Transfer of Heat — Hard Practice Quiz

A Physics cheat sheet for Transfer of Heat — every key formula with its symbols defined — plus a hard-level practice quiz to test recall.

Formulas & key concepts

Conduction: Heat \(Q\) conducted in time \(t\) through a material of thermal conductivity \(k\), area \(A\), and length \(L\), with temperature difference \(\Delta T\).

$$Q = \frac{(kA \Delta T)t}{L}$$

Where: \(k\) = thermal conductivity

Stefan-Boltzmann Law (Radiation): Radiant energy \(Q\) emitted in time \(t\) by an object with emissivity \(e\), area \(A\), and temperature \(T\).

$$Q = e\sigma T^4 At$$

Where: \(\sigma\) = Stefan-Boltzmann constant \((5.67 \times 10^{-8} J/(s\cdot m^2\cdot K^4))\)

Net Radiant Power: Net power \(P_{net}\) emitted/absorbed by an object at temperature \(T\) in an environment at \(T_0\).

$$P_{net} = e\sigma A (T^4 - T_0^4)$$

Practice quiz

  1. A spherical object of radius $R$ and emissivity $e$ is maintained at a constant temperature $T$ in a vacuum, radiating heat to an environment at $T_0$. Simultaneously, heat is supplied to the object via a conductive rod of length $L$, cross-sectional area $A_{rod}$, and thermal conductivity $k$, with a temperature difference $\Delta T_{rod}$ across it. If the object is in steady state, which of the following expressions correctly represents the temperature difference $\Delta T_{rod}$ required to maintain the object's temperature $T$?

    • $\Delta T_{rod} = \frac{e\sigma (4\pi R^2) L (T^4 - T_0^4)}{k A_{rod}}$
    • $\Delta T_{rod} = \frac{e\sigma (\pi R^2) L (T^4 - T_0^4)}{k A_{rod}}$
    • $\Delta T_{rod} = \frac{k A_{rod} (T^4 - T_0^4)}{e\sigma (4\pi R^2) L}$
    • $\Delta T_{rod} = \frac{e\sigma (4\pi R^2) (T^4 - T_0^4)}{k A_{rod} L}$

    Answer: $\Delta T_{rod} = \frac{e\sigma (4\pi R^2) L (T^4 - T_0^4)}{k A_{rod}}$

  2. Two identical rods of the same material and length are used to conduct heat. Rod A has a circular cross-section of radius $r$. Rod B has a square cross-section of side length $s$. If both rods conduct heat at the same rate for the same temperature difference, what is the ratio of $r$ to $s$?

    • $\frac{r}{s} = \frac{1}{\sqrt{\pi}}$
    • $\frac{r}{s} = \sqrt{\pi}$
    • $\frac{r}{s} = \frac{1}{\pi}$
    • $\frac{r}{s} = \pi$

    Answer: $\frac{r}{s} = \frac{1}{\sqrt{\pi}}$

  3. An object at absolute temperature $T$ radiates heat to an environment at $T_0$. If the object's absolute temperature $T$ is doubled, and its emissivity $e$ is halved, how does the net radiant power $P_{net}$ change, assuming $T_0$ is much smaller than $T$?

    • $P_{net}$ increases by a factor of $8$.
    • $P_{net}$ increases by a factor of $4$.
    • $P_{net}$ remains unchanged.
    • $P_{net}$ decreases by a factor of $2$.

    Answer: $P_{net}$ increases by a factor of $8$.

  4. A flat plate of area $A$ and emissivity $e$ is heated by a conductive rod of length $L$, cross-sectional area $A_{rod}$, and thermal conductivity $k$. The rod maintains a temperature difference $\Delta T_{rod}$ across its ends, with one end at the plate's temperature $T$ and the other at a higher temperature. The plate radiates heat to an environment at $T_0$. If the plate is in thermal equilibrium, which of the following expressions correctly represents the plate's temperature $T$?

    • $T = \left( T_0^4 + \frac{k A_{rod} \Delta T_{rod}}{L e\sigma A} \right)^{1/4}$
    • $T = \left( T_0^4 - \frac{k A_{rod} \Delta T_{rod}}{L e\sigma A} \right)^{1/4}$
    • $T = \left( \frac{k A_{rod} \Delta T_{rod}}{L e\sigma A} \right)^{1/4}$
    • $T = T_0 + \left( \frac{k A_{rod} \Delta T_{rod}}{L e\sigma A} \right)^{1/4}$

    Answer: $T = \left( T_0^4 + \frac{k A_{rod} \Delta T_{rod}}{L e\sigma A} \right)^{1/4}$

  5. A wall of thickness $L$ and area $A$ conducts heat at a rate $P_1$ when the temperature difference across it is $\Delta T$. If the wall is replaced by two identical walls, each of thickness $L/2$ and area $A$, stacked one after another (in series), what is the new heat conduction rate $P_2$ for the same total temperature difference $\Delta T$ across the combined two walls?

