Welcome to the World of Electric Current!
Hello future electrical engineers! This chapter, "Electric Current," is foundational to all of electricity. Don't worry if you sometimes feel lost in circuits; we will break down the flow of charge, energy transfer, and the essential rules governing every wire and component.
Understanding these concepts is key not just for your AS Level exams, but also for grasping how everything from your phone charger to the national grid works! Let's get started.
9.1 The Nature of Electric Current
What Exactly is Electric Current?
An electric current, symbol \(I\), is simply the movement of charge carriers. These are usually electrons in metals, but they can be ions in electrolytes or gases.
- Definition: Current is the rate of flow of charge.
- Unit: The SI base unit for current is the Ampere (A).
Charge, Current, and Time
The most basic relationship linking charge \((Q)\), current \((I)\), and time \((t)\) is:
$$Q = It$$
Where:
- \(Q\) is the total charge passed (measured in Coulombs, C).
- \(I\) is the current (A).
- \(t\) is the time (s).
The Quantisation of Charge
You need to understand that charge is quantised. This means charge comes in discrete, fixed packets. The smallest packet is the magnitude of the charge on a single electron, \(e\).
Think of it like money: you can have 1 cent, 2 cents, 3 cents, but you can't have 1.5 cents. Charge only exists in integer multiples of \(e\).
$$Q = N e$$ Where \(N\) is an integer (number of carriers) and \(e\) is the elementary charge (\(1.60 \times 10^{-19} \text{ C}\)).
The Microscopic View: Current and Drift Velocity
While the electrons move incredibly fast randomly, the actual net movement along the wire is slow—this is the drift velocity, \(v\).
For a current-carrying conductor, the relationship between the macroscopic current \(I\) and the microscopic properties of the conductor is given by:
$$I = Anvq$$
Let's break down this powerful equation:
- \(A\): Cross-sectional area of the wire (\(m^2\)).
- \(n\): Number density of charge carriers (\(m^{-3}\)). This is the number of charge carriers per unit volume.
- \(v\): Average drift velocity of the charge carriers (\(m/s\)).
- \(q\): Charge of a single carrier (usually \(e\)).
Analogy Alert: Imagine traffic flowing through a tunnel.
- \(I\) is the total traffic rate (cars per second).
- \(A\) is the size of the tunnel entrance.
- \(n\) is the car density (how tightly packed the cars are).
- \(v\) is how fast the cars are *drifting* forward.
Quick Takeaway 9.1:
The current $I$ is the flow of charge $Q$ ($Q=It$). For a conductor, the current is determined by the concentration and speed of the charge carriers ($I=Anvq$).9.2 Potential Difference and Power
Defining Potential Difference (p.d.)
The Potential Difference (p.d.), or voltage (\(V\)), across a component is defined as the energy transferred (or converted) per unit charge passing through it.
$$V = \frac{W}{Q}$$
Where:
- \(V\) is the potential difference (measured in Volts, V).
- \(W\) is the energy transferred (or work done) (measured in Joules, J).
- \(Q\) is the charge (C).
This definition tells you that 1 Volt is 1 Joule of energy transferred per 1 Coulomb of charge.
Electric Power
Power (\(P\)) is the rate at which energy is transferred. In a circuit, power is the rate at which the source supplies energy or the rate at which a component dissipates energy (usually as heat or light).
Since $P = W/t$ and $W = VQ$, we can derive the key power formulas:
$$P = VI$$ (Power = Voltage $\times$ Current)
Using Ohm's Law (\(V=IR\)), we can derive two more incredibly useful forms:
$$P = I^2 R$$
$$P = \frac{V^2}{R}$$
10.1 Practical Circuits: E.M.F. vs. P.D.
These terms are often confused, but they describe two different parts of the energy transfer process.
Electromotive Force (e.m.f.), $\mathcal{E}$
The e.m.f. of a source (like a battery or generator) is the total energy transferred per unit charge in driving charge around a complete circuit.
- It is the *maximum* voltage the source can supply.
- It's the energy conversion from chemical/mechanical energy *into* electrical energy.
Distinguishing E.M.F. and P.D.
Analogy: A water pump and pipes.
| Concept | Description | Water Analogy |
|---|---|---|
| E.M.F. ($\mathcal{E}$) | Energy supplied by the source (per unit charge). | The energy supplied by the pump to lift the water. |
| P.D. ($V$) | Energy dissipated/used by a component (per unit charge). | The energy lost as water flows down a waterfall or through a narrow pipe. |
The Effect of Internal Resistance
Real sources of e.m.f. (like batteries) are not perfect. They have their own internal opposition to current flow called internal resistance (\(r\)).
