Answer to Question #183636 in Electric Circuits for Lianne

Question

Show all the solutions. Round off answers to two decimal places. Be consistent with the units

2. a)  Considering the two wires of the same material and cross-sectional area, how does the resistance vary with the length?

  b)  Considering the two wires of the same material and length, how does the resistance vary with the cross-sectional area?

Answer 1

Explanations & calculations

• To analyse the behaviour in each case, the equation to be used is $\small R =\Large\frac{\rho\cdot L}{A}$ .
• For the case number 1, the material & the cross sectional area are identical for both wires.
• Since material is the same, the quality dependant on material: $\rho$ is the same.
• Then we can re-write the equation in proportional form

$\qquad\qquad \begin{aligned} \small R&=\small (\frac{\rho}{A})\cdot L\\ &\propto kL \end{aligned}$

• Then it is obvious that the resistance then changes linearly with the length of the wire such that as the length increases, the resistance increases.

2.

• For this case, length & the resistivity: $\rho$ are identical for both wires.
• Then performing a similar analysis yeilds the following,

$\qquad\qquad \begin{aligned} \small R&= \small \frac{(\rho L)}{A}\\ &=\small \frac{k_1}{A}\\ &\propto\frac{1}{A} \end{aligned}$

• Then it could be seen that the resistance is inversly proportional to the cross sectional area such that as the cross sectional area is increased, the resistance is decreased.