# If the intensity of $X-$rays becomes $\dfrac{I}{3}$ from $I$ after travelling $3.5\text{ cm}$ inside a target, then it’s intensity after travelling next $7\text{ cm}$ will be

A. $\dfrac{I}{6}$

B. $\dfrac{I}{12}$

C. $\dfrac{I}{9}$

D. $\dfrac{I}{27}$

Answer

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**Hint:**To solve these types of questions we need to know about the relationship between initial intensity, final intensity, absorption coefficient and the distance travelled by $X-$rays. In this question we are not given the value of absorption coefficient of the $X-$rays, hence first we will have to find that and then proceed further.

**Formula used:**

$I={{I}_{0}}{{e}^{-\mu x}}$

Here $I$ is the final intensity of $X-$rays after travelling a distance $x$.

${{I}_{0}}$ is the initial intensity of the $X-$rays.

And $\mu $ is the absorption coefficient of the $X-$rays.

**Complete step-by-step solution:**

We know that the final intensity of $X-$rays after travelling a distance $x$ is given by the following formula:

$I={{I}_{0}}{{e}^{-\mu x}}$

In the question, the value of absorption coefficient is not given. Hence our first task would be to calculate the value of absorption coefficient. We know that the intensity of $X-$rays becomes $\dfrac{I}{3}$ from $I$ after travelling $3.5\text{ cm}$ inside a target, hence on substituting the values in the formula we get:

$\begin{align}

& \dfrac{I}{3}=I{{e}^{-3.5\mu }} \\

& \Rightarrow \dfrac{1}{3}={{e}^{-3.5\mu }} \\

& \Rightarrow {{e}^{3.5\mu }}=3 \\

& \Rightarrow 3.5\mu =\ln 3 \\

& \Rightarrow \mu =\dfrac{\ln 3}{3.5} \\

& \Rightarrow \mu =0.31389\text{ c}{{\text{m}}^{-1}} \\

\end{align}$

Now that we have calculated the value of the absorption coefficient, we can the intensity of $X-$rays after travelling next $7\text{ cm}$. The total distance that the $X-$rays would travel will be:

$\begin{align}

& x=\left( 3.5+7 \right)\text{ cm} \\

& \Rightarrow \text{x=10}\text{.5 cm} \\

\end{align}$

On substituting the values in the formula, we get:

$\begin{align}

& I'=I{{e}^{-0.31389\times 10.5}} \\

& \Rightarrow I'=I{{e}^{-3.2958}} \\

& \therefore I'=\dfrac{I}{27} \\

\end{align}$

The intensity of $X-$rays after travelling next $7\text{ cm}$will be $\dfrac{I}{27}$. Hence, the correct option is $D$.

**Note:**To solve these types of questions we need to remember the relation between final intensity and initial intensity that depends on various factors like distance travelled by $X-$rays and the absorption coefficient. The final intensity is not affected by any other factors.

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