(a) ![1.72\cdot 10^{-5} \Omega m](https://tex.z-dn.net/?f=1.72%5Ccdot%2010%5E%7B-5%7D%20%5COmega%20m)
The resistance of the rod is given by:
(1)
where
is the material resistivity
L = 1.20 m is the length of the rod
A is the cross-sectional area
The radius of the rod is half the diameter:
, so the cross-sectional area is
![A=\pi r^2=\pi (2.85\cdot 10^{-3} m)^2=2.55\cdot 10^{-5} m^2](https://tex.z-dn.net/?f=A%3D%5Cpi%20r%5E2%3D%5Cpi%20%282.85%5Ccdot%2010%5E%7B-3%7D%20m%29%5E2%3D2.55%5Ccdot%2010%5E%7B-5%7D%20m%5E2)
The resistance at 20°C can be found by using Ohm's law. In fact, we know:
- The voltage at this temperature is V = 15.0 V
- The current at this temperature is I = 18.6 A
So, the resistance is
![R=\frac{V}{I}=\frac{15.0 V}{18.6 A}=0.81 \Omega](https://tex.z-dn.net/?f=R%3D%5Cfrac%7BV%7D%7BI%7D%3D%5Cfrac%7B15.0%20V%7D%7B18.6%20A%7D%3D0.81%20%5COmega)
And now we can re-arrange the eq.(1) to solve for the resistivity:
![\rho=\frac{RA}{L}=\frac{(0.81 \Omega)(2.55\cdot 10^{-5} m^2)}{1.20 m}=1.72\cdot 10^{-5} \Omega m](https://tex.z-dn.net/?f=%5Crho%3D%5Cfrac%7BRA%7D%7BL%7D%3D%5Cfrac%7B%280.81%20%5COmega%29%282.55%5Ccdot%2010%5E%7B-5%7D%20m%5E2%29%7D%7B1.20%20m%7D%3D1.72%5Ccdot%2010%5E%7B-5%7D%20%5COmega%20m)
(b) ![8.57\cdot 10^{-4} /{\circ}C](https://tex.z-dn.net/?f=8.57%5Ccdot%2010%5E%7B-4%7D%20%2F%7B%5Ccirc%7DC)
First of all, let's find the new resistance of the wire at 92.0°C. In this case, the current is
I = 17.5 A
So the resistance is
![R=\frac{V}{I}=\frac{15.0 V}{17.5 A}=0.86 \Omega](https://tex.z-dn.net/?f=R%3D%5Cfrac%7BV%7D%7BI%7D%3D%5Cfrac%7B15.0%20V%7D%7B17.5%20A%7D%3D0.86%20%5COmega)
The equation that gives the change in resistance as a function of the temperature is
![R(T)=R_0 (1+\alpha(T-T_0))](https://tex.z-dn.net/?f=R%28T%29%3DR_0%20%281%2B%5Calpha%28T-T_0%29%29)
where
is the resistance at the new temperature (92.0°C)
is the resistance at the original temperature (20.0°C)
is the temperature coefficient of resistivity
![T=92^{\circ}C](https://tex.z-dn.net/?f=T%3D92%5E%7B%5Ccirc%7DC)
![T_0 = 20^{\circ}](https://tex.z-dn.net/?f=T_0%20%3D%2020%5E%7B%5Ccirc%7D)
Solving the formula for
, we find
![\alpha=\frac{\frac{R(T)}{R_0}-1}{T-T_0}=\frac{\frac{0.86 \Omega}{0.81 \Omega}-1}{92C-20C}=8.57\cdot 10^{-4} /{\circ}C](https://tex.z-dn.net/?f=%5Calpha%3D%5Cfrac%7B%5Cfrac%7BR%28T%29%7D%7BR_0%7D-1%7D%7BT-T_0%7D%3D%5Cfrac%7B%5Cfrac%7B0.86%20%5COmega%7D%7B0.81%20%5COmega%7D-1%7D%7B92C-20C%7D%3D8.57%5Ccdot%2010%5E%7B-4%7D%20%2F%7B%5Ccirc%7DC)