Question

When X-rays of wavelength $0.5A$pass through $7mm$ thick aluminium sheet, then their intensity reduces to one fourth. The coefficient of absorption of aluminium for these X-rays will be given as,
$\begin{align}
  & A.0.198m{{m}^{-1}} \\
 & B.0.227m{{m}^{-1}} \\
 & C.0.752m{{m}^{-1}} \\
 & D.0.539m{{m}^{-1}} \\
\end{align}$

Answer 1

Hint: The new intensity of the x rays incident will be equal to the old intensity of falling multiplied by the exponential of the product of coefficient of absorption and thickness of the aluminium sheet. This equation should be used here in order to find the answer. The equation is to be rearranged and all the values have to be substituted. Hope these all may help you to solve this question.

Complete step by step answer:
The equation for the intensity of the incident X-ray falling is given as,
$I={{I}_{0}}{{e}^{\mu x}}$
Where $I$be the new intensity of the radiation which is mentioned in the question as the one by fourth of old intensity. This can be written as,
$I=\dfrac{{{I}_{0}}}{4}$
${{I}_{0}}$ be the intensity of the X-ray falling at the initial condition or the old condition.
$x$ be the thickness of an aluminium sheet. The value of the thickness is given as,
$x=7mm$
$\mu $be the coefficient of absorption we want to find.
These all values has to be substituted in the above cited equation,
$\begin{align}
  & I={{I}_{0}}{{e}^{\mu x}} \\
 & \dfrac{{{I}_{0}}}{4}={{I}_{0}}{{e}^{\mu \times 7}} \\
\end{align}$
Rearranging the equation will give the coefficient of absorption.
That is,
$\begin{align}
  & 7\mu =\ln \dfrac{I}{{{I}_{0}}}=\ln \dfrac{1}{4}=\ln \left( 0.25 \right) \\
 & \mu =\ln \left( 0.25 \right)\times \dfrac{1}{7}=0.198m{{m}^{-1}} \\
\end{align}$
Therefore the value of coefficient of absorption has been obtained.

So, the correct answer is “Option A”.

Note: The absorption coefficient is described as the measure of how far into an object the light of a specific wavelength can penetrate before it is getting absorbed. In a substance with a low absorption coefficient, light absorbed will be poor. If the surface is thin, it will be looking transparent to that wavelength. The absorption coefficient depends on the substance and also on the wavelength of light which is used for absorption.