Question

Solid sphere of radius r is getting cooled by radiation rate of cooling nd radius

Answer 1

What is the value of N if a solid sphere of radius R is getting cooled by radiation, and the rate of cooling is directly proportional to R^n? Is it 2, or (-1)? Can you explain it if it is (-1)?If that's the complete question here's the answer--

Since the heat loss is through radiation, the best we have is Stefan-Boltzmann law which states - heat energy radiated per unit surface area per unit time of an object is proportional to fourth power of it's absolute temperature.

According to Stefan-Boltzmann law

QAt=eσT4

Where emissivity, e=1 for a perfectly blackbody which we will can assume in our case. σ is Stefan-Boltzmann constant.

Surface area of sphere, A=4πR2, therefore

Qt=4πR2σT4 J/sec

dQdt=4πR2σT4 J/sec

…… eqn 1

This gives heat energy radiated per unit time, or emission power of the sphere. If we can convert energy radiated per unit time to change(fall) in temperature per unit time, then we have our rate of cooling.

The relation between heat energy absorbed/emitted, and rise/fall in temperature is through specific heat capacity of the material. It is defined as the quantity of heat energy required per unit mass of the substance to increase it's temperature by 1 ℃.

Let C be the specific heat capacity of the material our sphere is made from. This means it takes C joules of heat energy to increase temperature of 1 kg of this material by 1 °K. So to change temperature by T °K by the mass m=ρ43πR3 of our sphere, it will have to emit mCT joule of energy. Therefore

Q=mCT

Differentiating both sides w.r.t time,

dQdt=mCdTdt

dQdt=ρ43πR3CdTdt

Substitute this is eqn 1 above -

ρ43πR3CdTdt=4πR2σT4

dTdt=3σT4ρRC

dTdt=3σT4R−1ρC

This gives rate of cooling, and it has turned out proportional to −1 power of R. So N=−1.