Calculate the angles at which the first dark band and the next bright band are formedin the Fraunhofer diffraction pattern of a slit 0.3 mm wide (λ = 5890Å).

To calculate the angles for the first dark band and the next bright band in the Fraunhofer diffraction pattern, we can use the formula for single-slit diffraction:

θ = λ / b

where:

• θ is the angle of the diffraction pattern,
• λ is the wavelength of the light, and
• b is the width of the slit.

First, we need to convert the given values to the same units. The width of the slit, b, is given as 0.3 mm, which is 0.3 x 10^-3 m. The wavelength, λ, is given as 5890 Å, which is 5890 x 10^-10 m.

1. To find the angle for the first dark band (m=1), we use the formula:

θ_dark = mλ / b θ_dark = (1 * 5890 x 10^-10 m) / (0.3 x 10^-3 m) θ_dark = 1.963 x 10^-3 radians

To convert this to degrees, we multiply by (180/π):

θ_dark = 1.963 x 10^-3 * (180/π) = 0.112 degrees

2. To find the angle for the next bright band (m=2), we use the formula:

θ_bright = (mλ / b) - (λ / 2b) θ_bright = ((2 * 5890 x 10^-10 m) / (0.3 x 10^-3 m)) - ((5890 x 10^-10 m) / (2 * 0.3 x 10^-3 m)) θ_bright = 3.926 x 10^-3 - 0.9815 x 10^-3 = 2.9445 x 10^-3 radians

Again, to convert this to degrees, we multiply by (180/π):

θ_bright = 2.9445 x 10^-3 * (180/π) = 0.169 degrees

So, the first dark band is formed at an angle of approximately 0.112 degrees, and the next bright band is formed at an angle of approximately 0.169 degrees.