2.42 Calculating Current & Drift Velocity

Drift Velocity

• In a conductor, the current is due to the movement of charge carriers
  • The charge carriers can be negative or positive
  • However current is always taken to be in the same direction

• Drift velocity is the average velocity of the charge carriers travelling through the conductor
• In conductors, the charge carrier is usually free electrons
  • Free electrons only travel small distances before colliding with a metal ion
  • Therefore they have a relatively slow drift velocity of ∼ 10⁻³ m s⁻¹

• In the diagram below, the current in each conductor is from right to left
  • In diagram A (positive charge carriers), the drift velocity is in the same direction as the current
  • In diagram B (negative charge carriers), the drift velocity is in the opposite direction to the current

Conduction in a current-carrying conductor

• The density n represents the number of free charge carriers (electrons) per unit volume
  • Conductors, such as metals, have a high value of n
  • Insulators, such as plastics, have a low value of n
• Since the density of charge carriers is so large in conductors, the flow of current flow appears to happen instantaneously

The Transport Equation

• The current can be expressed in the transport equation:

• Where:
  • I = current (A)
  • n = number density (m⁻³)
  • q = the charge of the charge carrier (C)
  • v = drift velocity (m s⁻¹)
  • A = cross sectional area of the wire (m²), calculated using A = πr²

• The same equation is used whether the charge carriers are positive or negative
  • A negative value for v will indicate current in the opposite direction to the charge carriers
• The transport equation shows that v is inversely proportional to n
  • Since the more charge carriers available per unit volume the more the density will slow down their speed through the conductor
• The transport equation also shows that I is directly proportional to n 
  • Greater n means a greater charge is flowing and therefore a larger current I
• When the value of n is lower, the charge carriers must travel faster to carry the same current

Worked Example

Resistivity

• The transport equation tells us that current, I ∝ number of charge carriers, n
  • Therefore, the larger the number of charge carriers, the greater the current will be for the same applied voltage
  • This is because resistivity has decreased with more charge carriers available
• Different materials have different numbers of charge carriers

• Insulators have few charge carriers:
  
  • They have such a high resistivity that virtually no current will flow through them
  • A perfect insulator would have no charge carriers, n = 0 
  • A perfect insulator would have a current of zero regardless of the voltage applied

• Conductors have a large number of charge carriers
  • Metals are good conductors because they have free electrons
  • Free electrons are the atoms from the outer shell of each atom
  • Therefore there are lots of charge carriers per unit volume
  • This means resistivity is low

• Semiconductors have a small number of free electrons
  • There are fewer delocalised electrons in a semiconductor than in a metal 
  • There are a greater number of free electrons at a higher temperature
  • Resistivity changes in a semiconductor, due to the variation with temperature in free electrons which are available as charge carriers 
  • Silicon is an example of a semiconductor

The resistivity of some materials at room temperature

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