Compton Experiment and Resolving Power

The Compton experiment is a famous experiment that demonstrated the wave-particle duality of light. It involves the scattering of X-rays (or in this case, ultraviolet light) from electrons. The scattering of light causes a change in wavelength, known as the Compton shift.

The resolving power of an optical instrument refers to its ability to distinguish between two close wavelengths. In the context of the Compton experiment, we need an instrument with sufficient resolving power to measure the small change in wavelength caused by the scattering.

Given Data:
- Wavelength of incident ultraviolet light (λ₀) = 2430 Å (angstroms)
- Observation angle (θ) = 1800 (with respect to the direction of incident light)
- Correct answer = 50206

Calculating Compton Shift:
The Compton shift (Δλ) can be calculated using the equation:

Δλ = λ₁ - λ₀ = (h / mₑc) * (1 - cos(θ))

where:
- Δλ is the change in wavelength
- λ₀ is the initial wavelength
- λ₁ is the final wavelength (after scattering)
- h is the Planck's constant
- mₑ is the mass of an electron
- c is the speed of light
- θ is the scattering angle

Given that the electron is at rest, we can assume its initial wavelength (λ₀) is the same as the incident wavelength (2430 Å).

Calculating Resolving Power:
The resolving power (R) of an optical instrument is given by the equation:

R = λ / Δλ

where:
- R is the resolving power
- λ is the wavelength
- Δλ is the change in wavelength

To measure the Compton shift accurately, we need the resolving power to be at least equal to the change in wavelength.

Solution:
1. Calculate the Compton shift (Δλ) using the given equation and values:
Δλ = (6.63 x 10⁻³⁴ J.s / (9.11 x 10⁻³¹ kg) x (3 x 10⁸ m/s)) * (1 - cos(1800))
Δλ = 2.42 x 10⁻¹⁰ m

2. Convert the Compton shift to Å (angstroms):
Δλ = 2.42 x 10⁻¹⁰ m * (1 x 10¹⁰ Å / 1 m)
Δλ = 2.42 Å

3. Calculate the resolving power (R) using the Compton shift (Δλ) and incident wavelength (λ₀):
R = 2430 Å / 2.42 Å
R = 1004

The minimum resolving power required to measure the Compton shift is approximately 1004, which is different from the given correct answer of 50206. It is possible that there was an error in the calculation or that additional information or equations are missing. Please double-check the provided values and equations to ensure accuracy.