Derive Newton's law of cooling to show that the rate of loss of heat from the body is proportional to the temperature difference between the body and its surroundings.

Explanation

Newtion.s law of cooling :
When a hot body is cooled in air, the rate of loss of heat from the body is proportional to the temperature difference between the body and its surroundings.
Rate of loss of heat `alpha` temperature difference
Let at any instant temperature of the body is T and `T_(0)` is the temperature of the surroundings then rate of heat loss
`" "(-dQ)/(dt) propto DeltaT`
`" "(-dQ)/(dt) propto (T-T_(0))[because DeltaT=(T-T_(0))]`
`" "(-dQ)/(dt)=k(T-T_(0))" ...(1)"`
where, k is a constant
Consider a body of mass m and specific heat capacity s.
`because " "dQ=msDeltaT" ....(2)"`
By equations (1) and (2),
`" "(-msdT)/(dt)=k(T-T_(0))`
`" "(dt)/(T-T_(0))=(-k)/(ms)dt=-Kdt" ...(3)"`
where `" "K=k/(ms)`
`therefore " "(-dT)/(dt)=K(T-T_(0))`
Therefore rate of loss of heat is proportional to temperature difference between the body and its surroundings.