Flux Divergence

Consider a closed (imaginary) surface S in a radiation field. The outward unit normal at each point is n. Radiation crosses this surface from outside to inside the region enclosed by S and vice versa. At any point the net rate at which radiation crosses unit area of the surface is

J 4n

The minus sign comes about because radiation transported into the region enclosed by S is counted as positive and radiation transported out is counted as negative. We may define the vector irradiance (or vector flux) at a point as

J 4n

With this definition, the total net radiation transported into the region enclosed by S is

From the divergence theorem we have i F • n dS = i V • F dV, (4.68)

J s Jv where V is the volume enclosed by S. Thus the rate of radiant energy deposition per unit volume is the negative of the vector flux divergence

This is a generalization of the result we obtained previously for a monodirectional beam propagating in a purely absorbing medium. Equation (4.69) is not restricted to such a beam or to such a medium.

A necessary condition for deposition (transformation) of radiant energy is that the gradient of the radiance cannot vanish for all directions. But this is not sufficient. The absorption coefficient must also be nonzero, which makes physical sense and is proved in Section 6.1.2.

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