Question

In a double-slit experiment, two beams of coherent light traveling different paths arrive on a screen some distance away. What is the path difference between the two waves corresponding to the third bright band out from the central bright band?

In a double-slit experiment, two beams of coherent light traveling different paths arrive on a screen some distance away. What is the path difference between the two waves corresponding to the third bright band out from the central bright band?

	The path difference between the two waves is one and one-half wavelengths.
	The path difference between the two waves is four wavelengths.
	The path difference between the two waves is one-half of a wavelength.
	The path difference between the two waves is two wavelengths.
	The path difference between the two waves is one wavelength.
	The path difference between the two waves is three wavelengths.

Answer 1

Concepts and reason

The concept used to solve this problem is path difference between the two waves.

Find the path difference between the two waves for the third bright band in terms of wavelength from the double slit experiment.

Fundamentals

The double slit experiment demonstrates the wave nature and nature of particles.

This experiment involves the passing of a monochromatic light beam through two slits that are placed apart at a distance. The light passes through the slits, and it undergoes interference. As a result, the interference pattern is obtained at the screen.

Path difference is the difference in the distance travelled by two waves in the double slit experiment, which makes them to be out of phase.

“Bright bands are formed by the constructive interference and they are the positions of maximum intensity”.

Interference occurs because the light waves after passing through the slits will have a path difference. This is given as follows:

$n\lambda = \frac{{{x_n}d}}{D}$

Here, $n$ is a number, $\lambda$ is the wavelength, $D$ is the distance of the screen from the slit, $d$ is the distance between the slits, and ${x_n}$ is the position of the bright band.

Expression for the path difference is as follows:

$n\lambda = \frac{{{x_n}d}}{D}$

Dark bands will be obtained on the screen, if $n$ takes half integral values.

Bright bands will be obtained on the screen, if $n$ takes integral values.

For the third bright band, the value of $n$ is $3$ .

Substitute 3 for n in the expression for the path difference.

$3\lambda = \frac{{{x_3}d}}{D}$

Here, ${x_3}$ is the distance of the third bright band.

Re-arrange the above expression to get ${x_3}$ .

${x_3} = 3\lambda \frac{D}{d}$

From the above explanation, the incorrect options are as follows:

• The path difference between the two waves in the double slit experiment is one and one-half wavelength.

The above option is wrong because this involves half integral values of wavelength, which results in dark bands.

• The path difference between the two waves in the double slit experiment is four wavelengths.

The above option is wrong because the path difference has the multiplication factor four, which is for the fourth bright band.

• The path difference between the two waves in the double slit experiment in the double slit experiment is one-half of a wavelength.

The above option is wrong because it involves half integral value. This will be a path difference for the dark band.

• The path difference between the two waves in the double slit experiment is two wavelengths.

The above option gives path difference for the second bright band. Hence, it is an incorrect option.

• The path difference between the two waves in the double slit experiment is one wavelength.

This is the condition for the first bright band. Hence, it is incorrect.

Therefore, the correct option is as follows:

• The path difference between the two waves in the double slit experiment is three wavelengths.

The above option is correct because the expression for the path difference is $3\lambda = \frac{{{x_3}d}}{D}$ .

Ans:

The path difference between the two waves in the double slit experiment is three wavelengths.