Find the total time elapsed for a hollow copper sphere of inner radius 3 cm outer radius 6 cm, density ρ = 9 × 10 3 kg/m 3 , specific heat s = 4 × 10 3 J/kg K and emissivity e = 0.4 to cool from 727°C to 227°C when the surrounding temperature is 0 K.(for inner surface e = 1 Stefan’s constant σ = 5.6 × 10 -8 W/m 2 K 4 )
Text Solution
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(6.56 × 10 4 sec)
To find the total time elapsed for a hollow copper sphere to cool from 
to 
when the surrounding temperature is 
, we can follow these steps:
Convert temperatures to Kelvin
The temperatures given in Celsius need to be converted to Kelvin.
Calculate the volume of the hollow sphere
The volume 
of the hollow sphere can be calculated using the formula for the volume of a sphere:
Where:
Calculating the volume:
Calculate the mass of the hollow sphere
The mass 
of the hollow sphere can be calculated using the formula:
Where:
Calculating the mass:
Calculate the area of the outer surface
The area 
of the outer surface of the hollow sphere is given by:
Calculating the area:
Apply Stefan-Boltzmann Law
According to Stefan-Boltzmann Law, the rate of heat loss 
is given by:
Where:
Calculating the rate of heat loss:
Calculating 
:
Now substituting back:
Relate heat loss to temperature change
Using the relation 
:
Integrate to find time
Integrating from 
to 
:
This gives:
Substituting the values:
Calculating the final time:
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