Question
One end of a thermally insulated rod is kept at a temperature $T_{1}$ and the other end at $T_{2}$. The rod is composed of two sections of length $l_{1}$ and $l_{2}$ and thermal conductivities $K_{1}$ and $K_{2}$, respectively. The temperature at the interface of the two sections is(A) $\left(K_{2} l_{2} T_{1}+K_{1} l_{1} T_{2}\right) /\left(K_{1} l_{1}+K_{2} l_{2}\right)$(B) $\left(K_{2} l_{1} T_{1}+K_{1} l_{2} T_{2}\right) /\left(K_{2} l_{1}+K_{1} l_{2}\right)$(C) $\left(K_{1} l_{2} T_{1}+K_{2} l_{1} T_{2}\right) /\left(K_{1} l_{2}+K_{2} l_{1}\right)$(D) $\left(K_{1} l_{1} T_{1}+K_{2} l_{2} T_{2}\right) /\left(K_{1} l_{1}+K_{2} l_{2}\right)$
Step 1
Since the two sections of the rod are in series, the rate of flow of heat in them is equal. Therefore, we can write the equation for the rate of heat flow as: \[K_{1}A\frac{T_{1}-T}{l_{1}} = K_{2}A\frac{T-T_{2}}{l_{2}}\] Show more…
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A rod of length $l$ (laterally thermally insulated) of uniform cross-sectional area $A$ consists of a material whose thermal conductivity varies with temperature as $K=\frac{K_{o}}{a+b T}$, where $K_{0}, a$ and $b$ are constants. $T_{1}$ and $T_{2}\left(<T_{1}\right)$ are the temperature of two ends of rod. Then rate of flow of heat across the rod is (A) $\frac{A K_{0}}{b l}\left(\frac{a+b T_{1}}{a+b T_{2}}\right)$ (B) $\frac{A K_{0}}{b l}\left(\frac{a+b T_{2}}{a+b T_{1}}\right)$ (C) $\frac{A K_{0}}{b l} \ln \left[\frac{a+b T_{1}}{a+b T_{2}}\right]$ (D) $\frac{A K_{0}}{a l} \ln \left[\frac{a+b T_{2}}{a+b T_{1}}\right]$
Two rods $P$ and $Q$ have equal lengths. Their thermal conductivities are $K_{1}$ and $K_{2}$ and cross-sectional areas are $A_{1}$ and $A_{2}$. When the temperature at ends of each rod are $T_{1}$ and $T_{2}$ respectively, the rate of flow of heat through $P$ and $Q$ will be equal, if (a) $\frac{A_{1}}{A_{2}}=\frac{K_{2}}{K_{1}}$ (b) $\frac{A_{1}}{A_{2}}=\frac{K_{2}}{K_{1}} \times \frac{T_{2}}{T_{1}}$ (c) $\frac{A_{1}}{A_{2}}=\sqrt{\frac{K_{1}}{K_{2}}}$ (d) $\frac{A_{1}}{A_{2}}=\left(\frac{K_{1}}{K_{1}}\right)^{2}$
Heat and Kinetic Theory of Gases
Round 1
"A rod of length I (laterally thermally insulated) of the uniform cross-sectional area A consists of a material whose thermal conductivity varies with temperature as K=K/a+bT , where Ko, a and b are constants. Tį and T2 (< Tı) are the temperatures of two ends of the rod. Then, the rate of flow of heat across the rod is"
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