Question

Fringes from two slits are observed, using light of wavelengths $\lambda _{_{1}}$ =370nm and $\lambda _{_{2}}$ =740nm. At what values for the orders of m, if any, of the two wavelengths do the bright bands (fringes) of the two colors coincide?

Answer 1

Position of m^th order bright fringe is given by

$\small y_m = \frac{mL\lambda }{d}$

L = distance of screen from slit

d = distance between 2 slits

$\small \lambda$ = Wavelength of light wave used

Let us assume that m₁ order bright fringe of $\small \lambda_1$   coincide with m₂ order bright fringe of $\small \lambda_2$

this implies

$\small y_{m_1} = y_{m_2}$

$\small \frac{m_1L\lambda_1 }{d}= \frac{m_2L\lambda_2 }{d}$

$\small \frac{m_1L*370 }{d}= \frac{m_2L*740}{d}$

$\small m_1*370 = m_2*740$

$\small \frac{ m_1}{ m_2}=\frac{740}{370 }$

$\small \frac{ m_1}{ m_2}=\frac{2}{1}$

Above equation implies that all the orders of   $\small \lambda_1$ and $\small \lambda_2$ which are in ratio of 2:1 respectively coincide . For example 2^nd order of $\small \lambda_1$ coincides with 1^st order of $\small \lambda_2$ similarly 4^th order of $\small \lambda_1$ coincides with 2^nd order of $\small \lambda_2$ and so on.