Question

Two spherical balls $A$ and $B$ made of the same material, are heated to the same temperature. They are then placed in identical surroundings. If the diameter of $A$ is twice that of $B$, ratio of the rates of cooling of $A$ and $B$ will be

• A. $1:1$
• B. $2:1$
• C. $4:1$
• D. $1:4$

Answer 1

The correct option is D $1:4$

Given, sphere $A,B$ are of some martial

Also heated to same temperature

given that $d_A=2d_B$ ------- $\left(1\right)$

($d=$ diameter of sphere)

we know that

Rate of cooling = $\sigma eA{T}^4$ (Stepons law of cooling)

Since $A,B$ are of same material $\Rightarrow \sigma A=\sigma B$

Also given $T_A=T_B$.

$\Rightarrow Rateofcoding\alpha A\left(Area\right)$

$\Rightarrow \frac{RateofcoolingofA}{RateofcoolingofB}=\frac{\left(Areaeffective\right)A}{\left(Areaeffective\right)B}=\frac{r^2}{r^2}=\frac{d^2}{d^2}$

$\Rightarrow \frac{Q_A}{Q_B}={\left(\frac{d_A}{d_B}\right)}^2={\left(\frac{2d_B}{d_B}\right)}^2=\frac{4}{1}$

$Q_A:Q_B=4:1$