To determine the angular magnification [tex]\( M_c \)[/tex] of the eyepiece for a compound microscope, follow these steps:
1. Define the given parameters:
- The barrel length [tex]\( L \)[/tex] is 159.0 mm.
- The focal length of the objective lens [tex]\( f_{objective} \)[/tex] is 3.570 mm.
- The total angular magnification [tex]\( M_{total} \)[/tex] is -409.0.
2. Calculate the magnification of the objective lens [tex]\( M_o \)[/tex]:
The formula for the magnification of the objective lens is:
[tex]\[
M_o = -\frac{L}{f_{objective}}
\][/tex]
Substituting the given values:
[tex]\[
M_o = -\frac{159.0 \text{ mm}}{3.570 \text{ mm}} = -44.537815126050425
\][/tex]
3. Calculate the magnification of the eyepiece [tex]\( M_c \)[/tex]:
The total magnification [tex]\( M_{total} \)[/tex] is the product of the magnifications of the objective and the eyepiece:
[tex]\[
M_{total} = M_o \times M_c
\][/tex]
Solving for [tex]\( M_c \)[/tex]:
[tex]\[
M_c = \frac{M_{total}}{M_o}
\][/tex]
Substituting the known values:
[tex]\[
M_c = \frac{-409.0}{-44.537815126050425} = 9.183207547169811
\][/tex]
So, the angular magnification [tex]\( M_c \)[/tex] of the eyepiece is approximately [tex]\( 9.183 \)[/tex].
Thus,
[tex]\[
M_e \approx 9.183
\][/tex]
