Animated Solution for Physics - Atoms and Nuclei: The intensity of gamma radiation from a given source is $I$. On passing through $36\text{ mm}$ of lead, it is reduced to $I/8$. The thickness of lead, which will reduce the intensity to $I/2$ will be

Imagine a beam of high-energy gamma rays striking a thick, dense block of lead. As these photons travel through the material, they collide with atoms and get absorbed or scattered. This means the intensity of the beam decreases as it penetrates deeper.

But this decrease isn't linear. The probability of a photon being absorbed in any thin layer is proportional to the number of photons present. This constant fractional decrease leads to an exponential decay, governed by the beautiful attenuation law:

Here, $I$ is the initial intensity, $I^{\prime }$ is the transmitted intensity, $x$ is the thickness of the material, and $\mu$ is the absorption coefficient, which tells us how effectively the material (like lead) stops the radiation.

The problem gives us a perfect scenario to find our unknowns. We are told that when the thickness $x=36\text{ mm}$, the intensity is reduced to one-eighth of its original value, meaning $I^{\prime }=\frac{I}{8}$.

Let's plug this into our master equation:

We can immediately cancel the initial intensity $I$ from both sides. This leaves us with:

To make this equation easier to handle, let's express $\frac{1}{8}$ as a power of $2$. We know that $\frac{1}{8}=2^{-3}$.

Taking the natural logarithm ($\ln$) on both sides allows us to bring down the exponents:

Let's hold onto this result. Now, we need to find the new thickness, let's call it $x^{\prime }$, that reduces the intensity to exactly half, so $I^{\prime }=\frac{I}{2}$. We set up the equation again:

Canceling $I$ and writing $\frac{1}{2}$ as $2^{-1}$, we get:

Taking the natural logarithm once more yields:

Now we have two elegant equations: 1. $3\ln 2=36\mu$ 2. $\ln 2=\mu x^{\prime }$

We can simply substitute the value of $\ln 2$ from the second equation into the first equation:

Notice how the absorption coefficient $\mu$ beautifully cancels out from both sides! We don't even need to know its exact value. We are left with a simple linear equation:

Dividing by $3$, we find our final answer:

A Quick Intuitive Check: If $12\text{ mm}$ halves the intensity (this is the "half-value layer"), then passing through another $12\text{ mm}$ (total $24\text{ mm}$) will halve it again to $\frac{1}{4}$. Passing through a third $12\text{ mm}$ layer (total $36\text{ mm}$) will halve it once more to $\frac{1}{8}$. The math perfectly aligns with our physical intuition!