A beam of light consisting of wavelengths \(650\,\text{nm}\) and \(550\,\text{nm}\) illuminates
Young’s double slits with separation \(d=2\,\text{mm}\) such that the interference fringes are formed on
a screen placed at a distance \(D=1.2\,\text{m}\) from the slits. The least distance from the central maximum,
where the bright fringes due to both wavelengths coincide, is ________ \(\times10^{-5}\,\text{m}\).
Given:
\[
\lambda_1=650\,\text{nm},\quad \lambda_2=550\,\text{nm},\quad
d=2\times10^{-3}\,\text{m},\quad D=1.2\,\text{m}
\]
Show Hint
For coincidence of bright fringes in YDSE, always equate \(n_1\lambda_1=n_2\lambda_2\) and choose the smallest integers.
To determine where the bright fringes due to both wavelengths coincide, we need to find the least order \(m\) such that the path difference for both wavelengths is an integer multiple of both the wavelengths (\(\lambda_1\) and \(\lambda_2\)). The condition for a bright fringe in Young’s double-slit experiment is given by: \(d\sin\theta = m\lambda\) where \(m\) is the order of the fringe. For small angles, \(\sin\theta \approx \tan\theta \approx \frac{x}{D}\). Thus: \(x = \frac{m\lambda D}{d}\) We want the least \(x\) such that: \(m_1\lambda_1 = m_2\lambda_2\) Finding the least common multiple for integers: \(\frac{m_1}{m_2} = \frac{\lambda_2}{\lambda_1} = \frac{550}{650} = \frac{11}{13}\) This suggests \(m_1 = 11k\) and \(m_2 = 13k\), where \(k\) is the smallest integer to match this ratio with integer orders. Now substitute in the condition for \(x\): \(x = \frac{m_1\lambda_1 D}{d} = \frac{11k \times 650 \times 10^{-9}\, \text{m} \times 1.2\, \text{m}}{2\times10^{-3}\, \text{m}}\) Simplifying, we get: \(x = \frac{11k \times 650 \times 1.2}{2} \times 10^{-6}\, \text{m}\) Substituting \(k=1\) (smallest integer), \(x = \frac{11 \times 650 \times 1.2}{2} \times 10^{-6} = \frac{8580}{2} \times 10^{-6} = 4290 \times 10^{-6}\, \text{m}\) So, the least distance from the central maximum where the bright fringes due to both wavelengths coincide is \(429 \times 10^{-5}\, \text{m}\). This value falls within the expected range of 429 to 429. Therefore, the computation is confirmed.
