2}{R}\)
To find the total energy transferred (\(W\)) over a time interval \(t\):
\(W = VIt\)
Key Takeaway for Section 2
Key Takeaway: e.m.f. puts energy into the circuit (\(\text{chemical} \to \text{electrical}\)), while P.D. takes energy out (\(\text{electrical} \to \text{other forms}\)). Resistance is the ratio \(R = \frac{V}{I}\).
3. Component Characteristics and Non-Ohmic Behaviour
1. Ohmic Conductor (Fixed Resistor at Constant Temperature)
• Characteristic: The \(I\text{-}V\) graph is a straight line passing through the origin.
• Explanation: Resistance remains constant because the temperature does not change.
2. Filament Lamp
• Characteristic: The \(I\text{-}V\) graph curves, flattening out as \(V\) increases (the gradient decreases).
• Explanation: As current increases, the filament heats up. The metal ions in the filament vibrate with greater amplitude, colliding more frequently with the drifting electrons. This increases resistance, making the filament lamp a non-ohmic component.
3. Semiconductor Diode
• Characteristic: No current flows until a specific threshold voltage is reached in the forward direction. Once reached, current rises sharply with very low resistance. In the reverse direction, resistance is extremely high and negligible current flows.
4. Negative Temperature Coefficient (NTC) Thermistor
• Characteristic: As temperature increases, the resistance of an NTC thermistor decreases.
• Explanation: Thermal energy frees more charge carriers (increasing the number density \(n\) in \(I = nqvA\)), which significantly lowers the resistance.
Quick Review: Components Summary
• Fixed Resistor: Constant \(R\) (Straight line \(I\text{-}V\))
• Filament Lamp: Temperature up \(\implies R\) up (S-shaped curve)
• NTC Thermistor: Temperature up \(\implies R\) down
• Diode: Conducts only in one direction above threshold voltage
4. Resistivity and Core Practical 2
Understanding Resistivity
Resistance depends on the size and shape of a conductor (its length \(l\) and cross-sectional area \(A\)). Resistivity (\(\rho\)) is an intrinsic property of the material itself, regardless of its dimensions:
\(R = \frac{\rho l}{A}\)
Where:
• \(R\) = Resistance (\(\Omega\))
• \(\rho\) = Resistivity (\(\Omega\text{ m}\))
• \(l\) = Length of wire (\(\text{m}\))
• \(A\) = Cross-sectional area (\(\text{m}^2\))
Core Practical 2: Determining Electrical Resistivity
Goal: Measure the resistivity of a metal wire.
Method:
1. Measure the diameter of the wire at several different positions and orientations using a micrometer screw gauge. Calculate the mean diameter \(d\).
2. Calculate the cross-sectional area: \(A = \pi \left(\frac{d}{2}\right)^2 = \frac{\pi d^2}{4}\).
3. Connect the wire into a circuit with a power supply, ammeter, and voltmeter across a variable length \(l\) using crocodile clips.
4. Measure current \(I\) and potential difference \(V\) for different lengths \(l\) (e.g., \(0.20\text{ m}\) to \(1.00\text{ m}\)).
5. Calculate \(R = \frac{V}{I}\) for each length.
Graphical Analysis:
Rearranging \(R = \frac{\rho l}{A}\) into the straight-line equation form \(y = mx + c\):
\(R = \left(\frac{\rho}{A}\right)l + 0\)
• Plot a graph of \(R\) on the y-axis against \(l\) on the x-axis.
• The line should be straight and pass through the origin.
• \(\text{Gradient} = \frac{\rho}{A}\)
• Calculate resistivity: \(\rho = \text{Gradient} \times A\)
Common Pitfalls in Resistivity Calculations
• Diameter vs. Radius: Always remember to halve the diameter to find radius before calculating \(A = \pi r^2\).
• Unit Conversions: Micrometers read in millimeters (\(\text{mm}\)). Convert to meters before squaring! (\(1\text{ mm} = 1 \times 10^{-3}\text{ m}\), so \(1\text{ mm}^2 = 1 \times 10^{-6}\text{ m}^2\)).
5. Circuit Laws: Kirchhoff's Laws
Kirchhoff's First Law (Conservation of Charge)
Definition: The total current entering any junction is equal to the total current leaving that junction.
\(\sum I_{\text{in}} = \sum I_{\text{out}}\)
Physical Principle: This law is a direct consequence of the conservation of electric charge. Charge cannot simply appear or disappear at a junction.
Kirchhoff's Second Law (Conservation of Energy)
Definition: In any closed loop of a circuit, the sum of the electromotive forces (e.m.f.s) is equal to the sum of the potential differences.
\(\sum \varepsilon = \sum V\)
Physical Principle: This law is a direct consequence of the conservation of energy. The total electrical energy supplied by the source must equal the total energy transferred across the components in the loop.
Combining Resistors
Using Kirchhoff's laws, we derive the rules for combining resistors:
• Resistors in Series: The current is the same through each resistor, and the total P.D. is the sum of individual P.D.s.
