TITLE.PM5

840 ENGINEERING THERMODYNAMICS

dharm

\M-therm\Th15-4.pm5

This law holds true for more real substances ; there is however some deviation in the case of

a metallic conductor where the product (λmax.T) is found to vary with absolute temperature. It is

used in predicting a very high temperature through measurement of wavelength.

A combination of Planck’s law and Wien’s displacement law yields the condition for the maxi-

mum monochromatic emissive power for a black body.

() ()

exp exp

max max

max

E C

C

T

T

λb

λ

λ

=

L

N

M

O

Q

P−

=

× ×

F

HG

I

KJ

×

×

L

N

M

M

O

Q

P

P

−

−

− −

−

−

−

1 5

2

15 3

5

1

010 898 10

1

.374 2.

1.4388 10

2.898 10

2

3

or ()Eλbmax= 1.285 × 10–5 T^5 W/m^2 per metre wavelength ...(15.81)

15.5.9. Intensity of Radiation and Lambert’s Consine Law

15.5.9.1. Intensity of Radiation

When a surface element emits radiation, all of it will be intercepted by a hemispherical sur-

face placed over the element. The intensity of radiation (I) is defined as the rate of energy leaving

a surface in a given direction per unit solid angle per unit area of the emitting surface normal to the

mean direction in space. A solid angle is defined as a portion of the space inside a sphere enclosed by

a conical surface with the vertex of the cone at the centre of the sphere. It is measured by the ratio of

the spherical surface enclosed by the cone to the square of the radius of the sphere ; it unit is

steradian (sr). The solid angle subtended by the complete hemisphere is given by :

2 2

2

πr
r
= 2π.
Fig. 15.51 (a) shows a small black surface of area dA (emitter) emitting radiation in different
directions. A black body radiation collector through which the radiation pass is located at an angu-
lar position characterised by zenith angle θ towards the surface normal and angle φ of a spherical
coordinate system. Further the collector subtends a solid angle dω when viewed from a point on the
emitter.
Let us now consider radiation from the elementary area dA 1 at the centre of a sphere as
shown in Fig. 15.51. Suppose this radiation is absorbed by a second elemental area dA 2 , a portion of
the hemispherical surface.
The projected area of dA 1 on a plane perpendicular to the line joining dA 1 and dA 2 = dA 1 cos θ.

θ

In

I = I cosθ n θ

Radiation

collector

Normal Emitted radiation

Black surface

emitter

dA

φ

dω

θ

(a) Special distribution of radiations emitted from a surface.