Question

The intensity of gamma radiation from a given source is I. On passing through 36mm of lead, intensity is reduced to ${\displaystyle \frac{I}{8}}$. The thickness of lead which reduces the intensity to ${\displaystyle \frac{I}{2}}$ is
A 6 mm
B 9 mm
C 18 mm
D 12 mm

Answer 1

Hint: Intensity of a ray passing through ‘x’ thickness of lead is given by I=IO $e^{-\mu x}$. Where IO is the intensity from the source, x is the thickness of lead.

Complete step by step answer:
Given, intensity of gamma radiation from the source, IO= I
We know that the intensity of a ray passing through ‘x’ thickness of lead is given by I=IO $e^{-\mu x}$
Given that when the thickness of lead i.e. x=36 mm, the intensity i.e. I=${\displaystyle \frac{I}{8}}$
Substituting the values given in the equation, we get, ${\displaystyle \frac{1}{8}}=I{e}^{-\mu 36}$
Taking logarithm on both sides we get,
${\log }_e{\displaystyle \frac{1}{8}}={{\log }_e}^{-\mu 36}$
Using the values ${\log }_e{\displaystyle \frac{1}{8}}=-2.079$ we get,
$-2.079=-\mu 36$
$⟹\mu ={\displaystyle \frac{2.079}{36}}$
$⟹\mu =0.0578$
Now let the thickness of lead box be x so that intensity becomes, I= ${\displaystyle \frac{I}{2}}$
Substituting the value of I and $\mu$ in the equation we get,
${\displaystyle \frac{I}{2}}=I{e}^{-0.0578x}$
Then the above equation becomes, ${\displaystyle \frac{1}{2}}=e^{-0.0578x}$
Taking logarithm of both sides, ${\log }_e{\displaystyle \frac{1}{2}}=-0.0578x$
We know that ${\log }_e{\displaystyle \frac{1}{2}}=-0.693$
Then the equation becomes, -0.693=-0.0578x
$x={\displaystyle \frac{0.693}{0.0578}}$
$\therefore x=12mm.$

So, the correct answer is “Option D”.

Additional Information:
The penetration power of gamma radiation is very high. They can easily pass through the body and thus pose a formidable radiation protection challenge, requiring shielding made from dense materials such as lead or concrete.

Note:
Logarithm value of common terms should be learnt by the students to solve such types of questions. Students should be familiar with some common results of logarithm. For example-${\log }_{e}e=1$, ${\log }_e{e}^x=x{\log }_{e}e=x$ etc.