10.5.2 Derivation of the bsinθ Formula (Beyond the Syllabus)

The single slit pattern is the result of superposition of an infinite number of rays propagating from an infinite number of points along the slit width. The outcome obviously depends on the phase difference among the rays.

Notice that the path difference between the two rays from the edge of the slit (labelled as A and C in the diagrams) is \delta =b\sin \theta , b being the slit width. Consider the case when b\sin \theta =\lambda . You may be thinking of C.I. occurring between A and C since their path difference is l. Or you may be thinking of D.I. occurring between A and B since their path difference is \displaystyle \frac{\lambda }{2}.  And what about all the other rays? You may be thinking, “what a mess”.

Surprisingly, there is a simple way to sort out this mess. In our minds, we can split the slit into two halves. The top half sends out rays A1 to AN while the second half send out rays B1 to BN. If A1 and B1 are in anti-phase with each other, so are A2 and B2, A3 and B3, and every pair of rays until AN and BN. Pairing up the rays this way makes it clear that all the rays will superpose to zero when A and B are in antiphase, which occurs when A and C are in-phase. So we conclude that a complete D.I. occurs when b\sin \theta =\lambda .

If we now mentally divide the single slit into 2 slits of width \displaystyle \frac{b}{2} each, it is easy to realize that the next complete D.I. occurs when \displaystyle \frac{b}{2}\sin \theta =\lambda .

We can also mentally divide the single slit into 3 slits of width \displaystyle \frac{b}{3} each, and realize that complete D.I. also occurs when \displaystyle \frac{b}{3}\sin \theta =\lambda .

In fact, there is nothing to stop us from mentally dividing the single slit of width b into n number of slits of width \displaystyle \frac{b}{n}, and conclude that complete D.I. occurs when

\displaystyle \frac{b}{n}\sin \theta =\lambda , n=1,2...

Video Explanation   

The bsinθ Formula

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s