Question 196: Wave Optics

Question

Two wavelengths $$\lambda_1$$ and $$\lambda_2$$ are used in Young's double slit experiment. $$\lambda_1=450 \mathrm{~nm}$$ and $$\lambda_2=650 \mathrm{~nm}$$. The minimum order of fringe produced by $$\lambda_2$$ which overlaps with the fringe produced by $$\lambda_1$$ is $$n$$. The value of $$n$$ is _______.

Accepted Answer

In Young's double slit experiment, the condition for constructive interference (bright fringes) is given by:
$$d \sin 	heta = n \lambda$$
where:
$$d$$ is the distance between the slits
$$	heta$$ is the angle of the fringe relative to the central maximum
$$n$$ is the order of the fringe (an integer)
$$\lambda$$ is the wavelength of the light
We are given two different wavelengths:
$$\lambda_1 = 450 \, 	ext{nm}$$
$$\lambda_2 = 650 \, 	ext{nm}$$
For the fringes produced by these two wavelengths to overlap, the path difference must be an integer multiple of both wavelengths. This means:
$$d \sin 	heta = m \lambda_1 = n \lambda_2$$
where $$m$$ and $$n$$ are the orders of the fringes for $$\lambda_1$$ and $$\lambda_2$$, respectively.
To find the minimum order of fringe $$n$$ for $$\lambda_2$$ that coincides with a fringe for $$\lambda_1$$, we need to find the least common multiple (LCM) of these wavelengths in terms of their smallest integers. This can be formulated as:
$$m \lambda_1 = n \lambda_2$$
Dividing both sides by $$\lambda_1$$ and $$\lambda_2$$, we get:
$$\frac{m}{\lambda_2} = \frac{n}{\lambda_1}$$
Cross-multiplying, we get:
$$m \lambda_1 = n \lambda_2$$
Using the given wavelengths:
$$m 	imes 450 = n 	imes 650$$
Simplifying this equation, we get:
$$\frac{m}{n} = \frac{650}{450}$$
$$\frac{m}{n} = \frac{13}{9}$$
For the fringes to overlap, $$m$$ and $$n$$ must be integers. The smallest integers that satisfy this ratio are:
$$m = 13$$
$$n = 9$$
Therefore, the minimum order of fringe produced by $$\lambda_2$$ which overlaps with the fringe produced by $$\lambda_1$$ is:
$$n = 9$$

Wave Optics

Two wavelengths $\lambda_1 and \lambda_2 are used in Young's double slit experiment. \lambda_1=450 \mathrm{~nm} and \lambda_2=650 \mathrm{~nm}. The minimum order of fringe produced by \lambda_2 which overlaps with the fringe produced by \lambda_1 is n. The value of n$ is _______.

Integer / Numeric Answer Type