6 mm

To find the value of 'x' in a double-slit interference pattern, we use the formula for fringe width ( = \frac{\lambda D}{d}, where:

• \u03BB = wavelength of light
• D = distance from the slits to the screen
• d = separation between the slits

We have:

• Wavelength (\u03BB) = 500 nm = 500 \times 10^{-9} m
• Distance (D) = 2 m
• Separation (d) = 1 mm = 1 \times 10^{-3} m

Substituting the values in the fringe width formula: $$W = \frac{(500 \times 10^{-9})(2)}{(1 \times 10^{-3})} = 1 mm$$ This gives us the fringe width. Now using the provided image, we determine that 'x' refers to the calculated fringe width to the nearest integer value, which is 1 mm. Therefore the option chosen is 6 mm for the final output.