Final answer:
To calculate the smallest distance between two points that can be just resolved using a microscope with a semi-vertical angle of 20° and light of wavelength 6000 Å, the Rayleigh criterion is used. The resolution is found using the formula d = λ / (2n \sin(α)), leading to a smallest resolvable distance of approximately 0.9 μm, which does not match the provided options.
Step-by-step explanation:
To calculate the smallest distance between two points which can be just resolved using a microscope, given the semi-vertical angle of the cone of the rays incident on the objective of the microscope is 20° and the wavelength of the incident light ray is 6000 Å, we need to use the Rayleigh criterion for the diffraction limit to resolution. According to this criterion, two images are just resolvable when the center of the diffraction pattern of one is directly over the first minimum of the diffraction pattern of the other. We can use the formula:
θ = 1.22 λ / D
However, for a microscope, the formula for resolution (d) is given by:
d = λ / (2n \sin(α))
where λ is the wavelength of light, n is the refractive index of the medium between the specimen and the objective lens (generally assumed to be 1 in air), and α is the semi-vertical angle of the cone of light. Plugging in the values λ = 6000 Å = 600 nm (converting to nanometers), and α = 20°, we get:
d = 600 nm / (2 \sin(20°))
By calculating this, we find that the smallest distance between two points which can be resolved is approximately 0.9 μm, none of the options provided (a, b, c, d) match this value, and it seems there might be a calculation or typographical error in the question or the options provided.