Chapter 18, Problem 15P

To determine

The diameter of the tungsten wire.

Answer to Problem 15P

Yes, a $2.5mm$ diameter coper wire has the same resistance as a tungsten wire of the same length. The numerical detail: $4.56mm$

Explanation of Solution

Given info:

Diameter of copper wire, $D_{Cu}=2.5mm$

Formula used:

It is known that resistance:

  $R=\frac{\rho L}{A}$

Where,

  $\rho$ is the resistivity.

  $L$ is the length of wire

  $A$ is the cross-sectional area of the wire.

Calculation:

Formula of resistivity, $R=\frac{\rho L}{A}$

Resistance of the wires is equal. So, $R_W=R_{Cu}$ . Where, $W$ is the chemical symbol of tungsten.

Now, Replace the formula above for the expressions for resistivity,

  ${\rho }_W\frac{L}{A_w}={\rho }_{Cu}\frac{L}{A_{Cu}}$

Now, divide both sides by $L$ (since their lengths are equal).

  $\frac{{\rho }_W}{A_w}=\frac{{\rho }_{Cu}}{A_{Cu}}$

  ${\rho }_W{A}_{Cu}={\rho }_{Cu}{A}_w\left(Bycrossmulti.\right)$

Use that the cross-sectional area of a wire is $A=\pi {\left(\frac{D}{2}\right)}^2=\pi \frac{D^2}{4}$ . Where, $D$ is the diameter.

Now, solve the diameter of the tungsten wire by substituting given values in the formula:

  $D_w=\sqrt{\frac{{\rho }_W}{{\rho }_{Cu}}}{D}_{Cu}$

  $D_w=\sqrt{\frac{\left(5.6\times {10}^{-8}\Omega \cdot m\right)}{\left(1.68\times {10}^{-8}\Omega \cdot m\right)}}\left(2.5mm\right)=4.56mm$

Conclusion:

They can, if the diameter of the tungsten wire is $4.56mm$ .