Physics (9702) | Electricity

Welcome to the World of Electricity!

Hello future Physicists! This chapter, Electricity and D.C. Circuits, is one of the most fundamental and useful topics in AS Level Physics. Don't worry if complex circuits look daunting—we will break down the flow of charge, energy, and resistance into simple, manageable steps.
Understanding these principles allows you to comprehend everything from your phone battery to large-scale power grids. Let's get started!

1. Electric Current and Charge Flow (Syllabus 9.1)

1.1 What is Electric Current?

The definition of electric current is straightforward: it is the rate of flow of charge carriers. These carriers are usually electrons in metals.

• Charge Quantisation: The charge carried by charge carriers (like electrons) is quantised. This means charge only comes in discrete packets (multiples of the elementary charge, \(e\)).
• Definition of Current (\(I\)): Current is the charge (\(Q\)) passing a point per unit time (\(t\)).
  $$Q = I t$$
• Unit: The SI unit of charge is the Coulomb (C). The SI unit of current is the Ampere (A). \(1 \text{ A} = 1 \text{ C s}^{-1}\).

1.2 The Microscopic View: Drift Velocity

In a conductor, charge carriers (electrons) do not move in a straight line; they bounce randomly off lattice ions. When a voltage is applied, they gain a very slow overall movement in one direction, known as drift velocity (\(v\)).

The relationship between current (\(I\)) and the microscopic properties of the conductor is given by:

$$I = A n v q$$

• \(A\): Cross-sectional area of the conductor (\(m^2\))
• \(n\): Number density of charge carriers (\(m^{-3}\)). This is the number of mobile charge carriers per unit volume.
• \(v\): Average drift velocity (\(m s^{-1}\))
• \(q\): Charge on each carrier (C) (usually \(e = 1.60 \times 10^{-19} \text{ C}\) for electrons)

Did you know? The actual drift velocity of electrons in a household wire is often less than 1 mm per second! The signal (the electric field) travels at nearly the speed of light, but the individual electron movement is incredibly slow.

2. Potential Difference, Electromotive Force (e.m.f.), and Power (Syllabus 9.2 & 10.1)

2.1 Defining Potential Difference (p.d.)

Potential Difference (p.d.) (\(V\)) is defined as the energy transferred (work done) per unit charge when charge moves between two points in a circuit. It measures how much energy is converted from electrical energy into other forms (like heat or light) by a component.

$$V = \frac{W}{Q}$$

• Unit: The SI unit is the Volt (V). \(1 \text{ V} = 1 \text{ J C}^{-1}\).

2.2 Distinguishing e.m.f. and p.d.

This is a common exam area! Both are measured in Volts, but they represent different things in terms of energy flow:

• Electromotive Force (e.m.f.), \(E\): The total energy transferred per unit charge by a source (like a battery or generator) in driving charge around a complete circuit. This is the energy converted from chemical (or other) energy to electrical energy.
• Potential Difference (p.d.), \(V\): The energy transferred per unit charge when current passes through a load component (like a resistor). This is energy converted from electrical energy to heat, light, etc.

Analogy: Think of electricity as water in a pipe system. The e.m.f. is the powerful pump that raises the water (supplies total energy). The p.d. is the drop in pressure across a section of the pipe (energy used by a component).

2.3 Electrical Power

Power (\(P\)) is defined as the rate of energy transfer, or work done per unit time: \(P = W/t\).

Since \(W = VQ\) and \(Q = It\), we can substitute these into the power equation:

1. Basic Power Formula: $$P = V I$$

Using Ohm's Law (\(V = IR\)), we derive two more essential forms:

2. In terms of I and R: (Substitute \(V = IR\)) $$P = I^2 R$$
3. In terms of V and R: (Substitute \(I = V/R\)) $$P = \frac{V^2}{R}$$

Key Takeaway: e.m.f. is the total energy supplied by the source; p.d. is the energy used up by a component. Power measures the rate at which this energy transfer occurs.

3. Resistance, Resistivity, and Ohm's Law (Syllabus 9.3)

3.1 Defining Resistance and Ohm's Law

Resistance (\(R\)) is the opposition a component offers to the flow of electric current.

$$R = \frac{V}{I}$$

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

Ohm’s Law:

Ohm's Law states that for a metallic conductor kept at a constant temperature, the current (\(I\)) passing through it is directly proportional to the potential difference (\(V\)) across it.

