Compound Microscope: Solving for Objective Focal Length

[map7s]

A compound microscope has the objective and eyepiece mounted in a tube that is 18.0 cm long. The focal length of the eyepiece is 2.08 cm, and the near-point distance of the person using the microscope is 25.0 cm. If the person can view the image produced by the microscope with a completely relaxed eye, and the magnification is -4350, what is the focal length of the objective? (Include the sign.)

I tried using the equations

d i =d- fo

M = (d i * N)/(fo*fe)

since I could not find d i b/c I didn't know fo, I substituted d-fo for d i in the second equation. When I isolated fo, I got

fo=(dN)/(Mfe +N)

I thought that I was doing this right...what did I mess up?

 

[map7s]

did I do an algebraic mistake or...I assumed that the 18 cm = d...was that a wrong assumption to make?

 

[m2003]

Thank you for sharing your approach to solving for the focal length of the objective in this compound microscope setup. Your use of the equations for image distance and magnification is correct, but there seems to be a mistake in your substitution for the image distance. You correctly identified that the image distance (d i) is equal to the tube length (d) minus the focal length of the objective (fo). However, when you substituted this into the second equation, you should have used d - fo instead of d i. This is because d i represents the distance from the objective lens to the image, which is different from the tube length (d).

Using the corrected equation, we can solve for the focal length of the objective as follows:

M = (d - fo) * N / (fo * fe)

-4350 = (18.0 cm - fo) * (-25.0 cm) / (fo * 2.08 cm)

-4350 = (-450 cm + 25.0 cm * fo) / (2.08 cm)

-4350 * 2.08 cm = -450 cm + 25.0 cm * fo

-9030 cm = -450 cm + 25.0 cm * fo

-9030 cm + 450 cm = 25.0 cm * fo

-8580 cm = 25.0 cm * fo

-8580 cm / 25.0 cm = fo

-343.2 cm = fo

Therefore, the focal length of the objective is -343.2 cm. The negative sign indicates that the objective lens is a diverging lens, which is common in compound microscopes.