(b) I = Anev , where n is the number density of charge carriers

The idea

This equation links current to the microscopic motion of charge carriers. I is the current in amperes, A is the cross-sectional area of the conductor in m², n is the number density of charge carriers (number per m³), e is the elementary charge 1.60 × 10⁻¹⁹ C, and v is the mean drift velocity in m s⁻¹. Rearranged, v = I / (Ane). A useful check is that Anev has units m² × m⁻³ × C × m s⁻¹ = C s⁻¹ = A. In a multiple-choice question you may be asked to find v, n or A, or to predict how v changes when A doubles at constant current: v halves. Remember n is a number density, not a mass density, and e is the charge on one carrier, not the total charge.

What earns the marks

• States I = Anev and identifies each symbol with its unit: I in A, A in m², n in m⁻³, e in C, v in m s⁻¹.
• Rearranges correctly to v = I / (Ane) or n = I / (Aev) as required by the question.
• Uses e = 1.60 × 10⁻¹⁹ C for electrons or the appropriate carrier charge for ions.
• Applies proportionality reasoning, for example at constant I, doubling A halves v because v ∝ 1/A.
• Checks that the product Anev has units of current, confirming dimensional consistency.

Common mistakes and how to put them right

• Treating n as a mass density in kg m⁻³.

  Put it right: n is a number density of charge carriers per m³, so it is a pure number per unit volume.

• Using e as the total charge or as 1 C.

  Put it right: e is the elementary charge 1.60 × 10⁻¹⁹ C carried by one electron or proton.

• Forgetting to convert area from mm² or cm² to m² before substituting.

  Put it right: For example 1 mm² = 1 × 10⁻⁶ m².

• Assuming v is the speed of individual electrons along the wire.

  Put it right: v is the mean drift velocity, which is very small compared with random thermal speeds.

Exam tips

• Write the rearranged equation before substituting numbers so the examiner can follow your method.
• Convert all lengths to metres and areas to m² at the start of the calculation.
• Use the unit check Anev → C s⁻¹ = A to confirm you have not inverted the relationship.
• When comparing two wires, reason with ratios rather than recalculating from scratch.