To determine the distance from the source where the intensity of the waves is equal to [tex]\(2I\)[/tex], let's consider the relationship between intensity and distance. The intensity of waves from a point source follows the inverse square law, which states:
[tex]\[ I \propto \frac{1}{d^2} \][/tex]
where [tex]\( I \)[/tex] is the intensity and [tex]\( d \)[/tex] is the distance from the source.
Given:
- At distance [tex]\( d \)[/tex], the intensity is [tex]\( I \)[/tex].
- We need to find the new distance [tex]\( d_{\text{new}} \)[/tex] where the intensity is [tex]\( 2I \)[/tex].
Using the inverse square law, the relationship between the initial and new intensities and distances can be described as:
[tex]\[ \frac{I_{\text{new}}}{I} = \left(\frac{d}{d_{\text{new}}}\right)^2 \][/tex]
Substitute the given intensities:
[tex]\[ \frac{2I}{I} = \left(\frac{d}{d_{\text{new}}}\right)^2 \][/tex]
This simplifies to:
[tex]\[ 2 = \left(\frac{d}{d_{\text{new}}}\right)^2 \][/tex]
Taking the square root of both sides:
[tex]\[ \sqrt{2} = \frac{d}{d_{\text{new}}} \][/tex]
Solving for [tex]\( d_{\text{new}} \)[/tex]:
[tex]\[ d_{\text{new}} = \frac{d}{\sqrt{2}} \][/tex]
Therefore, the distance from the source where the intensity is equal to [tex]\( 2I \)[/tex] is:
[tex]\[ \frac{d}{\sqrt{2}} \][/tex]
Hence, the correct answer is:
[tex]\[ \boxed{\frac{d}{\sqrt{2}}} \][/tex]
