A microscope is focused on a mark on a table. When the mark is calculated by a glass plate of 3.00cm thick,the microscope has to be raised 1.1cm from the mark to be once more in focus.calculate the refractive index of the glass.
To calculate the refractive index of the glass, we need to use the formula for thin lenses:
1/f = (n - 1) * [(1/R1) - (1/R2)]
where:
n is the refractive index of the glass
f is the focal length of the lens
R1 is the radius of curvature of the first surface of the lens
R2 is the radius of curvature of the second surface of the lens.
In this case, we are given that the microscope is raised 1.1 cm from the mark on the table when a glass plate of 3.00 cm thick is placed in between. This means that the focal length of the lens changes by 1.1 cm when the glass is added.
Now, let's break down the steps to calculate the refractive index of the glass:
Step 1: Calculate the original focal length of the microscope without the glass plate.
To do this, consider the lens equation:
1/f = (n - 1) * [(1/R1) - (1/R2)]
For a microscope, since it is focused on the mark on the table, it means the object distance is very large (at infinity). Therefore, 1/f ≈ 0, and we can ignore the term involving R2.
So, the equation simplifies to:
0 = (n - 1) * (1/R1)
Since the lens is focused on a finite object, the focal length is negative, so R1 is positive. Therefore, (1/R1) is positive.
Step 2: Calculate the new focal length of the microscope with the glass plate.
The new focal length (f') is the original focal length (f) minus the change in the focal length when the glass is added (Δf).
f' = f - Δf
Step 3: Calculate the refractive index of the glass.
Since the microscope is focused on the mark at infinity, the sum of the focal lengths of the lens without glass (f) and the glass plate (f') will be equal to zero.
So, we have:
f + f' = 0
Substituting the values of f and f' from the previous steps:
[(n - 1) * (1/R1)] + [(n - 1) * (1/R1')] = 0
Since the thickness of the glass plate (d) is given as 3.00 cm and the microscope is raised by 1.1 cm, we can write:
d = (R1 - R1')
Simplifying further:
(1/R1) - (1/R1') = 1/d
Now, substituting this into the equation:
[(n - 1) * (1/R1)] + [(n - 1) * (1/R1)] = 1/d
2 * (n - 1) * (1/R1) = 1/d
(n - 1) * (1/R1) = 1/ (2 * d)
Finally, solving for n, we have:
n - 1 = (1/ (2 * d)) * (1/R1)
n = 1 + (1 / (2 * d * R1))
Now, substitute the values given for d (3.00 cm) and R1 (unknown) to calculate the refractive index (n) of the glass.