Show that in interference the maximum intensity pattern is four times the intensity due to each slit if amplitude of light emerging from slits is same

To understand how the maximum intensity in an interference pattern can be four times the intensity due to each individual slit, we need to delve into the principles of wave interference and the relationship between amplitude and intensity.

The Basics of Interference

When two coherent light waves meet, they can interfere with each other. This interference can be constructive or destructive, depending on the phase difference between the waves. In the case of constructive interference, the waves add together, leading to an increase in amplitude.

Amplitude and Intensity Relationship

The intensity \(I\) of a wave is proportional to the square of its amplitude \(A\). Mathematically, this relationship is expressed as:

I = kA²

where \(k\) is a constant that depends on the medium and other factors. This means that if we double the amplitude, the intensity increases by a factor of four, since:

(2A)² = 4A²

Interference from Two Slits

Now, let’s consider a scenario where we have two slits, each emitting light waves with the same amplitude \(A\). When these two waves meet at a point on a screen, the total amplitude \(A_{total}\) at that point during constructive interference is:

A_{total} = A + A = 2A

Calculating Maximum Intensity

Using the relationship between amplitude and intensity, we can find the maximum intensity \(I_{max}\) at the point of constructive interference:

I_{max} = k(A_{total})² = k(2A)² = k(4A²) = 4kA²

Now, the intensity due to each individual slit, denoted as \(I_{single}\), is:

I_{single} = kA²

Comparing Intensities

To find the relationship between the maximum intensity and the intensity from a single slit, we can express \(I_{max}\) in terms of \(I_{single}\):

I_{max} = 4I_{single}

This shows that the maximum intensity observed in the interference pattern is indeed four times the intensity due to each slit when the amplitudes of light emerging from the slits are the same.

Visualizing the Concept

Think of it like a crowd cheering at a concert. If two groups of fans (representing the two slits) cheer independently, their individual cheers create a certain sound level (intensity). However, when they cheer together in perfect harmony (constructive interference), their combined cheer is much louder—specifically, four times as loud if they are both cheering at the same volume (amplitude).

In summary, the maximum intensity in an interference pattern is four times the intensity from each slit due to the way amplitudes combine during constructive interference. This relationship highlights the fascinating nature of wave behavior and how it can lead to dramatic changes in observed intensity.