GATE 2024 MT – Question 59
A long cylindrical metal rod of radius $r$, length $L\gg r$, and resistivity $\rho_e$ carries current $I$ in vacuum and loses heat only by radiation. Its steady temperature is ______ K (nearest integer). Given: Stefan–Boltzmann constant $\sigma=5.667\times10^{-8}$ W m$^{-2}$ K$^{-4}$; $r=0.1$ mm, $L=1$ m, $\rho_e=10^{-8}$ Ω m, $I=0.3$ A, ambient $T_0=300$ K. Ignore radiation from the flat ends and take emissivity 1.
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Correct answer: 305 to 308
Explanation
Joule heating is $I^2R=I^2\rho_eL/(\pi r^2)$.
Side-surface radiation is $2\pi rL\sigma(T^4-T_0^4)$.
Equating them and cancelling $L$ gives $T=[T_0^4+I^2\rho_e/(2\pi^2\sigma r^3)]^{1/4}$.
Use $r=10^{-4}$ m and the stated constants: $T\approx307.2$ K.
The answer is **307 K**.