Calculating Current & Drift Velocity (Edexcel IAL Physics)

Spec point — 73

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:

$I=nqvA$

• 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 proportiona l 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 The number density of conduction electrons in a copper wire is 9.2 × 10²⁸ m⁻³. The wire carries a current of 3.5 A and it has a cross-sectional area of 1.5 mm².

Determine the average drift velocity of the electrons.

Answer:

Step 1: Consider the situation

• A copper wire is a conductor, and the free electrons are charge carriers
• Use the transport equation I = nqvA

Step 2: Rearrange the equation for drift speed v

$v=\frac{I}{nqA}$

Step 3: Substitute in values

• Current, I = 3.5 A
• Cross-sectional area,A = 1.5 mm² = 1.5 ÷ 1000² = 1.5 × 10⁻⁶ m²
• Number density of conduction electrons, n = 9.2 × 10²⁸ m⁻³
• Charge on an electron, q = 1.60 × 10⁻¹⁹ C (From the data sheet)

$v = \frac{3.5}{( 9.2 \times 10^{28 ) \times ( 1.60 \times 10^{-19} } ) \times ( 1.5 \times 10^{- 6} )} = 0.16 \times 10^{- 3}$ m s⁻¹

$v=0.16$ mm s⁻¹ (2 s.f.)

The Large Range of Material Resistivities

Spec point — 73

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

Exam Hint Remember that the cross-sectional area is in m², the drift velocity is in m s^-1 and the number density is in m^-3.

Therefore, sometimes unit conversions from cm or mm may be required, so make sure you're comfortable with these.