Only components that follow this rule are called Ohmic conductors (e.g., a resistor at constant temperature).

3.2 Resistivity (\(\rho\))

While resistance depends on the shape and size of a wire, resistivity (\(\rho\)) is a property of the material itself.

The resistance \(R\) of a wire is directly proportional to its length \(L\) and inversely proportional to its cross-sectional area \(A\):

$$R = \frac{\rho L}{A}$$

• \(R\): Resistance (\(\Omega\))
• \(L\): Length (\(m\))
• \(A\): Cross-sectional area (\(m^2\))
• \(\rho\): Resistivity (\(\Omega \text{ m}\)). A low resistivity means the material is a good conductor.

Memory Aid: If you need to make a long journey (large \(L\)), the resistance is high. If you drive down a wide motorway (large \(A\)), resistance is low.

4. I-V Characteristics and Variable Components (Syllabus 9.3)

4.1 I-V Graphs for Different Components

The I-V characteristic shows how the current flowing through a component varies with the potential difference across it.

1. Metallic Conductor (at Constant Temperature):
   • Shape: Straight line passing through the origin.
   • Conclusion: This is an Ohmic conductor. \(V \propto I\), and R is constant (the gradient \(I/V\) is constant).
2. Filament Lamp:
   • Shape: A curve that gets shallower as \(V\) and \(I\) increase (S-shaped curve).
   • Explanation: As current increases, the filament heats up. Because metals have positive temperature coefficients, an increase in temperature causes the resistance to increase.
3. Semiconductor Diode:
   • Shape: Almost zero current until a certain small positive voltage (forward bias voltage) is reached. Resistance then drops drastically, and current increases exponentially. In the negative voltage direction (reverse bias), resistance is almost infinite (no current flow).
   • Conclusion: This is a non-ohmic conductor used to allow current flow in only one direction.

Common Mistake: Remember that resistance is \(R=V/I\). On an I-V graph, the gradient is \(I/V\), so resistance is the reciprocal of the gradient (\(R = 1/\text{gradient}\)).

4.2 Variable Resistors: Thermistors and LDRs

These components are non-ohmic and act as sensors because their resistance changes significantly with environmental factors.

• Thermistor (Negative Temperature Coefficient - NTC):
  • Function: Resistance decreases as temperature increases.
  • Application: Used in temperature sensors (e.g., thermostats, fire alarms).
• Light-Dependent Resistor (LDR):
  • Function: Resistance decreases as light intensity increases.
  • Application: Used in light sensors (e.g., automatic street lights, darkness detectors).

Key Takeaway: Ohm's Law only applies if R is constant. Filament lamps heat up (R increases). Diodes only conduct one way. Thermistors and LDRs are resistance sensors.

5. Practical Circuits, Internal Resistance, and Terminal P.D. (Syllabus 10.1)

5.1 Internal Resistance

Real sources of e.m.f. (like batteries or power supplies) are not perfect. They themselves have some resistance, known as internal resistance (\(r\)).

When charge flows through the source, some electrical energy is wasted (turned into heat) within the source itself due to this internal resistance.

5.2 Terminal Potential Difference

The terminal potential difference (\(V\)) is the actual P.D. measured across the terminals of the source when current is flowing.

The energy provided by the e.m.f. (\(E\)) is split between the energy wasted internally (\(V_r\)) and the energy delivered to the external circuit load (\(V\)).

Applying the principle of energy conservation:

$$E = V + V_r$$

Since \(V_r = Ir\) (p.d. across the internal resistor) and \(V\) is the p.d. across the external load (where \(V=IR_{load}\)):

$$E = V + I r$$
$$V = E - I r$$

This equation shows that the terminal P.D. (\(V\)) is always less than the e.m.f. (\(E\)) when current (\(I\)) is flowing, because some voltage is 'dropped' internally. If \(I=0\) (open circuit), then \(V=E\).

6. Kirchhoff's Laws (Syllabus 10.2)

Kirchhoff's laws are essential tools for solving complex circuits that cannot be easily simplified using series and parallel rules. They are based on fundamental conservation laws.