\(R_{\text{total}} = R_1 + R_2 + R_3 + \dots\)
• Resistors in Parallel: The P.D. across each branch is identical, and the total current is the sum of currents in each branch.
\(\frac{1}{R_{\text{total}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots\)
6. Potential Dividers
What is a Potential Divider?
A potential divider is a simple circuit with two or more resistors in series that "divides" the input voltage into a smaller, useful output voltage.
The voltage across resistor \(R_2\) is given by the formula:
\(V_{\text{out}} = \left(\frac{R_2}{R_1 + R_2}\right) \times V_{\text{in}}\)
Analogy: Sharing a Pizza
Analogy: Think of \(V_{\text{in}}\) as a pizza being shared between two people. The person with the bigger appetite (larger resistance) takes a larger share of the slices (\(V_{\text{out}}\)). If \(R_2\) becomes much larger than \(R_1\), \(R_2\) takes almost all the voltage!
Sensor Circuits Using Potential Dividers
By replacing one of the fixed resistors with a sensor (like an NTC thermistor), we create circuits that automatically respond to the environment:
• Temperature Sensor: When temperature drops, the resistance of an NTC thermistor rises. If the thermistor is \(R_2\), its increased resistance causes \(V_{\text{out}}\) across it to increase, which can trigger a heating system.
7. Internal Resistance and Core Practical 3
What is Internal Resistance?
Real-world sources of e.m.f. (such as chemical cells or batteries) are not perfect. The chemicals and materials inside the cell offer resistance to the flow of charge. This is called internal resistance (\(r\)).
The Key Equations
When a current \(I\) flows through the cell, some energy is transferred into heat inside the cell itself. The voltage lost internally is called lost volts (\(Ir\)).
The voltage available to the external circuit is called the terminal potential difference (\(V\)).
\(\varepsilon = I(R + r)\)
Rearranging this gives:
\(V = \varepsilon - Ir\)
Where:
• \(\varepsilon\) = Electromotive force of the cell (\(\text{V}\))
• \(V\) = Terminal potential difference across external load (\(\text{V}\))
• \(I\) = Current flowing through circuit (\(\text{A}\))
• \(R\) = External load resistance (\(\Omega\))
• \(r\) = Internal resistance of the cell (\(\Omega\))
Note: When no current flows (\(I = 0\)), there are no lost volts (\(Ir = 0\)), so \(V = \varepsilon\). A high-resistance voltmeter placed across an open cell measures the e.m.f. directly.
Core Practical 3: Determining e.m.f. and Internal Resistance
Goal: Determine \(\varepsilon\) and \(r\) of an electrical cell.
Method:
1. Connect the cell in series with a switch, an ammeter, and a variable resistor.
2. Connect a voltmeter directly across the terminals of the cell to measure terminal P.D. (\(V\)).
3. Vary the resistance of the variable resistor to obtain several pairs of current (\(I\)) and terminal P.D. (\(V\)) readings.
4. Turn off the switch between readings to prevent the cell from running down and heating up (which would change \(r\)).
Graphical Analysis of \(V = \varepsilon - Ir\)
Rearranging the equation to match \(y = mx + c\):
\(V = -rI + \varepsilon\)
• y-axis: Terminal P.D. (\(V\))
• x-axis: Current (\(I\))
• y-intercept: Electromotive force (\(\varepsilon\))
• Gradient: Negative of internal resistance (\(\text{Gradient} = -r\))
Examiner Warning: Internal Resistance Graphs
• A common exam mistake is forgetting that the gradient is negative. Since \(\text{Gradient} = -r\), internal resistance is simply the positive magnitude of the gradient: \(r = -\text{gradient}\).
• Another common trap is not checking if the x-axis starts at zero. If the graph does not include \(I = 0\), you cannot read \(\varepsilon\) directly off the vertical axis—you must use the equation \(y = mx + c\)!
Chapter Summary & Exam Checklist
Before moving on, make sure you can confidently do the following:
1. Define current (\(I = \frac{\Delta Q}{\Delta t}\)), P.D. (\(V = \frac{W}{Q}\)), and e.m.f. with correct units.
2. Apply the transport equation (\(I = nqvA\)) and recall that drift velocity is very slow.
3. Sketch and explain \(I\text{-}V\) graphs for a fixed resistor, filament lamp, diode, and NTC thermistor.
4. Calculate resistivity using \(R = \frac{\rho l}{A}\) and explain Core Practical 2.
5. State Kirchhoff's 1st Law (conservation of charge) and 2nd Law (conservation of energy).
6. Solve series and parallel resistor networks and potential divider circuits (\(V_{\text{out}} = \frac{R_2}{R_1 + R_2} V_{\text{in}}\)).
7. Describe Core Practical 3 and determine \(\varepsilon\) (y-intercept) and \(r\) (\(-\text{gradient}\)) from a \(V\text{-}I\) graph.
8. Always give final numerical answers to a consistent number of significant figures (typically 2 or 3 s.f.).