understand that the charge on charge carriers is quantised
1 Electric Current
1.1 Definition and basic equation
- Electric current is the rate at which electric charge passes a given point in a circuit.
- Mathematically \[ I=\frac{\Delta Q}{\Delta t}\qquad\text{(ampere, A)} \] where \(\Delta Q\) is the charge transferred (C) in the time interval \(\Delta t\) (s).
- Re‑arranging gives the useful relation \[ Q = I\,t \] which allows us to calculate the total charge that has passed in a time \(t\).
1.2 Charge carriers
The particles that actually move when a current flows are called charge carriers. Their nature depends on the material:
| Material | Charge carriers | Typical charge \(q\) |
|---|---|---|
| Metals | Free electrons | \(-e\) |
| Electrolytes | Ions (cations + anions) | \(\pm e\) (or multiples) |
| Semiconductors | Electrons and holes | \(-e\) (electron), \(+e\) (hole) |
1.3 Microscopic description of current
- Current density \[ J = \frac{I}{A}\qquad\text{(A m\(^{-2}\))} \] where \(A\) is the cross‑sectional area of the conductor.
- Microscopic expression (useful for metals and doped semiconductors) \[ I = A\,n\,v\,q \] where \(A\) = cross‑sectional area (m\(^2\)) \(n\) = number of charge carriers per unit volume (m\(^{-3}\)) \(v\) = drift speed of the carriers (m s\(^{-1}\)) \(q\) = charge on each carrier (C, usually \(\pm e\)).
Example – current in a copper wire
Given: copper wire of length 0.50 m, cross‑sectional area \(A = 1.0\times10^{-6}\,\text{m}^2\), free‑electron density \(n = 8.5\times10^{28}\,\text{m}^{-3}\), drift speed \(v = 2.0\times10^{-4}\,\text{m s}^{-1}\).
\[ I = A n v e = (1.0\times10^{-6})(8.5\times10^{28})(2.0\times10^{-4})(1.602\times10^{-19}) \approx 2.7\ \text{A} \]2 Potential Difference & Electrical Power
2.1 Potential difference (voltage)
Potential difference is the energy transferred per unit charge:
\[ V = \frac{W}{Q}\qquad\text{(volt, V)} \] where \(W\) is the work (or energy) done on the charge \(Q\).2.2 Electrical power
Power is the rate at which electrical energy is transferred:
\[ P = \frac{W}{t}=VI\qquad\text{(watt, W)} \] Using Ohm’s law (\(V = IR\)) we obtain two additional useful forms: \[ P = I^{2}R\qquad\text{and}\qquad P = \frac{V^{2}}{R} \]Example – a 12 V lamp drawing 2 A
\[ P = VI = (12\ \text{V})(2\ \text{A}) = 24\ \text{W} \]3 Resistance, Resistivity & Ohm’s Law
3.1 Macroscopic resistance
- Resistance \(R\) opposes the flow of charge and is defined by Ohm’s law: \[ V = I R\qquad\text{(Ω = V A\(^{-1}\))} \]
- For a uniform conductor of length \(L\) and cross‑sectional area \(A\): \[ R = \rho\,\frac{L}{A} \] where \(\rho\) is the material’s resistivity (Ω m).
3.2 I‑V characteristics (required by the syllabus)
| Component | I‑V shape | Key features |
|---|---|---|
| Ohmic conductor (e.g., copper wire) | Straight line through the origin | Constant resistance; \(V\propto I\) |
| Semiconductor diode | Very low current until a “turn‑on” voltage, then steep rise | Non‑linear; conducts only above threshold |
| Filament lamp | Curve that becomes steeper with increasing \(I\) | Resistance rises with temperature |
| LDR (light‑dependent resistor) | Flatter slope in light, steeper in dark | Resistance decreases with light intensity |
| Thermistor (NTC) | Curve bending downwards as \(I\) increases | Resistance falls as temperature rises |
Worked example – resistance of a nichrome wire
Length \(L = 0.30\ \text{m}\), cross‑section \(A = 2.0\times10^{-6}\ \text{m}^2\), resistivity \(\rho = 1.10\times10^{-6}\ \Omega\text{m}\).
\[ R = \rho\frac{L}{A}= (1.10\times10^{-6})\frac{0.30}{2.0\times10^{-6}} = 0.165\ \Omega \]4 Quantisation of Charge
4.1 Fundamental statement
All observable electric charge occurs in integer multiples of the elementary charge \(e\):
\[ q = n\,e\qquad n = \pm1,\pm2,\pm3,\dots \] \[ e = 1.602\,\times10^{-19}\ \text{C} \]4.2 Experimental evidence (syllabus requirement)
- Millikan oil‑drop experiment – Direct measurement of individual droplet charges showed they are integer multiples of \(e\).
- Photoelectric effect – Each emitted electron carries a charge of exactly \(-e\).
- Charge balance in chemical reactions – Ions exchange whole numbers of elementary charges.
4.3 Typical charge carriers
| Carrier | Symbol | Charge \(q\) | Typical medium |
|---|---|---|---|
| Electron | \(e^{-}\) | \(-e\) | Metals, semiconductors |
| Proton | \(p^{+}\) | \(+e\) | Atomic nuclei |
| Alpha particle | \(\alpha^{2+}\) | \(+2e\) | Radioactive decay |
| Sodium ion | \(\text{Na}^{+}\) | \(+e\) | Electrolytes |
| Chloride ion | \(\text{Cl}^{-}\) | \(-e\) | Electrolytes |
4.4 Linking quantised charge to macroscopic current
The number of charge carriers crossing a cross‑section each second is
\[ N = \frac{I}{e} \]Thus a current of \(1\ \text{A}\) corresponds to
\[ N = \frac{1\ \text{C s}^{-1}}{1.602\times10^{-19}\ \text{C}} \approx 6.24\times10^{18}\ \text{carriers s}^{-1} \]5 DC Circuits (AS‑Level requirement)
5.1 Circuit symbols (syllabus §6)
| Component | Symbol (hand‑drawn) |
|---|---|
| Battery (ideal emf) |  |
| Resistor | 
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