Copper winding

Calculate the current flowing through the copper winding at an operating temperature of 70°C. Celsius, if the winding diameter is 1.128 mm and the winding length is 40 m. The winding is connected to 8V.

Correct result:

I =  9.5174 A

Solution:

$$\begin{gathered} U=8\text{ V } \\ D=1.128\text{ mm}\to \text{ m}=1.128\mathrm{/}1000\text{ m}=0.00113\text{ m } \\ l=40\text{ m } \\ {t}_1=20^{\circ }\text{C } \\ {t}_2=70^{\circ }\text{C } \\ t=t_2-t_1=70-20=50^{\circ }\text{C } \\ \alpha =0.004 \\ {\rho }_0=1.75\cdot 10^{-8}\doteq 1.75\cdot 10^{-8}\mathrm{\Omega } \\ \rho ={\rho }_0\cdot (1+\alpha \cdot t)=1.75\cdot 10^{-8}\cdot (1+0.004\cdot 50)\doteq 2.1\cdot 10^{-8}\mathrm{\Omega } \\ r=D\mathrm{/}2=0.0011\mathrm{/}2\doteq 0.0006\text{ m } \\ S=\pi \cdot r^2=3.1416\cdot 0.0006^2\doteq 9.9933\cdot 10^{-7}{\text{m}}^2 \\ R=\rho \cdot {\displaystyle \frac{l}{S}}=2.1\cdot 10^{-8}\cdot {\displaystyle \frac{40}{9.9933\cdot 10^{-7}}}\doteq 0.8406\mathrm{\Omega } \\ I={\displaystyle \frac{U}{R}}={\displaystyle \frac{8}{0.8406}}=9.5174\text{ A} \end{gathered}$$

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