use, for a current-carrying conductor, the expression I = Anvq , where n is the number density of charge carriers

Electric Current (Cambridge IGCSE/A‑Level Physics 9702 – Topic 9.1)

1. Macroscopic definition

The electric current is the rate at which electric charge passes a given cross‑section of a conductor.

\[

I=\frac{Q}{t}\qquad\text{(definition, SI unit: ampere, A)}

\]

• I – scalar quantity; the sign is indicated by the direction of conventional current.
• Q – total charge that crosses the section (C).
• t – elapsed time (s).

2. Charge carriers and sign convention

• In metals the mobile carriers are electrons (negative charge, \(q=-e\)).
• In electrolytes, many semiconductors and some metals the carriers can be positive ions or “holes” (positive charge, \(q=+e\)).
• Each carrier carries an integer multiple of the elementary charge
  

  \[

  e = 1.60\times10^{-19}\ \text{C}

  \]

• Conventional current is defined as the direction that a positive charge would move. Hence, in a metal the current direction is opposite to the actual electron drift.

3. Microscopic expression for current

\[

I = A\,n\,v\,q

\]

3.1 Derivation

Consider a uniform cylindrical conductor of cross‑sectional area \(A\) and length \(L\).

1. During a time interval \(t\) the carriers travel a distance \(vt\).
2. The volume swept out is \(A\,vt\).
3. Number of carriers in that volume: \(n\,A\,vt\) (where \(n\) is the number density, m\(^{-3}\)).
4. Each carrier carries charge \(q\); the total charge that passes the cross‑section is

   \[

   \Delta Q = n\,A\,v\,t\,q .

   \]

5. Current is charge per unit time:

   \[

   I = \frac{\Delta Q}{t}=A\,n\,v\,q .

   \]

3.2 Meaning of each symbol

3.3 Worked example – copper wire

Find the drift speed of electrons in a copper wire of radius \(r=1.0\ \text{mm}\) carrying a current \(I=5.0\ \text{A}\).

• Data for copper: \(n = 8.5\times10^{28}\ \text{m}^{-3}\), \(q = -e = -1.60\times10^{-19}\ \text{C}\).

\[

A = \pi r^{2}= \pi(1.0\times10^{-3}\ \text{m})^{2}=3.14\times10^{-6}\ \text{m}^{2}

\]

\[

v = \frac{I}{A\,n\,|q|}= \frac{5.0}{(3.14\times10^{-6})(8.5\times10^{28})(1.60\times10^{-19})}

\approx 1.2\times10^{-4}\ \text{m s}^{-1}

\]

3.4 Example – aluminium (different \(n\))

Aluminium has a lower free‑electron density, \(n\approx1.8\times10^{28}\ \text{m}^{-3}\). For the same wire dimensions and current (5 A) the drift speed becomes:

\[

v = \frac{5.0}{(3.14\times10^{-6})(1.8\times10^{28})(1.60\times10^{-19})}

\approx 5.5\times10^{-4}\ \text{m s}^{-1}

\]

Because fewer carriers are available, each must move faster to carry the same current.

4. From drift speed to electric field – mobility

The drift speed is proportional to the electric field \(E\) applied along the conductor:

\[

v = \mu\,E

\]

• \(\mu\) – mobility of the charge carriers (m² V⁻¹ s⁻¹).
• Typical values: electrons in copper \(\mu \approx 4.5\times10^{-3}\ \text{m}^2\text{V}^{-1}\text{s}^{-1}\); holes in silicon \(\mu \approx 4.8\times10^{-4}\ \text{m}^2\text{V}^{-1}\text{s}^{-1}\).

Combining \(I = A n q v\) with \(v = \mu E\) gives

\[

I = A n q \mu E \qquad\Longrightarrow\qquad

V = IR \;\; \text{with}\;\; R = \frac{L}{A\sigma},

\]

where the conductivity \(\sigma = n q \mu\). This links the microscopic picture to the macroscopic Ohm’s law (topic 9.2).

5. Current in simple circuits

• Potential difference (voltage): \(V = IR\) (Ohm’s law – introduced in 9.2 but useful here to see how a driving voltage produces a current).
• Electrical power: \(P = IV = I^{2}R = \dfrac{V^{2}}{R}\). Power tells how much energy per unit time is converted to heat, light, etc.

6. Measuring electric current

Safety note: Ensure the instrument’s current rating exceeds the expected current, use proper insulation, and remember that high currents can cause heating and fire hazards.

7. Factors that affect the magnitude of current

• Cross‑sectional area (\(A\)) – larger area → more carriers can pass simultaneously.
• Number density (\(n\)) – metals have high \(n\); insulators have very low \(n\).
• Drift speed (\(v\)) – proportional to the applied electric field; \(v = \mu E\).
• Charge per carrier (\(q\)) – determines how much charge each carrier transports.
• Temperature – can change \(n\) (especially in semiconductors) and \(\mu\) (in all materials).

8. Assessment Objective (AO) mapping

9. Suggested diagram

10. Key points to remember

• The macroscopic current is the product of geometry, material properties and carrier dynamics: \(I = A n v q\).
• Drift speed is extremely small; the rapid propagation of an electric signal is due to the electric field, not the motion of individual carriers.
• Mobility links drift speed to the electric field and provides the bridge to Ohm’s law and resistance.
• Current direction follows the convention of positive charge flow, even when electrons are the actual carriers.
• Accurate measurement and safe handling of current are essential practical skills.