Current Affairs JEE Main & Advanced

(1) A motion, which repeat itself over and over again after a regular interval of time is called a periodic motion. Revolution of earth around the sun (period one year), rotation of earth about its polar axis (period one day), Motion of hour's hand of a clock (period 12-hour) etc are common example of periodic motion. (2) Oscillatory or vibratory motion is that motion in which a body moves to and fro or back and forth repeatedly about a fixed point in a definite interval of time. In such a motion, the body is confined with in well-defined limits on either side of mean position. Oscillatory motion is also called as harmonic motion. (i) Common examples are (a) The motion of the pendulum of a wall clock (b) The motion of a load attached to a spring, when it is pulled and then released. (c) The motion of liquid contained in U-tube when it is compressed once in one limb and left to itself. (d) A loaded piece of wood floating over the surface of a liquid when pressed down and then released executes oscillatory motion. (ii) Harmonic oscillation is that oscillation which can be expressed in terms of single harmonic function (i.e. sine or cosine function). Example : \[y=a\sin \omega \,t\] or \[y=a\cos \omega \,t\] (iii) Non-harmonic oscillation is that oscillation which can not be expressed in terms of single harmonic function. It is a combination of two or more than two harmonic oscillations. Example : \[y=a\sin \omega \,t+b\sin 2\omega \,t\] 

If R is the radius of the sun and T its temperature, then the energy emitted by the sun per sec through radiation in accordance with Stefan's law will be given by \[P=A\sigma {{T}^{4}}=4\pi {{R}^{2}}\sigma {{T}^{4}}\] In reaching earth this energy will spread over a sphere of radius r (= average distance between sun and earth); so the intensity of solar radiation at the surface of earth (called solar constant S) will be given by \[S=\frac{P}{4\pi {{r}^{2}}}=\frac{4\pi {{R}^{2}}\sigma {{T}^{4}}}{4\pi {{r}^{2}}}\] i.e.\[T={{\left[ {{\left( \frac{r}{R} \right)}^{2}}\frac{S}{\sigma } \right]}^{1/4}}\]  \[={{\left[ {{\left( \frac{1.5\times {{10}^{8}}}{7\times {{10}^{5}}} \right)}^{2}}\times \frac{1.4\times {{10}^{3}}}{5.67\times {{10}^{-8}}} \right]}^{1/4}}\tilde{-}5800\,K\] As \[r=1.5\times {{10}^{8}}\,km,\] \[R=7\times {{10}^{5}}\,km,\] \[S=2\frac{cal}{c{{m}^{2}}min}=1.4\frac{kW}{{{m}^{2}}}\] and \[\sigma =5.67\times {{10}^{-8}}\frac{W}{{{m}^{2}}{{K}^{4}}}\] This result is in good agreement with the experimental value of temperature of sun, i.e., 6000 K.

(1) The theoretical explanation of black body radiation was done by Planck. (2) According to Plank's atoms of the walls of a uniform temperature enclosure behave as oscillators, each with a characteristic frequency of oscillation. (3) These oscillations emits electromagnetic radiations in the form of photons (The radiation coming out from a small hole in the enclosure are called black body radiation). The energy of each photon is hn. Where n  is the frequency of oscillator and h is the Plank's constant. Thus emitted energies may be hn, 2hn, 3hn ... nhn but not in between. According to Planck's law \[{{E}_{\lambda }}d\lambda =\frac{8\pi hc}{{{\lambda }^{5}}}\,\frac{1}{[{{e}^{hc/\lambda KT}}-1]}\,d\lambda \] where c = speed of light and k = Boltzmann's constant. This equation is known as Plank's radiation law. It is correct and complete law of radiation (4) This law is valid for radiations of all wavelengths ranging from zero to infinite. (5) For radiations of short wavelength \[\left( \lambda \frac{hc}{KT} \right)\] Planck's law reduces to Rayleigh-Jeans energy distribution law \[{{E}_{\lambda }}d\lambda =\frac{8\pi KT}{{{\lambda }^{4}}}d\lambda \]  