When a current \(I\) flows, some of the energy supplied by the source is wasted overcoming this internal resistance. This causes a "lost voltage" or "lost p.d." (Ir).
The Terminal Potential Difference (\(V_{\text{term}}\)) is the actual voltage available to the external circuit.
The relationship is:
$$\mathcal{E} = V_{\text{term}} + Ir$$
Where \(V_{\text{term}} = IR\) (R is the external load resistance).
Key point: The terminal p.d. is always less than the e.m.f. when a current is flowing. If the circuit is open (\(I=0\)), then \(V_{\text{term}} = \mathcal{E}\).
Quick Takeaway 9.2 & 10.1:
P.D. is energy used per charge ($V=W/Q$). Power is the rate of energy transfer ($P=VI$). E.M.F. is the total energy supplied, which is reduced by internal resistance ($r$) to give the terminal p.d.9.3 Resistance and Resistivity
Defining Resistance
Resistance (\(R\)) is the measure of how much a component opposes the flow of electric current.
Definition: Resistance is the ratio of potential difference across a component to the current flowing through it.
$$R = \frac{V}{I}$$ (Measured in Ohms, \(\Omega\)).
Ohm's Law
Ohm's Law is a specific condition, not a universal law of nature!
Statement: For an ohmic conductor (or resistor), the current is directly proportional to the potential difference across it, provided that physical conditions (like temperature) remain constant.
Mathematically, this means \(R\) is constant for an ohmic component.
I-V Characteristics (Who Obeys Ohm's Law?)
Not all components are ohmic. We use I-V graphs (current on the y-axis, voltage on the x-axis) to show how resistance changes. Remember, resistance is $R = V/I$, or $1/R$ is the gradient of the V-I graph.
1. Metallic Conductor at Constant Temperature (Ohmic):
- Graph is a straight line through the origin.
- Gradient (and therefore resistance $R$) is constant.
2. Filament Lamp (Non-Ohmic):
- Graph starts straight but curves inwards (gradient decreases).
- Explanation: As the current increases, the filament heats up. Higher temperature causes the metal lattice ions to vibrate more vigorously. Electrons collide more frequently with these ions, causing an increase in resistance.
3. Semiconductor Diode (Non-Ohmic):
- Graph is almost flat near the origin, then rises steeply in the "forward bias" direction.
- It has extremely high resistance in the reverse bias direction (it acts like a one-way valve for current).
Resistivity: What Makes a Wire a Resistor?
Resistance depends not just on the material, but on the dimensions of the component. Resistivity (\(\rho\)) is a property of the material itself.
The resistance \(R\) of a uniform wire is:
$$R = \frac{\rho L}{A}$$
Where:
- \(L\) is the length of the wire (m).
- \(A\) is the cross-sectional area (\(m^2\)).
- \(\rho\) is the resistivity (measured in \(\Omega \text{ m}\)).
Memory Aid: If you want low resistance, you need a S.L.A.P. of wire: Short Length (\(L \downarrow\)), Large Area (\(A \uparrow\)), low $\rho$ (P).
Thermistors and LDRs (Sensors)
These are two crucial components whose resistance changes drastically with external conditions.
1. Light-Dependent Resistor (LDR):
- Function: Resistance changes with light intensity.
- Rule: Resistance decreases as light intensity increases.
- Application: Automatic lighting controls (streetlights turn on when resistance increases due to darkness).
- Function: Resistance changes with temperature.
- Assumption: You only need to know about Negative Temperature Coefficient (NTC) thermistors.
- Rule: Resistance decreases as temperature increases.
- Application: Temperature sensors, thermostats, fire alarms.
Quick Takeaway 9.3:
Resistance is opposition ($R=V/I$). Resistivity $\rho$ is a material property ($R = \rho L/A$). Know the specific $I-V$ graphs for ohmic conductors, filament lamps (resistance increases with heat), and diodes. LDRs and NTC Thermistors are variable resistors used as sensors.10.2 Kirchhoff's Laws: The Circuit Rules
Kirchhoff's Laws are fundamental rules for analyzing complex circuits. They are derived directly from the fundamental conservation laws of Physics.
Kirchhoff's First Law (The Current Law)
Statement: The sum of the currents entering a junction (node) must equal the sum of the currents leaving that junction.
$$\sum I_{\text{in}} = \sum I_{\text{out}}$$
Underlying Principle: Conservation of Charge. Since charge cannot be created or destroyed, the total amount of charge carriers flowing into a junction must equal the amount flowing out.
Analogy: A traffic junction. If 10 cars enter the intersection per minute, 10 cars must leave per minute, assuming no cars are stored there.
