A rod of length $L$ with sides fully insulated is of a material whose thermal conductivity varies with $\alpha$ temperature as $ K= \frac{\alpha }{T}$, where $\alpha$ is a constant. The ends of the rod are kept at temperature $T_1$ and $T_2$. The temperature $T$ at $x,$ where $x$ is the distance from the end whose temperature is $T_1$ is
A rod of length $L$ with sides fully insulated is of a material whose thermal conductivity varies with $\alpha$ temperature as $ K= \frac{\alpha }{T}$, where $\alpha$ is a constant. The ends of the rod are kept at temperature $T_1$ and $T_2$. The temperature $T$ at $x,$ where $x$ is the distance from the end whose temperature is $T_1$ is
- A
${T_1}{\left( {\frac{{{T_2}}}{{{T_1}}}} \right)^{\frac{x}{L}}}$
- B
$\frac{x}{L}\ln \frac{{{T_2}}}{{{T_1}}}$
- C
${T_1}{e^{\frac{{{T_2}x}}{{{T_1}L}}}}$
- D
${T_1} + \frac{{{T_2} - {T_1}}}{L}x$
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