Question

A double slit is illuminated simultaneously with orange light of wavelength $620\mathrm{nm}$ and light of an unknown wavelength. The $\mathrm{m}=4$ bright fringe of the unknown wavelength overlaps the $\mathrm{m}=3$ bright orange fringe. What is the unknown wavelength$?$

Short answer

Unknown wavelength is$465\mathrm{nm}$.

Step-by-step solution

Formula for position

The aim is to require use of very fact that the $\mathrm{m}=4$bright orange fringe has the identical position $\mathrm{y}$because $\mathrm{m}=3$bright orange fringe.

And we know that for the sunshine fringes within the double slit experiment, we may write the position as follows:

${\mathrm{y}}_{\mathrm{m}}=\frac{\mathrm{m\lambda L}}{\mathrm{d}}$

Calculation for unknown wavelength

We'll ask the unknown wavelength as ${\mathrm{\lambda }}_{\mathrm{un}}$.

And therefore the orange wavelength as ${\mathrm{\lambda }}_{\mathrm{or}}$.

When the positions of the two fringes are equalized, the result is

localid="1649168675266" $\frac{4{\mathrm{\lambda }}_{\mathrm{un}}\mathrm{L}}{\mathrm{d}}=\frac{3{\mathrm{\lambda }}_{\mathrm{or}}\mathrm{L}}{\mathrm{d}}$

Becauselocalid="1649168679741" $\mathrm{d}$andlocalid="1649168685122" $\mathrm{L}$are equal, they wipe out on equal sides. Hence

localid="1649167746395" ${\mathrm{\lambda }}_{\mathrm{un}}=\frac{3{\mathrm{\lambda }}_{\mathrm{or}}}{4}$

$=\frac{3\times 620\mathrm{nm}}{4}$