    • $P_2 = P_1$
    • $P_2 = 2P_1$
    • $P_2 = P_1/2$
    • $P_2 = 4P_1$

    Answer: $P_2 = P_1$

  6. An object of surface area $A$, mass $m$, specific heat capacity $c$, and emissivity $e$ is at temperature $T$ and radiates heat into an environment at $T_0$. If this net energy loss causes a temperature drop $\Delta T_{obj}$ in the object, what is the expression for the time $t$ it takes for this temperature drop to occur?

    • $t = \frac{mc \Delta T_{obj}}{e\sigma A (T^4 - T_0^4)}$
    • $t = \frac{e\sigma A (T^4 - T_0^4)}{mc \Delta T_{obj}}$
    • $t = \frac{mc \Delta T_{obj}}{e\sigma A T^4}$
    • $t = \frac{mc \Delta T_{obj}}{A (T^4 - T_0^4)}$

    Answer: $t = \frac{mc \Delta T_{obj}}{e\sigma A (T^4 - T_0^4)}$

  7. A small, perfectly black ($e=1$) sphere of radius $r$ is placed in a vacuum chamber whose walls are maintained at a constant temperature $T_w$. Heat is supplied to the sphere by a heater at a constant rate $P_{heater}$. Which of the following expressions correctly represents the equilibrium temperature $T_{sphere}$ of the sphere?

    • $T_{sphere} = \left( T_w^4 + \frac{P_{heater}}{4\pi r^2 \sigma} \right)^{1/4}$
    • $T_{sphere} = \left( T_w^4 - \frac{P_{heater}}{4\pi r^2 \sigma} \right)^{1/4}$
    • $T_{sphere} = \frac{P_{heater}}{4\pi r^2 \sigma} + T_w$
    • $T_{sphere} = \left( \frac{P_{heater}}{4\pi r^2 \sigma} \right)^{1/4}$

    Answer: $T_{sphere} = \left( T_w^4 + \frac{P_{heater}}{4\pi r^2 \sigma} \right)^{1/4}$

  8. A cylindrical rod of length $L$, radius $R$, and thermal conductivity $k_1$ conducts heat at a rate $P_1$ for a given temperature difference. A second rod, made of a material with thermal conductivity $k_2 = 2k_1$, has length $2L$ and radius $R/2$. What is the ratio of the heat conduction rate of the second rod ($P_2$) to the first rod ($P_1$) for the same temperature difference?

    • $\frac{P_2}{P_1} = \frac{1}{4}$
    • $\frac{P_2}{P_1} = \frac{1}{2}$
    • $\frac{P_2}{P_1} = 1$
    • $\frac{P_2}{P_1} = 2$

    Answer: $\frac{P_2}{P_1} = \frac{1}{4}$

  9. Two objects, A and B, have the same surface area $A$ and are at the same absolute temperature $T$. Object A has emissivity $e_A$ and is in an environment at $T_{0A}$. Object B has emissivity $e_B = 2e_A$ and is in an environment at $T_{0B} = T_{0A}/2$. What is the ratio of the net radiant power of object B to object A ($P_{net,B} / P_{net,A}$)? Assume $T_{0A}$ is not negligible compared to $T$.

    • $\frac{P_{net,B}}{P_{net,A}} = \frac{2(T^4 - T_{0A}^4/16)}{T^4 - T_{0A}^4}$
    • $\frac{P_{net,B}}{P_{net,A}} = \frac{2(T^4 - T_{0A}^4/2)}{T^4 - T_{0A}^4}$
    • $\frac{P_{net,B}}{P_{net,A}} = \frac{2(T^4 - T_{0A}^4)}{T^4 - T_{0A}^4/16}$
    • $\frac{P_{net,B}}{P_{net,A}} = 2$

    Answer: $\frac{P_{net,B}}{P_{net,A}} = \frac{2(T^4 - T_{0A}^4/16)}{T^4 - T_{0A}^4}$

  10. A metal sphere of radius $R$ and emissivity $e$ is heated internally by a constant power source $P_{source}$. It is surrounded by a thin insulating shell of thickness $L$ and thermal conductivity $k$. The outer surface of the insulating shell, with area $A_{shell}$, radiates heat to an environment at $T_0$. If the system is in steady state, and the temperature of the sphere is $T_{sphere}$ and the outer surface of the shell is $T_{shell}$, derive an expression for $T_{sphere}$ in terms of $T_{shell}$ and other given parameters.

    • $T_{sphere} = T_{shell} + \frac{L e\sigma (T_{shell}^4 - T_0^4)}{k}$
    • $T_{sphere} = T_{shell} - \frac{L e\sigma (T_{shell}^4 - T_0^4)}{k}$
    • $T_{sphere} = \frac{L e\sigma (T_{shell}^4 - T_0^4)}{k}$
    • $T_{sphere} = T_{shell} + \frac{k (T_{shell}^4 - T_0^4)}{L e\sigma}$

    Answer: $T_{sphere} = T_{shell} + \frac{L e\sigma (T_{shell}^4 - T_0^4)}{k}$

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