Problem 2.3  Source:   Problem 18 - page 161 -    HALLIDAY, RESNICK, WALKER, Jearl - Book: Fundamentos de Física - Vol 3 - Ed. LTC - 8ª edição - 2009.

A certain wire has a resistance R. What is the resistance of a second wire, made of the same material, with half of the length and half of the diameter?

Solution of the Problem 2.3

Let's name R₁ the resistance of the first wire and R₂ the resistance of the second wire. If the wires are made of the same material then we do not need to take into account the ρ of the material. So we'll work with the area and its length. Let's not forget that the area of a circle is given by:

So if the second wire is half the length of the first wire, it is logical that its resistance will be reduced to half in relation to the first wire. This is only due to the length. Now let's look at the diameter. In the equation above we see that the area is in the denominator. Therefore, if we reduce the diameter to half means that the denominator will be reduced by a factor four (4), because the diameter is squared. In this case, the resistance of the second wire will be quadrupled. As by reducing the length, we had a reduction two times, we conclude that the second wire will have twice the resistance of the first wire. Translating this into an equation, we have: