IB DP Physics: SL复习笔记5.2.5 Resistivity

Resistance, Length and Cross-Sectional Area

• The resistance of a sample depends on:
  • The material it is made of
  • The length of the sample
  • The cross-sectional area of the sample

• The resistance of a conductor (e.g. a wire) is:
  • Directly proportional to its length
  • Inversely proportional to its cross-sectional area

How the length and width of a wire affect its resistance

Defining Resistivity

• This leads to the definition of a new quantity, called resistivity
• Resistivity is a property describing the extent to which a material opposes the flow of electric current through it
• It is defined as follows:

The resistivity of a material is equal to the resistance per unit length of a material with unit cross-sectional area

• The equation for the resistivity is:

• Where:
  • ρ = resistivity in ohm-metres (Ω m)
  • R = resistance in ohms (Ω)
  • A = cross sectional area of material in square metres (m²)
  • L = length of material in meres (m)

Resistivity of Conductors and Insulators

• Resistivity is dependent on temperature
  • In conductors, the resistivity increases with increasing temperature
  • In insulators, the resistivity decreases with increasing temperature

• Since ρ ∝ R, when a wire or the filament inside a lamp heat up, their resistance increases

Resistivity of some materials at room temperature

• Conductors have the lowest values of resistivity
• Wires are made from copper because of its relatively low resistivity at room temperature
• Insulators have such a high resistivity that virtually no current will flow through them

Step 1: Write down the known quantities

• • Copper (Cu):
    • Diameter, dCu = 5 mm = 5 × 10^–3 m
    • Length, LCu = 8 mm = 8 × 10^–3 m
    • Resistivity, ρCu = 1.7 × 10^–8 Ω m

• • Aluminium (Al):
    • Diameter, dAl = 10 mm = 10 × 10^–3 m
    • Length, LAl = 16 mm = 16 × 10^–3 m
    • Resistivity, ρAl = 2.6 × 10^–8 Ω m

Step 2: A conductor is better if it has a lower resistance

Step 3: Write down the equation for resistivity

Step 4: Rearrange the above equation to calculate the resistance R

Step 5: Substitute the numbers into the above equation to calculate the resistance of copper RCu and the resistance of aluminium RAl

• • The cross-sectional area A is calculated from the diameter d as follows:
    • A = π(d/2)2

Step 6: Compare the two values of the resistance

• • The cylinder with the lower resistance is the better conductor

RCu = 6.9 × 10^–6 Ω

RAl = 5.3 × 10^–6 Ω

RAl < RCu

The aluminium cylinder is the better conductor.