According to Wien's law the product of wavelength corresponding to maximum intensity of radiation and temperature of body (in Kelvin) is constant, i.e. \[{{\lambda }_{m}}T=b=\text{constant}\] where b is Wien's constant and has value \[2.89\times {{10}^{-3}}m\text{-}K\]. As the temperature of the body increases, the wavelength at which the spectral intensity \[({{E}_{\lambda }})\]is maximum shifts towards left. Therefore it is also called Wien's displacement law. This law is of great importance in 'Astrophysic' as through the analysis of radiations coming from a distant star, by finding \[{{\lambda }_{m}}\] the temperature of the star \[T(=b/{{\lambda }_{m}})\] is determined.

A perfectly black body emits radiation of all possible wavelength. Langley and later on Lummer and Pringsheim investigated the distribution of energy amongst the different wavelengths in the thermal spectrum of a black body radiation. The results obtained are shown in figure. From these curves it is clear that (1) At a given temperature energy is not uniformly distributed among different wavelengths. (2) At a given temperature intensity of heat radiation increases with wavelength, reaches a maximum at a particular wavelength and with further increase in wavelength it decreases. (3) For all wavelengths an increase in temperature causes an increase in intensity. (4) The area under the curve will represent the total intensity of radiation at a particular temperature i.e. Area \[=E=\int{{{E}_{\lambda }}d\lambda }\] From Stefan's law \[E=\sigma {{T}^{4}}\Rightarrow \] Area under \[{{E}_{\lambda }}-\lambda \] curve (A) \[\propto {{T}^{4}}\] (5) The energy \[({{E}_{\max }})\] emitted corresponding to the wavelength of maximum emission \[({{\lambda }_{m}})\] increases with fifth power of the absolute temperature of the black body i.e.,  \[{{E}_{\max }}\propto {{T}^{5}}\]

If volume, radiating surface area, nature of surface, initial temperature and surrounding of water and given liquid are equal and they are allowed to cool down (by radiation) then rate of loss of heat and fall in temperature of both will be same. i.e.   \[{{\left( \frac{dQ}{dt} \right)}_{\text{water}}}={{\left( \frac{dQ}{dt} \right)}_{\text{liquid}}}\] \[({{m}_{W}}{{c}_{W}}+W)\frac{({{\theta }_{1}}-{{\theta }_{2}})}{{{t}_{1}}}=({{m}_{l}}{{c}_{l}}+W)\frac{({{\theta }_{1}}-{{\theta }_{2}})}{{{t}_{2}}}\] or \[\left[ \frac{{{m}_{W}}{{c}_{W}}+W}{{{t}_{1}}} \right]=\left[ \frac{{{m}_{l}}{{c}_{l}}+W}{{{t}_{2}}} \right]\] \[W={{m}_{c}}{{c}_{c}}=\] Water equivalent of calorimeter, where \[{{m}_{c}}\] and  \[{{c}_{c}}\] are mass and specific heat of calorimeter. If density of water and liquid is \[\rho \] and \[\rho '\] respectively then \[{{m}_{W}}=V{{\rho }_{W}}\] and \[{{m}_{l}}=V\rho {{\,}_{l}}\] Specific heat of liquid \[{{c}_{l}}=\frac{1}{{{m}_{l}}}\left[ \frac{{{t}_{l}}}{{{t}_{W}}}({{m}_{W}}{{c}_{W}}+W)-W \right]\]

(1) Curve between \[log(\theta -{{\theta }_{0}})\] and time As \[\frac{d\theta }{dt}\propto -(\theta -{{\theta }_{0}})\Rightarrow \,\frac{d\theta }{(\theta -{{\theta }_{0}})}=-Kdt\] Integrating \[{{\log }_{e}}(\theta -{{\theta }_{0}})=-Kt+C\] \[{{\log }_{e}}(\theta -{{\theta }_{0}})=-Kt+{{\log }_{e}}A\] This is a straight line with negative slope (2) Curve between temperature of body and time As  \[{{\log }_{e}}(\theta -{{\theta }_{0}})=-Kt+{{\log }_{e}}A\]\[\Rightarrow \]\[{{\log }_{e}}\frac{\theta -{{\theta }_{0}}}{A}=-Kt\] \[\Rightarrow \] \[\theta -{{\theta }_{0}}=A{{e}^{-kt}}\] which indicates temperature decreases exponentially with increasing time. (3) Curve between the rate of cooling (R) and body temperature \[(\theta )\].   \[R=K(\theta -{{\theta }_{0}})=K\theta -K{{\theta }_{0}}\] This is a straight line intercept R-axis at \[-K{{\theta }_{0}}\] (4) Curve between rate of cooling (R) and temperature difference between  body \[(\theta )\] and surrounding \[({{\theta }_{0}})\]  \[R\propto (\theta -{{\theta }_{0}})\]. This is a straight line passing through origin.  

