Two slits are 0.144 mm apart. A mixture of red light (wavelength = 665 nm) and yellow-green light (wavelength = 565 nm) falls on the slits. A flat observation screen is located 2.11 m away. What is the distance on the screen between the third-order red fringe and the third-order yellow-green fringe?

To find the distance between the third-order red fringe and the third-order yellow-green fringe on the screen, we can use the formula for the interference pattern created by a double-slit experiment.

The formula is:

d * sin(theta) = m * lambda

where:
- d is the distance between the slits (0.144 mm)
- theta is the angle between the central maximum and the fringe (which we can assume to be small, so we can use the small angle approximation and consider sin(theta) = tan(theta))
- m is the order of the fringe (in this case, we're looking for the third-order fringes)
- lambda is the wavelength of light (in this case, for red light, lambda = 665 nm = 665 * 10^-9 m and for yellow-green light, lambda = 565 nm = 565 * 10^-9 m)

First, let's calculate the angle theta for the third-order red fringe:

d * tan(theta_red) = m * lambda_red

0.144 mm * tan(theta_red) = 3 * 665 nm

Now, let's calculate the angle theta for the third-order yellow-green fringe:

d * tan(theta_yellow-green) = m * lambda_yellow-green

0.144 mm * tan(theta_yellow-green) = 3 * 565 nm

Next, we need to find the distance on the screen between these two fringes. We can consider a right-angled triangle formed by the distance between the slits and the screen. The difference between the distances from the slits to the two fringes can be considered as the base of the triangle, and the screen distance can be considered as the height.

Let's call the distance between the third-order red fringe and the third-order yellow-green fringe as "x".

Now, we can use the trigonometric identity tan(theta) = opposite / adjacent to find the relationship between x, theta_red, and theta_yellow-green:

tan(theta_red) = x / 2.11 m
tan(theta_yellow-green) = (x + D) / 2.11 m

where D is the difference in distances between the screen and the slits due to the additional path traveled by the yellow-green light compared to the red light to reach the third-order fringe.

By substituting and rearranging the equations, we can find the value of x.

Note: The small angle approximation is used here since we have assumed theta to be a small angle. If the angle is large, then we need to use a more accurate formula to calculate the fringe locations.