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:

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)

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)

1. Millikan oil‑drop experiment – Direct measurement of individual droplet charges showed they are integer multiples of \(e\).
2. Photoelectric effect – Each emitted electron carries a charge of exactly \(-e\).
3. Charge balance in chemical reactions – Ions exchange whole numbers of elementary charges.

4.3 Typical charge carriers

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)
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