Angular Magnification of Microscope

[LeeLoo]

Homework Statement

Hey, I'm having trouble with this question on microscopes.

A microscope is made up of an objective of power +100D and an eyepiece of power +50D, the two lenses being separated by 18cm. How far in front of the objective must an object be placed if the image of the object formed by the microscope is viewed by an emmetrope exercising 4D of accommodation? Assume the observer's eye is close to the eyepiece. What will be the angular magnification in this case?

I worked out the object distance to be -1.07cm but I can't figure out how to get the angular magnification (the answer is 204.6x).


Relevant equations

Gauss Law


The attempt at a solution

I drew a diagram with the two lenses and labelled the distance between them (18mm). Then I worked backwards by treating the final image distance as 1/-8 as the person has to accommodate by 8D and from there I found the object distance for lens 2 which is the image distance for lens 1. Then I found the initial object distance. So now I'm stuck on the angular magnification. Can someone please show me the working out?

Thanks! :)

 

[Redbelly98]

I'm not sure what role the 4D accomodation plays, but couldn't you take a slightly off-axis ray passing through the center of the 100D lens, and calculate at what angle it exits the 50D lens?

 

[baba89]

Dear student,

To calculate the angular magnification of a microscope, we can use the formula:

M = -fe/fo

Where M is the angular magnification, fe is the effective focal length of the eyepiece, and fo is the focal length of the objective.

In this case, the effective focal length of the eyepiece (fe) can be calculated using the formula:

fe = 1/(d - fo)

Where d is the distance between the two lenses (18cm) and fo is the focal length of the objective (+100D = +1/100m = 0.01m).

Substituting these values, we get:

fe = 1/(18cm - 0.01m) = 0.055m

Now, we can calculate the angular magnification using the first formula:

M = -fe/fo = -0.055m/0.01m = -5.5

However, this is the angular magnification for a distant object. To calculate the angular magnification for an object placed at a specific distance, we need to use the formula:

M = -fe/fo * (1 + do/fo)

Where do is the distance of the object from the objective. In this case, do = -1.07cm = -0.0107m.

Substituting this value, we get:

M = -0.055m/0.01m * (1 + (-0.0107m)/0.01m) = -5.5 * 0.9893 = -5.443

Finally, to find the angular magnification in terms of x, we can use the formula:

M = tan(x)

Where x is the angular magnification in radians. So, we have:

tan(x) = -5.443

x = tan^-1(-5.443) = -81.05°

Therefore, the angular magnification in this case is 81.05x or 204.6x, as stated in the question.

I hope this helps! Let me know if you have any further questions.

Best,