# Stefan’s Law

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#### Stefan’s Law:

• The heat energy radiated per unit time per unit area of a perfectly black body is directly proportional to the fourth power of its absolute temperature.
• Explanation: Let Eb, the heat radiated per unit time per unit area of a perfectly black body whose absolute temperature is T.

So by Stefan’s Law,

Eb ∝ T4

Eb   = σ  T4

where σ  is a constant known as Stefan’s constant.

The value of σ in S.I. system is 5.67 × 10-8 Jm-2 K-4s-1. or 5.67 x 10-8 Wm-2 K-4

The value of σ  in c.g.s system is 5.67 × 10-5 erg cm-2-4s-1.

Dimensions of σ are [M1L0T-3K-4]

#### Expression for the Rate of Loss of Heat to the Surrounding:

• Let T be the absolute temperature of a perfectly black body. Let To be the absolute temperature of the surrounding.

So by Stefan’s Law,

Heat radiated per unit time per unit area of a perfectly black body   = σ  T4

Let A be the surface area of the perfectly black body. Then,

Heat lost by the body  per   unit   time = A σ T4

where σ is a constant known as Stefan’s constant.

Heat   received  from the surrounding per unit time = A σ To4

Net rate of loss of heat = A σ T4 – A σ To4

= A σ(  T4 – To4)

This is an expression for the rate of loss of heat to the surrounding.

#### Newton’s Law of Cooling:

• The rate of loss of heat by a body is directly proportional to its excess temperature over that of the surroundings provided that this excess is small.
• Explanation: Let θ and θo, be the temperature of a body and its surroundings respectively. Let dQ / dt be the rate of loss of heat, So from Newton’s Law of Cooling, where k is a constant.

#### Alternate Statement:

• By Newton’s law of cooling, mathematically Where, θ andθ, are the temperature of the body and its surroundings respectively and

dQ / dt is the rate of. loss of heat. K is constant.

Let ‘m’ be the mass of the body, c be its specific heat. • Thus, the rate of fall of a temperature of a body is directly proportional to its excess temperature over that of the surroundings.

#### Derivation of Newton’s Law of Cooling from Stefan’s Law:

• Let us consider a body whose surface area is A having absolute temperature T and kept in the surrounding having absolute temperature To.  Let e be the emissivity (or coefficient of emission) of the surface of the body.

Let ( T  -To) =  x,  where  x  is Small.

∴ T   =  To  +    x.

Let dQ/ dt be the rate of loss of heat by the body. We know that

E / Eb = e

∴  E  = e Eb

Where E & Eb, are the emissive powers of the body and perfectly black body respectively.

Using Stefan’s Law we know that for a perfectly black body rate of loss of heat =  Aσ(  T4   –  To4 )

Therefore, for given body, As  x /To is small so higher powers of  x /To will be very small and hence those terms can be neglected. • This is Newton’s Law of cooling i.e. the rate of loss of heat of a body is directly proportional to its excess temperature over the surroundings provided the excess is small. Thus Newton’s Law of Cooling is derived (or deduced) from Stefan’s Law.

#### Limitations of Newton’s Law of Cooling:

• This law is applicable when the excess temperature of a body over the surroundings is very small (about 40OC)
• When body is cooling the temperature of the surrounding is assumed to be constant. which is not true.
• The law is applicable for higher temperature using forced convection.

#### Solar Constant:

• The solar constant is the rate at which solar radiant energy is intercepted by the earth per unit area at the outer limits of earth’s atmosphere at the earth-sun mean distance.
• The solar constant, S = 1353 W/m².

#### Calculation of Surface Temperature of the Sun:

• The central portion of the sun is very hot. It has a temperature of 107 K. It can be estimated using concepts of nuclear reactions.
• The outer surface of the sun is comparatively cooler this region is called the photosphere. Its temperature can be estimated using solar constant.
• Let T be the absolute temperature of the surface of the sun. Let Rs be its radius. By Stefan’s law the total power radiated per second is given by Where σ = Stefan’s constant

Let r be the earth-sun mean distance. r = 1.496 × 10¹¹ m,

Now the energy radiated by sun is distributed over a sphere of surface area 4πr²

By definition of solar constant Science > You are Here