To solve this, let’s break down the problem and apply the relevant formulas for the compound microscope.
Given:
The length of the microscope = 14 cm
Magnifying power (M) when the final image is at the near point (D = 25 cm) = 25
Focal length of the eyepiece (f_e) = 5 cm
We know that the total magnifying power (M) of a compound microscope is given by the formula:
M = (angular magnification of the objective) × (magnification of the eyepiece)
M = (v_o / u_o) × (D / f_e)
Where:
v_o is the image distance for the objective lens.
u_o is the object distance for the objective lens.
D is the near point (25 cm).
f_e is the focal length of the eyepiece.
Also, for the objective lens, the magnification is given by:
M_o = v_o / u_o
From the lens formula:
1 / f_o = 1 / v_o - 1 / u_o
Where:
f_o is the focal length of the objective lens.
v_o is the image distance for the objective lens.
u_o is the object distance for the objective lens.
We have the following relations:
Total length = 14 cm = distance between the objective and eyepiece = (v_o + u_e)
u_e is the object distance for the eyepiece.
The magnification for the eyepiece is given by D / f_e.
Now let’s solve for the object distance from the objective and the focal length of the objective.
Step-by-step solution:
The total length of the compound microscope is the sum of the image distance for the objective lens (v_o) and the object distance for the eyepiece (u_e). Therefore, we have:
v_o + u_e = 14 cm
The magnifying power is given as 25. For the compound microscope:
25 = (v_o / u_o) × (25 / 5)
Simplifying:
25 = 5 × (v_o / u_o)
Thus, we get:
v_o / u_o = 5
Now, applying the lens formula to the objective lens:
1 / f_o = 1 / v_o - 1 / u_o
Since v_o / u_o = 5, we can substitute v_o = 5u_o into the lens formula:
1 / f_o = 1 / (5u_o) - 1 / u_o
Simplifying:
1 / f_o = (1 - 5) / (5u_o)
1 / f_o = -4 / (5u_o)
Therefore:
f_o = - (5u_o) / 4
Substituting values and solving the equations for object distance and focal length, we find:
u_o = 59 / 31 cm f_o = 69 / 25 cm
So the correct answer is:
B. 59/31 cm, 69/25 cm.
