Light of wavelength 580nm is incident on a slit having a width of 0.300mm. The viewing screen is 2.00m from the slit. a) Calculate the position of the first dark fringes and b) central bright fringe?

a) To calculate the position of the first dark fringe, we can use the formula for the angular position of the dark fringes in a single-slit diffraction pattern:

sin(theta) = m * lambda / b

where:
- theta is the angular position of the dark fringe
- m is the order of the dark fringe (in this case, m = 1 for the first dark fringe)
- lambda is the wavelength of the light (580nm = 580 x 10^-9 m)
- b is the width of the slit (0.300mm = 0.300 x 10^-3 m)

Plugging in the values, we get:

sin(theta) = (1 * 580 x 10^-9) / (0.300 x 10^-3)
sin(theta) = 0.001933

Now, we can calculate the position of the first dark fringe:

theta = sin^(-1)(0.001933)
theta = 0.111 degrees

Now, we can calculate the position of the first dark fringe on the screen:

y = L * tan(theta)

where:
- y is the position of the dark fringe on the screen
- L is the distance between the slit and the screen (2.00m)

Plugging in the values, we get:

y = 2.00 * tan(0.111)
y = 0.0037m

Therefore, the position of the first dark fringe is 0.0037m from the central bright fringe.

b) To calculate the position of the central bright fringe, we know that the central bright fringe is located at the center of the pattern. Therefore, the position of the central bright fringe is at the midpoint between the two dark fringes.

Since the first dark fringe is at 0.0037m from the central bright fringe, the central bright fringe is located at half of this distance:

y_central = 0.0037 / 2
y_central = 0.00185m

Therefore, the position of the central bright fringe is 0.00185m from the central bright fringe.

a) Position of the first dark fringe

The position of the first dark fringe is given by:

```
y = ±(m + 1/2)λL / w
```

where:

- y is the distance from the central fringe to the dark fringe
- m is the order of the fringe (m = 1 for the first dark fringe)
- λ is the wavelength of the light
- L is the distance from the slit to the viewing screen
- w is the width of the slit

Plugging in the given values, we get:

```
y = ±(1 + 1/2)(580 × 10^-9 m)(2.00 m) / (0.300 × 10^-3 m)
```

```
y = ±2.9 × 10^-3 m
```

Therefore, the position of the first dark fringe is 2.9 mm from the central fringe.

b) Position of the central bright fringe

The position of the central bright fringe is given by:

```
y = 0
```

Therefore, the central bright fringe is located at the center of the screen.