Question

A coherent beam of light strikes a single slit and produces a spread-out diffraction pattern beyond. The number of photons detected per unit time a detector in the very center of the pattern isX. Now two more slits are opened nearby, the same width as the original. equally spaced on either side of it, and equally well. illuminated by the beam. How many photons will be detected per unit time at the center detector now: Why?

Short answer

The number of photons per unit time detected at the center detector is$9X$.

Step-by-step solution

Concept used

A low-intensity beam of light is sent toward a narrow single slit. On the far side, individual flashes are seen sporadically at detectors over a broad area that has an order of magnitude wider than the slit width. Two more slits equidistant from the given slit are opened on either side of the given slit. All the slits are equally illuminated (same magnitude of the electric field).

Diffraction Pattern

As the number of slits of equal width increases, the diffraction pattern becomes sharper and more intense. As the number of slits increases, the constructive and destructive interferences also increase with the maximum constructive interference occurring at the peak positions. The intensity of light in these peak positions is proportional to the number of photons that hit per unit of time. Also, the number of photons hitting per unit of time is proportional to the square of the sum of electric fields.

Number of Photons

For a single slit with an electric field E , the number of photons X is given as:

$$\begin{gathered} X\propto E^2 \\ X=k{E}^2 \end{gathered}$$

Where, K is the constant of proportionality.

Now, when there are three slits of equal illumination, the sum of electric fields is 3E.

So, the number of photons hitting per unit of time $X_1$ is given as:

$$\begin{gathered} X_1=k(3E{)}^2 \\ {X}_1=9k{E}^2 \\ {X}_1=9X \end{gathered}$$

Conclusion 

Therefore, the number of photons per unit time detected at the center detector is 9X when a total of three slits are used for observing the diffraction pattern.