Fyneman exercise II ch2-1

Q.

A copper wire of radius $a$ has a concentric insulating sheath with outer radius $b$. The wire carries an electric current that raises its temperature to $T_1$ while the outside of the insulation remains at $T_2$ near room temperature.

a)

What is $\nabla T$ inside the insulation? Give answer in terms of $a$, $b$, $T_1$, $T_2$.

b)

How big is the temperature difference $(T_1-T_2)$ if a current of 20 amps is sent through No. 10 gauage copper wire which is covered with a layer of rubber 0.2 cm thick whose thermal conductivity is $1.6\times 10^{-3} {watt \over cm ^\circ C}$?

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Trial.

a)

$\nabla T = {\Delta T \over \Delta s} = {T_2 - T_1 \over b-a}\tag{1}$

b)

Copper wire generate heat by Joule heating^[1]
$P=I^2R\tag{2}$

Let length of copper wire $l$, and cross-sectional area $A$, then

$R=\rho {l \over A}\tag{3}$
, where $\rho$ is electrical resistivity^[2].

From eq(2) and (3)
$P=I^2\rho {l \over A}\tag{4}$

Radius is $a$,
$A=\pi a^2\tag{5}$

Putting eq.(5) to eq(4) gives

$P={I^2\rho l \over \pi a^2}\tag{6}$

This heat trasfered throgh rubber layer, will be

$-P \propto {\Delta T \over \Delta R}\cdot S \tag{7}$
, where $S$ is area heat transfered, $\Delta R=b-a$(“-” means lose energy).

Suppose $b-a << l$, then
$S=2\pi al\tag{8}$

If proportionality constant is $\kappa$, from (7) & (8)

$-P=\kappa {\Delta T \over b-a} \cdot 2\pi al\tag{9}$

Using eq(6), (9),

$-\Delta T = {I^2\rho l \over \pi a^2}{b-a\over \kappa 2\pi al}\tag{10}$

$T_1-T_2={I^2\rho (b-a) \over \kappa 2\pi^2 a^3}\tag{11}$

mean	values
No. 10 gauage wire diameter[3]	$2.588\times 10^{-1}\ cm$
Electrical resistivity[2] of copper[4]	$16.78\times10^{-9}\Omega\cdot m$ (at 20 °C)

Using above table and eq.(11)

$T_1-T2 = {100 A^2\times 1.678\times10^{-6}\Omega\cdot cm\times 0.2cm \over 1.6\times 10^{-3}{watt \over cm^\circ C}\times 2 \times 3.14^2\times 1.294^3\times 10^{-3}cm^3}=0.491 ^\circ C$