Chapter 12.8, Problem 8P

To determine

To calculate: The radius of the resistance of a wire to the transition of electricity wires directly as the length of the wire and inversely as the square of the radius of a cross section of a wire.

Answer to Problem 8P

The radius of the resistance: $0.3$ cm.

Explanation of Solution

Given information:

The radius of the resistance of a wire to the transition of electricity wires directly as the length of the wire and inversely as the square of the radius of a cross section of a wire.

An $800$ m wire has a radius of $0.48$ cm and a resistance of $0.3125$ ohms. If another wire made from the same metal has a length of $500$ m and a resistance of $0.5$ ohms.

Formula used:

For varies directly and inversely:

  $R=K\frac{l}{r^2}$

Calculation:

Consider that, An $800$ m wire has a radius of $0.48$ cm and a resistance of $0.3125$ ohms. If another wire made from the same metal has a length of $500$ m and a resistance of $0.5$ ohms

Recall that varies directly and inversely:

  $R=K\frac{l}{r^2}$

The values are:

  $R=0.3125,l=800,r=0.48$

Therefore, the value of $k$ is,

  $\begin{array}{c}R=K\frac{l}{r^2}\\ 0.3125=K\frac{800}{{0.48}^2}\\ K=\frac{0.3125\times {0.48}^2}{800}\\ K=\frac{0.3125\times 0.2304}{800}\end{array}$

So simplified further the value of $k$ is,

  $\begin{array}{l}K=\frac{0.3125\times 0.2304}{800}\\ K=0.00009\end{array}$

Now to find the value of $r$ ,

  $R_2=K\frac{l_2}{r_2{}^2}$

The values are:

  $R=0.5,K=0.00009,l=500$

Hence, when:

  $\begin{array}{c}{R}_2=K\frac{l_2}{r_2{}^2}\\ 0.5=0.00009\times \frac{500}{r_2{}^2}\\ 0.5r_2{}^2=0.00009\times 500\\ r_2{}^2=\frac{0.00009\times 500}{0.5}\\ r_2{}^2=0.09\end{array}$

So,

  $\begin{array}{c}{r}_2{}^2=\sqrt{0.09}\\ r_2=0.3\end{array}$

Thus, The radius is: $0.3$ cm.