Two beams of light having intensities $$I$$ and $$4I$$ interfere to produce a fringe pattern on a screen. The phase difference between the two beams are $$\dfrac{\pi}{2}$$ and $$\dfrac{\pi}{3}$$ at points $$A$$ and $$B$$ respectively. The difference between the resultant intensities at the two points is $$xI$$. The value of $$x$$ will be ______.

Two beams of intensities $$I$$ and $$4I$$ interfere. We need to find the difference in resultant intensities at points $$A$$ (phase difference $$\pi/2$$) and $$B$$ (phase difference $$\pi/3$$).

The resultant intensity when two beams of intensities $$I_1$$ and $$I_2$$ interfere with phase difference $$\phi$$ is given by:

$$I_R = I_1 + I_2 + 2\sqrt{I_1 I_2}\cos\phi$$

Here $$I_1 = I$$ and $$I_2 = 4I$$, so:

$$I_R = I + 4I + 2\sqrt{I \times 4I}\cos\phi = 5I + 4I\cos\phi$$

At point A (phase difference $$\pi/2$$), the intensity is:

$$I_A = 5I + 4I\cos\frac{\pi}{2} = 5I + 4I(0) = 5I$$

At point B (phase difference $$\pi/3$$), the intensity is:

$$I_B = 5I + 4I\cos\frac{\pi}{3} = 5I + 4I\left(\frac{1}{2}\right) = 5I + 2I = 7I$$

The difference between these intensities is:

$$I_B - I_A = 7I - 5I = 2I$$

Since the difference is $$xI$$, we get $$x = 2$$.