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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 E_{b}, the heat radiated per unit time per unit area of a perfectly black body whose absolute temperature is T.

So by Stefan’s Law,

E_{b} ∝ T^{4}

E_{b} *= σ* T^{4}

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^{-4}s^{-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} C°^{-4}s^{-1}.

Dimensions of σ are [M^{1}L^{0}T^{-3}K^{-4}]

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

- Let T be the absolute temperature of a perfectly black body. Let T
_{o}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* T^{4}

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

Heat lost by the body per unit time = A σ T^{4}

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

Heat received from the surrounding per unit time = A σ T_{o}^{4}

Net rate of loss of heat = A σ T^{4} – A σ T_{o}^{4}

= A σ( T^{4} – T_{o}^{4})

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 θ_{o}*,*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_{o }*,* 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 T
_{o}. Let e be the emissivity (or coefficient of emission) of the surface of the body.

Let ( T -T_{o}) = x, where x is Small.

∴ T = T_{o} + x.

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

E / E_{b} = e

∴ E = e E_{b}

Where E & E**b**, 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( T^{4} – T_{o}^{4} )

Therefore, for given body,

As x /T_{o} is small so higher powers of x /T_{o} 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 10
^{7}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 > Physics > Radiation > You are Here |

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