Rate of Loss of Heat <img
Category : JEE Main & Advanced
(1) Rate of loss of heat (or initial rate of loss of heat) : If an ordinary body at temperature T is placed in an environment of temperature \[{{T}_{0}}({{T}_{0}} \[\Delta Q={{Q}_{\text{emission}}}-{{Q}_{\text{absorption}}}=A\varepsilon \,\sigma ({{T}^{4}}-T_{0}^{4})\] (2) Rate of loss of heat \[({{R}_{H}})=\frac{dQ}{dt}=A\varepsilon \,\sigma ({{T}^{4}}-T_{0}^{4})\] (i) If two bodies are made of same material, have same surface finish and are at the same initial temperature then \[\frac{dQ}{dt}\propto A\]\[\Rightarrow \]\[\frac{{{\left( \frac{dQ}{dt} \right)}_{1}}}{{{\left( \frac{dQ}{dt} \right)}_{2}}}=\frac{{{A}_{1}}}{{{A}_{2}}}\] (3) Initial rate of fall in temperature (Rate of cooling): If m is the body and c is the specific heat then \[\frac{dQ}{dt}=mc.\frac{dT}{dt}=mc\frac{d\theta }{dt}\] \[(\because \ Q=mc\,\Delta T\ \]and \[dT=d\theta )\] (i) Rate of cooling \[({{R}_{c}})=\frac{d\theta }{dt}=\frac{(dQ/dt)}{mc}\]\[=\frac{A\varepsilon \,\sigma }{mc}({{T}^{4}}-T_{0}^{4})\] \[=\frac{A\varepsilon \,\sigma }{V\rho \,c}({{T}^{4}}-T_{0}^{4})\]; where m = density \[(\rho )\times \]volume (V) (ii) for two bodies of the same material under identical environments, the ratio of their rate of cooling is \[\frac{{{({{R}_{c}})}_{1}}}{{{({{R}_{c}})}_{2}}}=\frac{{{A}_{1}}}{{{A}_{2}}}.\frac{{{V}_{2}}}{{{V}_{1}}}\] (4) Dependence of rate of cooling : When a body cools by radiation the rate of cooling depends on (i) Nature of radiating surface i.e. greater the emissivity, faster will be the cooling. (ii) Area of radiating surface, i.e. greater the area of radiating surface, faster will be the cooling. (iii) Mass of radiating body i.e. greater the mass of radiating body slower will be the cooling. (iv) Specific heat of radiating body i.e. greater the specific heat of radiating body slower will be cooling. (v) Temperature of radiating body i.e. greater the temperature of body faster will be cooling. (vi) Temperature of surrounding i.e. greater the temperature of surrounding slower will be cooling. Comparison of rate of heat loss \[({{R}_{H}})\] and rate of cooling \[({{R}_{C}})\] for different bodies
Body
Condition
Rate of heat loss \[{{R}_{H}}=\frac{dQ}{dt}\]
Rate of cooling \[{{R}_{c}}=\frac{dT}{dt}\]or\[\frac{d\theta }{dt}\]
Two solid sphere
\[T,\,{{T}_{0}},\,c,\,\rho \] are same
\[{{R}_{H}}\propto A\propto {{r}^{2}}\] Þ \[\frac{{{({{R}_{H}})}_{1}}}{{{({{R}_{H}})}_{2}}}=\frac{r_{1}^{2}}{r_{2}^{2}}\]
\[{{R}_{c}}\propto \frac{A}{V}\propto \] \[\propto \frac{{{r}^{2}}}{{{r}^{3}}}\propto \frac{1}{r}\]
Two solid sphere of diff. material
\[T,\,{{T}_{0}}-\] same
\[{{R}_{H}}\propto A\propto {{r}^{2}}\]
\[{{R}_{c}}\propto \frac{A}{V\rho \,c}\] \[\propto \frac{1}{r\rho \,c}\]
Different shape bodies like cube, sphere plate
\[T,\,{{T}_{0}},\,c,\,\rho -\] same
\[{{R}_{H}}\propto A\] \[{{A}_{\max }}\to \]Plate \[{{A}_{\min }}\to \]sphere
\[{{R}_{c}}\propto \frac{A}{V}\]
Bodies of different materials
\[T,\,{{T}_{0}},\,m,\,A\] are same but c diff.
\[{{R}_{H}}\to \] same for all. bodies
\[{{R}_{c}}\propto \frac{1}{c}\]
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