The two slits of a double slit each have a width of 0.140 mm and a distance between centers of 0.840 mm. (a) What orders are missing,and (b) what is the approximate intensity of orders m = 0 to m = 6?

To tackle the problem of the double slit experiment you've presented, we need to delve into the principles of interference patterns created by light passing through two closely spaced slits. The parameters you've given—slit width and distance between the slits—are crucial for determining the missing orders and the intensity of the various orders of interference.

Understanding Missing Orders

In a double slit setup, the interference pattern is influenced by both the width of the slits and the distance between them. The condition for constructive interference (bright fringes) is given by:

• Condition for Bright Fringes: \( d \sin(\theta) = m \lambda \)
• Condition for Dark Fringes: \( a \sin(\theta) = (m + 0.5) \lambda \)

Where:

• d: Distance between the slits
• a: Width of each slit
• m: Order of the interference (an integer)
• \(\lambda\): Wavelength of the light used
• \(\theta\): Angle of the interference pattern

To determine which orders are missing, we need to find the values of \(m\) for which the dark fringe condition is satisfied. This means we need to check when the angle \( \theta \) leads to a situation where the sine function exceeds 1, which is not physically possible.

Calculating the Missing Orders

Given:

• Width of each slit, \( a = 0.140 \, \text{mm} = 0.140 \times 10^{-3} \, \text{m} \)
• Distance between slits, \( d = 0.840 \, \text{mm} = 0.840 \times 10^{-3} \, \text{m} \)
• Assuming a wavelength \( \lambda \) (for example, \( 500 \, \text{nm} = 500 \times 10^{-9} \, \text{m} \))

For dark fringes, we set up the equation:

\( a \sin(\theta) = (m + 0.5) \lambda \)

To find the maximum order \( m \) that can exist, we set \( \sin(\theta) \leq 1 \):

\( a = (m + 0.5) \lambda \)

Rearranging gives:

\( m = \frac{a}{\lambda} - 0.5 \)

Substituting the values:

\( m = \frac{0.140 \times 10^{-3}}{500 \times 10^{-9}} - 0.5 \approx 0.28 - 0.5 \approx -0.22 \)

This indicates that the first order (m = 0) is not missing, but higher orders will be limited by the slit width.

Intensity of Orders from m = 0 to m = 6

The intensity of the interference pattern can be calculated using the formula:

\( I(m) = I_0 \left( \frac{\sin(\beta)}{\beta} \right)^2 \left( \cos(\alpha) \right)^2 \)

Where:

• β: \( \frac{\pi a \sin(\theta)}{\lambda} \)
• α: \( \frac{\pi d \sin(\theta)}{\lambda} \)
• I0: Maximum intensity

For each order \( m \) from 0 to 6, we can calculate the corresponding intensity by substituting the appropriate values of \( \theta \) derived from the interference conditions. The angles can be found using:

\( \sin(\theta) = \frac{m \lambda}{d} \)

As an example, for \( m = 0 \):

\( \sin(\theta) = 0 \Rightarrow \theta = 0 \)

Thus, \( β = 0 \) and \( α = 0 \), leading to:

\( I(0) = I_0 \)

For higher orders, as \( m \) increases, the values of \( β \) and \( α \) will change, and you will need to compute the sine and cosine values accordingly. This results in a decrease in intensity as \( m \) increases due to the diffraction effects from the slit width.

In summary, the missing orders are determined by the relationship between slit width and wavelength, while the intensity of each order can be calculated using the interference intensity formula, taking into account the contributions from both the slit width and the distance between the slits. This interplay creates the fascinating patterns we observe in the double slit experiment.