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.