**Q231. A wire of length 1 m and radius 10**

^{(-3)}m is carrying a heavy current and is assumed to radiate as a black body. At equilibrium, its temperature is 900 K while that of surrounding is 300 K. The resistivity of the material of the wire at 300 K is Ï€^{2}×10^{(-8)}ohm m and its temperature coefficient of resistance is 7.8×10^{(-3)}/°C (Stefan’s constant Ïƒ = 5.68 ×10^{(-8)}W/m^{2}K^{4})The resistivity of wire at 900 K is nearly

**Q232. A solid aluminium sphere and a solid lead sphere of same radius are heated to the same temperature and allowed to cool under identical surrounding temperatures. The specific heat capacity of aluminium=900 J/kg°C and that of lead = 130 J/kg°C. The density of lead =10**

^{4}kg/m^{3}and that of aluminum =2.7×10^{3}kg/m^{3}. Assume that the emissivity of both the spheres is the sameThe ratio of rate of fall of temperature of the aluminium sphere to the rate of heat loss from the lead sphere is

232(a) Heat emitted by the surface of sphere per unit time P_r'=ÏƒT

^{4}(Ï€R

^{2}) Since the radius of both spheres is equal, the rate of heat loss by aluminium sphere = rate of heat loss by lead sphere

**Q233. Assume that the thermal conductivity of copper is twice that of aluminium and four times that of brass. Three metal rods made of copper, aluminium and brass are each 15 cm long and 2 cm in diameter. These rods are placed end to end, with aluminium between the other two. The free ends of the copper and brass rods are maintained at 100°C and 0°C, respectively. The system is allowed to reach the steady state condition. Assume there is no loss of heat anywhere**

When steady state condition is reached everywhere which of the following statement is true?

233(d) Under steady state conditions, the temperatures at all section in the system remain constant and maintain a constant temperature gradient for a given material. The temperature gradient in copper, aluminium and brass will not be same however, the rate of heat conducted across all sections whether in copper or aluminium or brass will be the same

**Q234. A thin copper rod of uniform cross section A square metres and of length L metres has a spherical metal sphere of radius r metre at its one end symmetrically attached to the copper rod. The thermal conductivity of copper is Îµ. The free end of the copper rod is maintained at the temperature T kelvin by supplying thermal energy form a P watt source. Steady state conditions are allowed to be established while the rod is properly insulated against heat loss from its lateral surface. Surroundings are at 0°C. Stefan’s constant =Ïƒ W/m**

^{2}K^{4}After the steady state conditions are reached, the temperature of the spherical end rod, T_S is

234 (d) The distance from the end A of the copper rod (where power is supplied) and centre O of the metal sphere is (L+r). Hence, in the steady state conditions, P=(KA(T-T_s))/(L+r) Which gives T_S=T-(P(L+r))/KA (The heat is transmitted up to centre of the spherical end, and the sphere loses energy by radiation out of its spherical surface)

**Q235. An immersion heater, in an insulated vessel of negligible heat capacity brings 10 g of water to the boiling point from 16°C in 7 min. Then**

Power of heater is nearly

235 (b) In 7 min, temperature of 100 g of water is raised by (1000-16°C)=84°C. The amount of the heat provided by heater Q_W=C_W m_W ∆T=(1 cal/g°C)(100g)(84°C) =8.4×10

^{3}cal=(8.4×10

^{3}×4.186) J ≃3.5×10

^{4}J Power of heater=Q_w/t_1 =(8.4×10

^{3}cal)/((7×60) s)=20 cal/s =(20×4.18) J/s=83.6 W≈84 W

**Q236. A body of area 0.8×10**

^{(-2)}m^{2}and mass 5×10^{(-4)}kg directly faces the sum on a clear day. The body has an emissivity of 0.8 and specific heat of 0.8cal/kg K. The surrounding are at 27°C. (solar constant =1.4 kW/m^{2})The rate of rise of the body’s temperature is nearly

236 (b) We consider the leaf to be a black body. The rate of energy radiated at any instant (dQ/dt)_e=ÏƒeAT_0^4 (i) If m is the mass of the body and C is its specific heat then the heat gained dQ/dt=mc dT/dt (ii) Since intensity of sun beam is uniform, power incident on the body is SA heat absorbed (dQ/dt)_ab=SA e Rate of increases of temperature =((net power absorbed))/(thermal capacity of body) =((SAÎ±-ÏƒAeT

^{4}))/mc Given Ïƒ=5.67×10

^{(-8)}J/s m

^{2}K

^{4}A=0.8×10

^{(-2)}m

^{2};Î±=e=0.8 S=1.4×10

^{3}W/m

^{2}T=300 K m=5×10

^{(-4)}kg C=0.8 kcal/kg K=0.8×4.2 kJ/kg K dT/dt=([1.4×10

^{3}×0.8×10

^{(-2)}×0.8-5.67×10

^{(-8)}×0.8×10

^{(-2)}×0.8×(300)

^{4}])/(5×10

^{(-4)}×(0.8×4.2×10

^{3})) =((8.96-2.9))/1.68=3.6°C/s=3.6 K/s

**Q237. A copper collar is to fit tightly about a steel shaft that has a diameter of 6 cm at 20°C. The inside diameter of the copper collar at that temperature is 5.98 cm**

To what temperature must the copper collar be raised so that it will just slip on the steel shaft, assuming the steel shaft remains at 20°C?(Î±_copper=17×10^{(-6)}/K)

237 (c) l_1=l_0 (1+Î±t) (2Ï€×6)/2=l_0 (1+Î±t) (2Ï€×5.98)/2=l_0 (1+20Î±) 6/5.98=(1+Î±t)/(1+20Î±)=((1+17×10

^{(-6)}t))/(1+20×17×10

^{(-6)}) ∴t=216.8°C≈217°C

**Q238. Two insulated metal bars each of length 5 cm and rectangular cross section with sides 2 cm and 3 cm are wedged between two walls, one held at 100° C and the other at 0° C. The bars are made of lead and silver.〖 K〗_pb=350 W/mK,K_Ag=425 W/mK**

Thermal current through lead bar is

238(b) Resistance of lead bar R_Pb=l_Pb/(K_Pb A_Pb )=(5×10

^{(-2)})/(350×6×10

^{(-4)}) =10/21=5/21 K/W Thermal current through lead bar I_Pb=∆T/R_Pb =(100×21)/5=420 W