Question

In a microscope of the type shown in Fig. 34-20, the focal length of the objective is 4.00 cm, and that of the eyepiece is 8.00 cm. The distance between the lenses is 25.00 cm. (a) What is the tube length s? (b) If image I in Fig. 34-20 is to be just inside focal point F₁, how far from the objective should the object be? What then are (c) the lateral magnification m of the objective, (d) the angular magnification m_θ of the eyepiece, and (e) the overall magnification M of the microscope?

Short answer

1. Tube length is$s=13\text{cm}$ .
2. Distance of the object from the objective,$p=5.23\text{cm}$.
3. Lateral magnification,$m=-3.25$.
4. Angular magnification,$m_{\theta }=3.13$.
5. Overall magnification,$M=-10.2$ .

Step-by-step solution

Step 1: Given data

The focal length of the objective,$f_{ob}=4\text{cm}=\text{0.04}\text{m}$ .

The focal length of the eyepiece,$f_{ey}=8\text{cm}=\text{0.08}\text{m}$.

Distance between the lenses,$L=25\text{cm}=\text{0.25}\text{m}$ .

Determining the concept

From the fig.34.20, write an expression for the tube length. Inserting the given values will give its value. Using the lens law, find the distance between the objective and the object. Then using the corresponding formulae, find the magnification of the objective, angular magnification of the eyepiece, and overall magnification of the compound microscope.

Formula are as follows:

• Tube length,$\mathit{s}\mathbf{=}\mathit{L}\mathbf{-}{\mathit{f}}_{\mathbf{o}\mathbf{b}}\mathbf{-}{\mathit{f}}_{\mathbf{e}\mathbf{y}}$.
  
• Thin lens formula, $\frac{\mathbf{1}}{{\mathbf{f}}_{\mathbf{0}\mathbf{b}}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{p}}\mathbf{+}\frac{\mathbf{1}}{\mathbf{i}}$.
  
• Magnification, $\mathit{m}\mathbf{=}\mathbf{-}\frac{\mathbf{i}}{\mathbf{p}}$.
  
• Overall magnification,$\mathit{M}\mathbf{=}\mathit{m}{\mathit{m}}_{\mathbf{\theta }}$.
• Angular magnification, ${\mathit{m}}_{\mathbf{\theta }}\mathbf{=}\frac{\mathbf{25}}{{\mathbf{f}}_{\mathbf{e}\mathbf{y}}}$.

Where m is the magnification, p is the pole, f is the focal length, and i is the image distance, m_θis theangular magnification,s is the tube length.

Determining the tube length s

(a)

From figure 34.20,

$s=L-f_{ob}-f_{ey}$

Inserting the given values,

$s=0.25-0.04-0.08$

role="math" localid="1663019531134" $s=0.13\text{m}=\text{13}\text{cm}$

Hence, the Tube length is 13 cm.

Determining if the image I is to be just inside the focal point F1, then how far from the objective should the object be?

(b)

To calculate the distance between the objective and the object (p) needs the image distance I.

From the figure, write,

$$\begin{gathered} i=f_{ob}+s \\ i=0.04+0.13 \\ i=0.17\text{m} \end{gathered}$$

The lens formula is given by,

$\begin{array}{rcl}\frac{1}{f_{0b}}& =& \frac{1}{p}+\frac{1}{i}\\ \frac{1}{0.04}& =& \frac{1}{p}+\frac{1}{0.17}\\ \frac{1}{p}& =& \frac{1}{0.04}-\frac{1}{0.17}\\ p& =& 0.0523\text{m}=\text{5.23}\text{cm}\\ & & \end{array}$

Therefore, the distance between the objective and the object is 5.23 cm.

Determining the lateral magnification m.

(c)

Find the magnification of the objective using the values of i and p as follows:

$$\begin{gathered} m=-\frac{i}{p} \\ m=-\frac{0.17}{0.0523} \\ m=-3.25 \end{gathered}$$

Therefore, the magnification of the objective is -3.25.

 Determining the angular magnification mθ

(d)

The angular magnification of the eyepiece can be calculated as,

$$\begin{gathered} m_{\theta }=\frac{\text{25}\text{cm}}{f_{ey}} \\ {m}_{\theta }=\frac{\text{25}\text{cm}}{\text{8}\text{cm}} \\ {m}_{\theta }=3.13 \end{gathered}$$

Therefore, the angular magnification of the eyepiece is 3.13.

Determining the overall magnification M

(e)

Overall magnification of the microscope is,

$\begin{array}{rcl}M& =& m{m}_{\theta }\\ M& =& -3.25\times 3.13\\ M& =& -10.1725\\ & & ~-10.2\\ & & \end{array}$

Therefore, the overall magnification of the microscope is -10.2.

The tube length, the distance between the objective and object, magnification of the objective, angular magnification of the eyepiece, and overall magnification of the compound microscope can be found using the corresponding formulae.