When the temperature difference between the body and its surrounding is not very large i.e. \[T-{{T}_{0}}=\Delta T\]then \[{{T}^{4}}-T_{0}^{4}\] may be approximated as \[4T_{0}^{3}\Delta T\] By Stefan?s law, \[\frac{dT}{dt}=\frac{A\varepsilon \sigma }{mc}[{{T}^{4}}-T_{0}^{4}]\] Hence \[\frac{dT}{dt}=\frac{A\varepsilon \sigma }{mc}4T_{0}^{3}\Delta T\]\[\Rightarrow \]\[\frac{dT}{dt}\propto \Delta T\] or  \[\frac{d\theta }{dt}\propto \theta -{{\theta }_{0}}\] i.e., if the temperature of body is not very different from surrounding, rate of cooling is proportional to temperature difference between the body and its surrounding. This law is called Newton's law of cooling. (1) Greater the temperature difference between body and its surrounding greater will be the rate of cooling. (2) If \[\theta ={{\theta }_{0}}\], \[\frac{d\theta }{dt}=0\] i.e. a body can never be cooled to a temperature lesser than its surrounding by radiation. (3) If a body cools by radiation from \[\theta _{1}^{o}C\] to \[\theta _{2}^{o}C\] in time t, then \[\frac{d\theta }{dt}=\frac{{{\theta }_{1}}-{{\theta }_{2}}}{t}\] and \[\theta ={{\theta }_{av}}=\frac{{{\theta }_{1}}+{{\theta }_{2}}}{2}\]. The Newton's law of cooling becomes \[\left[ \frac{{{\theta }_{1}}-{{\theta }_{2}}}{t} \right]=K\left[ \frac{{{\theta }_{1}}+{{\theta }_{2}}}{2}-{{\theta }_{0}} \right]\]. This form of law helps in solving numericals. (4) Practical examples (i) Hot water loses heat in smaller duration as compared to moderate warm water. (ii) Adding milk in hot tea reduces the rate of cooling.  

(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	\[{{R}_{c}}\propto \frac{1}{c}\]

 

According to it the radiant energy emitted by a perfectly black body per unit area per sec (i.e. emissive power of black body) is directly proportional to the fourth power of its absolute temperature, i.e. \[E\propto {{T}^{4}}\] \[\Rightarrow \]\[E=\sigma {{T}^{4}}\] where \[\sigma \]  is a constant called Stefan's constant having dimension \[[M{{T}^{-3}}{{\theta }^{-4}}]\] and value \[5.67\times {{10}^{-8}}W/{{m}^{2}}{{K}^{4}}\] (i) For ordinary body : \[e=\varepsilon E=\varepsilon \sigma {{T}^{4}}\] (ii) Radiant energy : If Q is the total energy radiated by the ordinary body then \[e=\frac{Q}{A\times t}=\varepsilon \sigma {{T}^{4}}\] \[\Rightarrow \]\[Q=A\,\varepsilon \sigma {{T}^{4}}t\] (iii) Radiant power (P) : It is defined as energy radiated per unit area i.e. \[P=\frac{Q}{t}=A\varepsilon \sigma {{T}^{4}}\]. (iv) If an ordinary body at temperature T is surrounded by a body at temperature T0, then Stefan's law may be put as \[e=\varepsilon \,\sigma \,({{T}^{4}}-T_{0}^{4})\]