Exercise 1.2.22

Question

The resistance $R$ of a wire of fixed length is related to its diameter $x$ by an inverse square law, that is, by a function of the form $R(x)= kx^{-2}.$
    \begin{enumerate}[label=(\alph*)]
        \item A wire of fixed length and 0.005 meters in diameter has a resistance of 140 ohms. Find the value of $k.$
        \item Find the resistance of a wire made of the same material and of the same length as the wire in part (a) but with a diameter of 0.008 meters.
    \end{enumerate}

Khagan Gulusoy · Accepted Answer

Recall that the unit of electrical resistance ohm is often denoted by $\Omega$ in physics.
\begin{enumerate}[label=(\alph*)]
    \item From the exercise, we have $x=0.005$ meters and $R(0.005)=140 \Omega$. Therefore, we have:
    $$140\Omega=k\cdot 0.005^{-2}\frac{1}{\mathrm{~m}^2}.$$
    $$140\Omega=40000 \frac{1}{\mathrm{~m}^2}\cdot k.$$
    $$k=0.0035\ \Omega\cdot m^2.$$

\item Since the wire is made from the same material and is the same length as the wire in (a), we have the same $k$ as in (a). Hence,
    $$R(0.008)=0.0035\ \Omega\cdot m^2\cdot 15625\frac{1}{\mathrm{~m}^2}.$$
    $$R(0.008)=54.6875\approx 54.7\Omega.$$
\end{enumerate}

Exercise 1.2.22

Answers

Comments

Exercise 1.2.22

Answers

Comments

Add answer