Red light of 650 nm can be seen in three orders in a particular grating. About how many slits per centimeter does this grating have?

The formula for the diffraction grating is given by: d * sin(θ) = m * λ where, d = distance between slits (or the grating constant), θ = angle at which maximum intensity occurs, m = order, λ = wavelength of the light. We can rewrite the formula to find the d value: d = (m * λ) / sin(θ) We are given λ = 650 nm and we need to find d.

Since we are given that the red light can be seen in three orders, we will calculate the angle θ for each order (m = 1, 2, 3). For m = 1, the angle θ will be such that sin(θ) = λ / d. Since we have no information about θ, we can assume the value of θ to be small. In that case sin(θ) ≈ tan(θ) ≈ θ, and we can rewrite the formula as: d ≈ (1 * λ) / θ We can follow the same approximation for m = 2 and m = 3: For m = 2: d ≈ (2 * λ) / θ For m = 3: d ≈ (3 * λ) / θ

Now, we will use the three equations to eliminate the θ variable and find the grating constant (d). Divide the equation for m = 2 by the equation for m = 1: (2 * λ) / θ ≈ 2 * (λ / θ) which simplifies to d ≈ 2 * d Similarly, For m = 3, divide the equation for m = 3 by the equation for m = 1: (3 * λ) / θ ≈ 3 * (λ / θ) which simplifies to d ≈ 3 * d Now, we have two equations for d: For m = 2: d ≈ 2 * d For m = 3: d ≈ 3 * d Solving the two equations, we find that d must be such that it is an integral multiple of itself, mainly d = n * 650 nm, where n is a natural number. If we consider the lowest possible value for n (n = 1), we will have the minimum distance between the slits (d_min). d_min = 1 * 650 nm = 650 nm

The number of slits per centimeter can be found by converting d_min from nanometers to centimeters and then finding the reciprocal: d_min = 650 nm * (1 cm / 1e7 nm) = 6.5e-5 cm Number of slits per centimeter = 1 / d_min = 1 / 6.5e-5 cm = 15385 slits/cm The grating has approximately 15,385 slits per centimeter.