Kirchhoff's Second Law (The Voltage Law)
Statement: In any closed loop in a circuit, the sum of the electromotive forces (\(\mathcal{E}\)) must be equal to the sum of the potential differences (\(V\)) around that loop.
$$\sum \mathcal{E} = \sum V$$
Underlying Principle: Conservation of Energy. Any energy supplied by the sources (\(\sum \mathcal{E}\)) must be completely dissipated or converted by the components in that closed path (\(\sum V\)).
Analogy: A rollercoaster. The energy supplied by the lift hill (E.M.F.) must equal the total energy lost through friction and height drops (P.D.) when the car returns to the start.
Applying Kirchhoff's Laws to Resistance
1. Resistors in Series
If resistors \(R_1, R_2, R_3, \dots\) are connected in series, the total resistance \(R_{\text{total}}\) is:
$$R_{\text{total}} = R_1 + R_2 + R_3 + \dots$$
Derivation Principle (Using KVL):
- Current \(I\) is the same everywhere (KCL).
- The total voltage supplied (\(V_{\text{total}}\)) is split across the resistors: \(V_{\text{total}} = V_1 + V_2 + \dots\) (KVL).
- Using \(V=IR\): \(IR_{\text{total}} = IR_1 + IR_2 + \dots\)
- Dividing by \(I\) gives the series formula.
2. Resistors in Parallel
If resistors \(R_1, R_2, R_3, \dots\) are connected in parallel, the total resistance \(R_{\text{total}}\) is:
$$\frac{1}{R_{\text{total}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots$$
Derivation Principle (Using KCL):
- The voltage \(V\) is the same across all branches (KVL applied to loops).
- The total current \(I_{\text{total}}\) splits into the branches: \(I_{\text{total}} = I_1 + I_2 + \dots\) (KCL).
- Using \(I=V/R\): \(\frac{V}{R_{\text{total}}} = \frac{V}{R_1} + \frac{V}{R_2} + \dots\)
- Dividing by \(V\) gives the parallel formula.
Common Mistake to Avoid:
When calculating parallel resistance, remember that you calculate \(1/R_{\text{total}}\) first. Don't forget the final step of inverting the answer to get $R_{\text{total}}$!
Quick Takeaway 10.2:
KCL is Conservation of Charge ($\sum I_{\text{in}} = \sum I_{\text{out}}$). KVL is Conservation of Energy ($\sum \mathcal{E} = \sum V$). These principles allow us to find series ($R_{\text{total}} = R_1 + R_2$) and parallel resistance ($1/R_{\text{total}} = 1/R_1 + 1/R_2$).10.3 Potential Dividers
The Principle of the Potential Divider
A potential divider is a circuit consisting of two or more resistors connected in series to a source of voltage. Its purpose is to provide an output voltage (\(V_{\text{out}}\)) that is a fraction of the total supply voltage (\(V_{\text{in}}\)).
If two resistors, \(R_1\) and \(R_2\), are in series with supply voltage \(V_{\text{in}}\), the output voltage taken across \(R_2\) is:
$$V_{\text{out}} = V_{\text{in}} \times \left( \frac{R_2}{R_1 + R_2} \right)$$
This works because the potential difference across a resistor is proportional to its resistance, provided the current is constant (which it is in a series circuit).
Did you know? This principle is how the volume knob on an old stereo works. It's often a variable resistor (rheostat or potentiometer) acting as a potential divider!
Sensing Circuits using Potential Dividers
By replacing one of the fixed resistors with a sensor (LDR or Thermistor), we create a circuit where the output voltage depends on the physical environment (light or temperature).
1. LDR in a Potential Divider (Light Sensor):
- The LDR is \(R_2\).
- When light intensity increases, \(R_2\) decreases.
- Since \(R_2\) is smaller, the fraction $\frac{R_2}{R_1+R_2}$ decreases, so $V_{\text{out}}$ decreases.
- Application: Turning a device ON when it gets dark (requires a small $V_{\text{out}}$ to activate the device).
2. Thermistor in a Potential Divider (Temperature Sensor):
- The Thermistor (NTC type) is \(R_2\).
- When temperature increases, \(R_2\) decreases.
- Therefore, $V_{\text{out}}$ decreases as the temperature rises.
- Application: Turning a fan ON when it gets hot (requires a small $V_{\text{out}}$ to activate the device).
The Potentiometer (Comparing P.D.s)
A potentiometer (or "pot" for short) uses the potential divider principle across a long length of uniform resistance wire.
- It works by moving a sliding contact along the wire to tap off a variable potential difference.
- Use: A potentiometer is used in null methods to compare potential differences or to measure the e.m.f. of a cell accurately, without drawing any current from the cell being measured.
A galvanometer is used in a null method to detect when zero current flows. At the point where the galvanometer reads zero, the potential difference from the potentiometer is exactly equal to the potential difference (or e.m.f.) being measured.