A highly conducting solid sphere of radius `R` density `rho` and specific heat s is kept in an evacuted chamber. A parallel beam of thermal radiation of instensity I is incident on its surfcae Consider the sphere to be a perfectly black body and its temperature at certain instant considered at `t =0` is `T_(0)` [Take Stefan's constant as `sigma`] Answer the following questions based on above information
The maximum attainable temperature of the sphere is .

Options

A. `((I)/(4sigma))^(½)`
B. `((I)/(2sigma))^(⅓)`
C. `((I)/(4sigma))^(¼)`
D. Never occurs

Answer

C

Explanation

The rate at which energy is absorbed by the sphere is `P_(1) =piR^(2) I` Let at any time t the temperature of sphere be `T` than rate at which heat is being radiated is `P_(2) = piR^(2) I - 4pi sigma R^(2) T^(4)`
`rArr ms (dT)/(dt) = piR^(2) [I - 4sigma T^(4)]`
`rArr(dT)/(I-4sigmaT^(4))=(piR^(2)dt)/(4/(3)piR^(3)rhos)=(3dt)/(4Rrhos)`
`int_(T_(0))^(T)(dT)/(I-4sigmaT^(4))=int_(0)^(t)(3dt)/(4Rrhos)`
Solving this we get `T` as function of t For maximum temp `(dT)/(dt) = 0 ` We get `T_(max) = ((I)/(4sigma)) ^(1//4)` .