Higher tier: examples
Question
A 2 m length of wire is found to have a resistance of 36 \(\Omega\).
What is the resistance of each of the following wires of equal length and made of the same material but having different cross section areas?
a) Double the area of cross section.
b) Four times the area of cross section.
c) Quarter the areas of cross section.
Since each wire is identical, resistance and area of cross section will be inverse proportionTwo values are said to be inversely proportional, when one amount increases and the other amount decreases..
a) Doubling the area of cross section halves the resistance, so this wire will have half the resistance = \(\frac{1}{2}\times{36~}\Omega{~=~18~}\Omega\).
b) Four times the area of cross section will quarter the resistance, so this wire will have a quarter the resistance = \(\frac{1}{4}\times{36~}\Omega{~=~9~}\Omega\).
c) Quartering the area of cross section will increase the resistance by a factor of four, so this wire will have 4 times the resistance = \({4}\times{36~}\Omega{~=~144~}\Omega\).
Question
A wire of length 50 cm is found to have a resistance of 15 \(\Omega\).
What is the resistance of a wire made from the same material but 150 cm long and half the area of cross section?
Resistance is directly proportional to length.
The new wire is three times the length of the original and so this will increase the resistance by a factor of three.
Resistance is inversely proportional to area of cross section.
The new wire has half the area of cross section of the original and so this will increase the resistance by a factor of two.
The combined effect is to increase the resistance by a factor of three times two, or six = 6 x 15 = 90 \(\Omega\).
The resistance of the new wire is 90 \(\Omega\).