Two beams of light having intensities I and 4I interfere to produce a fringe pattern on a screen. The phase difference between the beams is π/2 at point A and π at point B. Then the difference between the resultant intensities at A and B is

To find the difference between the resultant intensities at points A and B, we can use the concept of interference.

Interference occurs when two or more waves superpose or combine with each other. In this case, we have two beams of light with intensities I and 4I.

At point A, the phase difference between the beams is π/2. This means that one beam is shifted by π/2 relative to the other beam.

At point B, the phase difference between the beams is π. This means that one beam is shifted by π relative to the other beam.

To calculate the resultant intensity at point A, we can use the formula for the intensity of interference:

I_resultant_A = I1 + I2 + 2√(I1 * I2) * cos(Δφ_A)

where I1 and I2 are the intensities of the two beams, and Δφ_A is the phase difference at point A.

Substituting the given values, we have:

I_resultant_A = I + 4I + 2√(I * 4I) * cos(π/2)

Simplifying this expression, we get:

I_resultant_A = 5I + 4√(4I^2) * cos(π/2)

Since cos(π/2) = 0, the expression simplifies further:

I_resultant_A = 5I

Therefore, the resultant intensity at point A is 5I.

Similarly, we can calculate the resultant intensity at point B using the same formula:

I_resultant_B = I + 4I + 2√(I * 4I) * cos(π)

Simplifying this expression, we get:

I_resultant_B = 5I + 4√(4I^2) * cos(π)

Since cos(π) = -1, the expression simplifies further:

I_resultant_B = 5I - 4√(4I^2)

Therefore, the resultant intensity at point B is 5I - 4√(4I^2).

To find the difference between the resultant intensities at A and B, we subtract the intensity at B from the intensity at A:

Difference = I_resultant_A - I_resultant_B

Difference = (5I) - (5I - 4√(4I^2))

Simplifying this expression, we get:

Difference = 4√(4I^2)

Therefore, the difference between the resultant intensities at points A and B is 4√(4I^2).