Question

calculate how thick an absorber needs to be absorbed half of the incoming ligh

Answer 1

Question #62662 - Physics - Atomic and Nuclear Physics

Calculate how thick an absorber needs to be absorbed half of the incoming light.

Solution.

Attenuation coefficient of the volume of a material characterizes how easily it can be penetrated by a beam of light. A large attenuation coefficient means that the beam is quickly "attenuated" (weakened) as it passes through the medium, and a small attenuation coefficient means that the medium is relatively transparent to the beam. The SI unit of attenuation coefficient is the reciprocal metre (m⁻¹). It is defined as

$$\mu = - \frac {1}{\Phi} \frac {d \Phi}{d z},$$

where $\Phi$ is the radiant flux, $z$ is the path length of the beam. This differential equation gives a solution:

$$\Phi = \Phi_ {0} \mathrm {e} ^ {- \mu z}.$$

If a half of incoming light was absorbed, then

$$\Phi = \frac {1}{2} \Phi_ {0} \rightarrow \frac {1}{2} \Phi_ {0} = \Phi_ {0} \mathrm {e} ^ {- \mu z} \Rightarrow \frac {1}{2} = \mathrm {e} ^ {- \mu z} \Rightarrow \mu z = \ln 2 \Rightarrow z = \frac {\ln 2}{\mu}.$$

**Answer**: if absorber has an attenuation coefficient equals $\mu$, its thickness for absorbing half of the incoming light should be $z = \frac{\ln 2}{\mu}$.

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