Question

In a biprism experiment, the distance between \(4^{th}\) and \(13^{th}\) bright band on the same side is '\(y\)' when light of wavelength \(6000\) Å is used. For a light of wavelength '\(λ\)' the distance between \(6^{th}\) and \(16^{th}\) bright band on the same side is again '\(y\)'. The value of '\(λ\)' in Å units is

• A. \(6500\)
• B. \(6300\)
• C. \(5400\)
• D. \(4800\)

Hint:

Fringe width is proportional to wavelength; count the number of fringe widths.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
In a biprism experiment the fringe width is \(\beta = \frac{\lambda D}{d}\), proportional to \(\lambda\) when \(D\) and \(d\) are fixed.

Step 2: Count the fringe widths:
The distance between the 4th and 13th bright bands on the same side is \(13 - 4 = 9\) fringe widths, so \(y = 9\beta_1\).
The distance between the 6th and 16th bright bands is \(16 - 6 = 10\) fringe widths, so \(y = 10\beta_2\).

Step 3: Equate:
\(9\lambda_1 = 10\lambda\), so
\[ \lambda = \frac{9\times6000}{10} = 5400\ \text{\AA} \]

Step 4: Why the other options are wrong.
6500 and 6300 \(\text{\AA}\) are larger than 6000 \(\text{\AA}\), but more fringes within the same distance mean narrower fringes, so \(\lambda\) must be smaller. 4800 \(\text{\AA}\) would correspond to a ratio \(\frac{9}{10}\) mistaken for \(\frac{4}{5}\).

Final Answer:
The wavelength is 5400 angstrom, option (C). \[ \boxed{5400\ \text{\AA}} \]