A point source S emits unpolarized light uniformly in all directions. At two points A and B, the ratio 𝑟 = 𝐼_𝐴/𝐼_𝐵 of the intensities of light is 2. If a set of two polaroids having 45° angle between their pass-axes is placed just before point B, then the new value of 𝑟 will be _____.

Let the initial intensity of the unpolarized light at point A be $I_A$ and at point B be $I_B$.

We are given that the initial ratio of the intensities at A and B is:
$r=\frac{I_A}{I_B}=2$
This means $I_A=2I_B$.

Now, a set of two polaroids is placed just before point B. Let's analyze how the light intensity changes as it passes through these polaroids:
1. The light incident on the first polaroid is unpolarized light of intensity $I_B$. When unpolarized light of intensity $I_B$ passes through the first polaroid, it becomes linearly polarized, and its intensity is reduced by half. Thus, the intensity after the first polaroid is:
$I_1=\frac{I_B}{2}$
2. The angle between the transmission axes (pass-axes) of the two polaroids is given as $\theta ={45}^{\circ }$. According to Malus's Law, when polarized light of intensity $I_1$ passes through the second polaroid, the transmitted intensity $I_B^{\prime }$ is given by:
$I_B^{\prime }=I_1{\cos }^2\left(\theta \right)$

Substituting the values of $I_1$ and $\theta ={45}^{\circ }$:
$I_B^{\prime }=\left(\frac{I_B}{2}\right){\cos }^2\left({45}^{\circ }\right)$
Since $\cos \left({45}^{\circ }\right)=\frac{1}{\sqrt{2}}$, we have:
${\cos }^2\left({45}^{\circ }\right)={\left(\frac{1}{\sqrt{2}}\right)}^2=\frac{1}{2}$
Therefore, the new intensity at point B is:
$I_B^{\prime }=\frac{I_B}{2}·\frac{1}{2}=\frac{I_B}{4}$

The intensity at point A remains unchanged, so $I_A^{\prime }=I_A$.
The new ratio of the intensities $r^{\prime }$ will be:
$r^{\prime }=\frac{I_A^{\prime }}{I_B^{\prime }}=\frac{I_A}{I_B/4}=4\left(\frac{I_A}{I_B}\right)$

Since the initial ratio $\frac{I_A}{I_B}=2$:
$r^{\prime }=4·2=8$

Thus, the new value of the ratio is 8.