Chapter 8.2, Problem 19P

(a) Consider a light beam traveling downward into the ocean. As the beam progresses, it is partially absorbed and its intensity decreases. The rate at which the intensity is decreasing with depth at any point is proportional to the intensity at that depth. The proportionality constant $\mu$ is called the linear absorption coefficient. Show that if the intensity at the surface is $I_0,$ the intensity at a distance $s$ below the surface is $I=I_0{e}^{-\mu s}.$ The linear absorption coefficient for water is of the order of ${10}^{-2}{\text{ft}}^{-1}$ (the exact value depending on the wavelength of the light and the impurities in the water). For this value of $\mu ,$ find the intensity as a fraction of the surface intensity at a depth of $1\text{ft}$, $50\text{ft},500\text{ft},1$ mile. When the intensity of a light beam has been reduced to half its surface intensity $\left(I=\frac{1}{2}{I}_0\right),$ the distance the light has penetrated into the absorbing substance is called the half-value thickness of the substance. Find the half-value thickness in terms of $\mu .$ Find the half-value thickness for water for the value of $\mu$ given above.

(b) Note that the differential equation and its solution in this problem are mathematically the same as those in Example 1, although the physical problem and the terminology are different. In discussing radioactive decay, we call $\lambda$ the decay constant, and we define the half-life $T$ of a radioactive substance as the time when $N=\frac{1}{2}{N}_0$ (compare half-value thickness). Find the relation between $\lambda$ and $T$.