Chapter 35, Problem 1Q

Does the spacing between fringes in a two-slit interference pattern increase, decrease, or stay the same if (a) the slit separation is increased, (b) the color of the light is switched from red to blue, and (c) the whole apparatus is submerged in cooking sherry? (d) If the slits are illuminated with white light, then at any side maximum, does the blue component or the red component peak closer to the central maximum?

To determine

To find:

a) Does the spacing between the fringes in a two-slit interference pattern increase, decrease, or stay the same if the slit separation is increased?

b) Does the spacing between the fringes in a two-slit interference pattern increase, decrease, or stay the same if the color of the light is switched from red to blue?

c) Does the spacing between the fringes in a two-slit interference pattern increase, decrease, or stay the same if the whole apparatus is submerged in cooking sherry?

d) If the slits are illuminated with white light, then at any side maximum does the blue component or the red component peak closer to the central maximum?

Answer to Problem 1Q

Solution:

a) The spacing between the fringes in a two-slit interference pattern decreases if the slit separation is increased.

b) The spacing between the fringes in a two-slit interference pattern decreases if the color of the light is switched from red to blue.

c) The spacing between the fringes in a two-slit interference pattern decreases if the whole apparatus is submerged in cooking sherry.

d) If the slits are illuminated with white light, then at any side maximum the blue component peak will be closer to the central maximum.

Explanation of Solution

1) Concept:

We use the concept of double slit experiment. Using the equation, we can determine whether the spacing between the fringes will increase, decrease or stay same. For part c), we use the relation between the initial and the new wavelength.

2) Formulae:

$∆y=\frac{\lambda D}{d}$

3) Calculations:

a) Does the spacing between the fringes in a two-slit interference pattern increase, decrease, or stay the same if the slit separation is increased?

Using the equation,

$∆y=\frac{\lambda D}{d}$

In this equation, $∆y$ is spacing between in the fringes, $\lambda$ is wavelength, $D$ is distance between the slits and screen and $d$ is the separation between the slits.

From the above equation if we increase the slit separation d, the spacing between the fringes decreases.

b) Does the spacing between the fringes in a two-slit interference pattern increase, decrease, or stay the same if the color of the light is switched from red to blue?

We know that the wavelength of the light spectrum decreased from red to blue.

${\lambda }_r>{\lambda }_b$

$∆y=\frac{\lambda D}{d}$

So, here, if $\lambda$ decreases, then $∆y$ also decreases.

c) Does the spacing between the fringes in a two-slit interference pattern increase, decrease, or stay the same if the whole apparatus is submerged in cooking sherry?

We know,

$n=\frac{{\lambda }_i}{{\lambda }_f}$

We can write the wavelength as

${\lambda }_f=\frac{{\lambda }_i}{n}$

We get,

$∆y=\frac{{\lambda }_{f}D}{d}$

$∆y=\frac{{\lambda }_i}{n}\frac{D}{d}$

So, $∆y$ is inversely proportional to n. Therefore, $∆y$ decreases.

d) If the slits are illuminated with white light, then at any side maximum does the blue component or the red component peak closer to the central maximum?

Here, the blue component peak will be closer to the central maximum because

$∆y\propto \lambda$

And for the lowest value of $∆y$ the value of $\lambda$ must be minimum, which is the blue light.

Hence, the blue component will be closer to the central maximum.

Conclusion:

We can use the concept of double slit experiment to determine the spacing between the fringes.