A solid copper sphere of density `rho` specific heat capacity `C` and radius `r` is initially at 200K. It is suspended inside a chamber whose walls are at 0 K. The time required (in `mu` s) for the temperature of the sphere to drop to 100 K is
(`sigma` is Stefan's constant and all the quantities are in SI units.)

Options

A. `48 (r rhoC)/sigma`
B. `1/48 (r rhoC)/sigma`
C. `27/7 (r rhoC)/sigma`
D. `7/27 (r rhoC)/sigma`

Answer

B

Explanation

Here, `T= 200 K and T_(0) = 0K`
As the rate of fall of temperature,
`(DeltaT)/(Deltat) = (sigmaA e(T^(4) - T_(0)^(4)))/(ms)`
where,` sigma = ` Stefan.s constant, A = area of sphere, and `e = ` emissivity = 1 and S = specific heat capacity
So `t=(msDeltaT)/(sigma A(T^(4) - T_(0)^(4))) " " (because T_(0) = 0K)`
`rArr t = ((rhoV)C(200K-100K))/(sigma (A) (200^(4)-0^(4)))`
`rArr t = (rho4/3 pir^(3) C xx 100)/(sigma4pir^(2) xx (200)^(4))`
`rArr t = 1/48 (rrho C)/sigma xx 10^(-6) s=1/48 (rhorC)/sigma mus`
Hence, the correct option is (b)