Part A) The overall magnification of the microscope is equal to the product of the magnification of the objective lens and the magnification of the eyepiece:
M = M_objective x M_eyepiece
M = 64 x 10
M = 640
Therefore, the overall magnification of the microscope is 640×.
Part B) The magnification of the eyepiece is given by:
M_eyepiece = f_objective / (f_objective + d)
where f_objective is the focal length of the objective lens, and d is the distance between the objective lens and the eyepiece. Since the image of the objective lies very close to the focal point of the eyepiece, we can assume that d is equal to the focal length of the eyepiece, which we will denote as f_eyepiece. Substituting these values into the equation above, we get:
10 = f_objective / (f_objective + f_eyepiece)
We can rearrange this equation to solve for f_eyepiece:
f_eyepiece = f_objective / 10 - 1
The magnification of the objective lens is given by:
M_objective = f_eyepiece / (f_eyepiece - d)
Substituting the value of d, we get:
M_objective = f_eyepiece / (f_eyepiece - f_eyepiece)
M_objective = 1
Therefore, the magnification of the objective lens is 1×. We can use this value to solve for f_objective:
M = M_objective x M_eyepiece
640 = 1 x 10 x (f_objective / (f_objective + f_eyepiece))
Substituting the expression for f_eyepiece that we derived earlier, we get:
640 = 10 x f_objective / (f_objective / 9)
640 = 90
Solving for f_objective, we get:
f_objective = 14.06 cm
Therefore, the focal length of the objective lens is 14.06 cm.