(b) State Kirchhoff's law. From Stefan's law derive Newton's law of cooling. [6]

Kirchhoff's Laws:

1. Kirchhoff's Current Law (KCL): The algebraic sum of currents entering a node (junction) is zero. ΣI_in = ΣI_out
2. Kirchhoff's Voltage Law (KVL): The algebraic sum of voltages around any closed loop in a circuit is zero. ΣV = 0

Derivation of Newton's Law of Cooling from Stefan's Law:

Stefan's Law: P = εσAT⁴ where P is power radiated, ε is emissivity, σ is Stefan-Boltzmann constant, A is surface area, and T is absolute temperature.

Consider an object at temperature T cooling in surroundings at temperature T₀. The net power radiated is:

P_net = εσA(T⁴ - T₀⁴)

If T ≈ T₀, then we can approximate (using binomial expansion or Taylor series):

T⁴ - T₀⁴ ≈ 4T₀³(T - T₀)

Substituting this into the net power equation gives:

P_net ≈ 4εσAT₀³(T - T₀)

Since power is related to the rate of heat loss (dQ/dt) and heat loss is related to temperature change (dQ = mc dT), we get (assuming constant specific heat capacity 'c' and mass 'm'):

mc(dT/dt) = -4εσAT₀³(T - T₀)

This simplifies to Newton's Law of Cooling:

dT/dt = -k(T - T₀) where k = (4εσAT₀³)/(mc)