Deriving and using I = nAve

Overview

Recall questions on the meaning of each symbol, the derivation from the volume of charge passing in one second, typical drift velocities and why they are so slow, why a lamp lights immediately despite that, and worked calculations comparing conductors.

What you need to know

Answer each one before you reveal it.

Questions from this subtopic

Each one with the model answer underneath.

1. 1

   State the equation I = nAve and define every symbol with its unit.

   Model answer

   In I = nAve, I is the current in A, n is the free carrier number density in m⁻³, A is the cross-sectional area in m², v is the drift velocity in m s⁻¹ and e is the charge on a carrier in C.

2. 2

   Outline how the equation I = nAve is derived for free electrons in a wire.

   Model answer

   In a time t every electron travels a distance vt, so all the electrons in a cylinder of volume Avt pass a chosen cross-section. That cylinder holds nAvt electrons carrying a charge nAvte, and dividing by t gives I = nAve.

3. 3

   Calculate the drift velocity in a copper wire of cross-sectional area 1.0 mm² carrying 1.0 A, taking n = 8.5 × 10²⁸ m⁻³.

   Model answer

   Rearranging, v = I/(nAe) = 1.0 ÷ (8.5 × 10²⁸ × 1.0 × 10⁻⁶ × 1.60 × 10⁻¹⁹) = 7.4 × 10⁻⁵ m s⁻¹. Substituting 7.35 × 10⁻⁵ m s⁻¹ back into nAve returns 1.0 A.

4. 4

   A copper wire of diameter 0.80 mm carries 2.0 A. Calculate the drift velocity, taking n = 8.5 × 10²⁸ m⁻³.

   Model answer

   The area is A = π × (0.40 × 10⁻³ m)² = 5.03 × 10⁻⁷ m², so v = 2.0 ÷ (8.5 × 10²⁸ × 5.03 × 10⁻⁷ × 1.60 × 10⁻¹⁹) = 2.9 × 10⁻⁴ m s⁻¹. Substituting 2.93 × 10⁻⁴ m s⁻¹ back into nAve returns 2.0 A.

5. 5

   A wire of area 2.0 × 10⁻⁷ m² carries 0.50 A with a drift velocity of 1.5 × 10⁻⁴ m s⁻¹. Calculate the carrier number density.

   Model answer

   Rearranging, n = I/(Ave) = 0.50 ÷ (2.0 × 10⁻⁷ × 1.5 × 10⁻⁴ × 1.60 × 10⁻¹⁹) = 1.0 × 10²⁹ m⁻³. Substituting 1.04 × 10²⁹ m⁻³ back into nAve returns 0.50 A.

6. 6

   A copper wire carries 3.0 A with a drift velocity of 2.5 × 10⁻⁴ m s⁻¹ and n = 8.5 × 10²⁸ m⁻³. Calculate its cross-sectional area.

   Model answer

   Rearranging, A = I/(nve) = 3.0 ÷ (8.5 × 10²⁸ × 2.5 × 10⁻⁴ × 1.60 × 10⁻¹⁹) = 8.8 × 10⁻⁷ m². Substituting 8.82 × 10⁻⁷ m² back into nAve returns 3.0 A.

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