6.1 Kirchhoff's First Law (The Junction Rule)

• Statement: The sum of currents entering any junction in a circuit must equal the sum of currents leaving that junction.
• Conservation Principle: This is a direct consequence of the conservation of charge. Charge cannot be created or destroyed at a junction.
• Mathematical form: \(\sum I_{in} = \sum I_{out}\)

6.2 Kirchhoff's Second Law (The Loop Rule)

• Statement: In any closed loop in a circuit, the sum of the electromotive forces (e.m.f.s) must equal the sum of the potential differences (p.d.s).
• Conservation Principle: This is a direct consequence of the conservation of energy. Any energy supplied by sources must be entirely dissipated by the resistors and loads in that loop.
• Mathematical form: \(\sum E = \sum IR\) (Note: Careful sign convention must be used when applying this).

Key Takeaway: K1 is about charge conservation at junctions. K2 is about energy conservation around loops. Master these two laws, and no D.C. circuit problem is unbeatable!

7. Combined Resistance (Series and Parallel) (Syllabus 10.2)

We can use Kirchhoff's laws to derive the standard formulae for combining resistors.

7.1 Resistors in Series

Components are in series if they are connected end-to-end, so the current flowing through each component is the same.

• Using Kirchhoff's Second Law (Energy Conservation): The total voltage supplied is shared among the resistors: \(V_T = V_1 + V_2\).
• Since \(V = IR\) and \(I_T = I_1 = I_2\), we substitute: \(I R_T = I R_1 + I R_2\).
• Combined Resistance Formula: $$R_{total} = R_1 + R_2 + ...$$

The total resistance in series is always greater than the largest individual resistance.

7.2 Resistors in Parallel

Components are in parallel if they are connected across the same two points, so the potential difference across each component is the same.

• Using Kirchhoff's First Law (Charge Conservation): The total current splits and rejoins: \(I_T = I_1 + I_2\).
• Since \(I = V/R\) and \(V_T = V_1 = V_2\), we substitute: \(V/R_T = V/R_1 + V/R_2\).
• Combined Resistance Formula: $$\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + ...$$

The total resistance in parallel is always smaller than the smallest individual resistance. Adding more paths reduces the overall opposition to current flow.

8. Potential Dividers (Syllabus 10.3)

8.1 The Principle of a Potential Divider

A potential divider is a circuit consisting of two or more resistors connected in series to a source of P.D. Its purpose is to divide the source P.D. into smaller voltages.

If two resistors, \(R_1\) and \(R_2\), are connected in series to a source \(V_{in}\), the voltage across \(R_2\) (\(V_{out}\)) can be calculated using the principle that voltage drops are proportional to resistance in a series circuit.

$$V_{out} = V_{in} \left( \frac{R_2}{R_1 + R_2} \right)$$

This formula is crucial for solving potential divider problems.

8.2 The Potentiometer and Null Methods

A potentiometer is simply a potential divider that uses a long wire or a variable resistor (rheostat) to provide a continuously adjustable output voltage.

• Use: A potentiometer can be used as a means of comparing potential differences, such as comparing the e.m.f.s of two cells.
• Null Method: This involves using a galvanometer to detect when the current is zero (the "null point"). At the null point, the P.D. supplied by the potentiometer is exactly equal to the P.D. being measured (e.g., the e.m.f. of the cell). Null methods are highly desirable because when the galvanometer reads zero, no current is drawn from the measured source, meaning the internal resistance of that source has no effect on the measurement.

8.3 Potential Dividers as Sensors (LDRs and Thermistors)

By replacing one fixed resistor in a potential divider with a sensor (LDR or thermistor), we create a circuit whose output voltage (\(V_{out}\)) is dependent on the environmental condition.

Example: Using an NTC Thermistor in the \(R_2\) position (connected to \(V_{out}\)):

1. If temperature increases, the thermistor resistance (\(R_2\)) decreases.
2. Since \(V_{out} \propto R_2\), the output P.D. (\(V_{out}\)) decreases.
3. Application: This low output voltage could switch on a cooling fan or trigger a warning signal when the temperature exceeds a threshold.

Key Takeaway: Potential dividers split voltage proportionally. Sensors like LDRs and thermistors allow the output voltage to become sensitive to light or temperature changes, forming the basis of many electronic control systems.