Question

For a microscope with objective \( f_o = 1 \, \text{cm} \), eyepiece \( f_e = 10 \, \text{cm} \), and image distance \( v = 25 \, \text{cm} \), find total magnification.

Hint:

The total magnification of a microscope is the product of the magnifications of the objective and the eyepiece.

Answer 1

Step 1: Understanding the magnification formula.
The total magnification of a microscope is the product of the magnifications produced by the objective lens and the eyepiece. The magnification produced by the objective lens is given by: \[ M_{\text{objective}} = \frac{v}{u} \] where \( v \) is the image distance and \( u \) is the object distance for the objective lens. The magnification produced by the eyepiece is given by: \[ M_{\text{eyepiece}} = 1 + \frac{D}{f_e} \] where \( D \) is the least distance of distinct vision (usually taken as 25 cm), and \( f_e \) is the focal length of the eyepiece.
Step 2: Substituting the given values.
We are given \( f_o = 1 \, \text{cm} \), \( f_e = 10 \, \text{cm} \), and \( v = 25 \, \text{cm} \). First, calculate the magnification by the eyepiece: \[ M_{\text{eyepiece}} = 1 + \frac{25}{10} = 3.5 \] Step 3: Finding the object distance \( u \) using the lens formula.
We can use the lens formula for the objective lens: \[ \frac{1}{f_o} = \frac{1}{v_o} - \frac{1}{u_o} \] where \( v_o \) is the image distance for the objective and \( u_o \) is the object distance for the objective. Using this, we can find \( u \). For simplicity, let’s assume an appropriate value for \( u \) and calculate \( M_{\text{objective}} \).
Step 4: Conclusion.
Thus, the total magnification \( M \) is the product of \( M_{\text{objective}} \) and \( M_{\text{eyepiece}} \).

Answer 2

Step 1: A microscope's total magnification is the objective lens's magnification multiplied by the eyepiece's magnification.

Step 2: The eyepiece magnification is 1 + (D/f_e), using D = 25 cm and f_e = 10 cm, giving 1 + 25/10 = 3.5.

Step 3: The objective magnification is v/u, using the image distance v (25 cm) and the object distance u found via the lens formula with f_o = 1 cm.

Step 4: Multiplying the two magnifications together gives the microscope's total magnification.

Answer 3

Step 1: For a compound microscope with a very short-focus objective, a common approximation treats the given image distance as the microscope's tube length, \( L \approx v = 25 \, \text{cm} \).
Step 2: The objective's magnification is then approximated as \( M_o \approx \dfrac{L}{f_o} = \dfrac{25}{1} = 25 \).
Step 3: The eyepiece magnification is \( M_e = 1 + \dfrac{D}{f_e} = 1 + \dfrac{25}{10} = 3.5 \).
Step 4: Total magnification: \[ M = M_o \times M_e = 25 \times 3.5 \] \[ \boxed{M = 62.5} \]