Question

95 through 100. Three-lens systems. In Fig. 34-49, stick figure O (the object) stands on the common central axis of three thin, symmetric lenses, which are mounted in the boxed regions. Lens 1 is mounted within the boxed region closest to O, which is at object distance $p_1$. Lens 2 is mounted within the middle boxed region, at distance $d_{12}$from lens 1. Lens 3 is mounted in the farthest boxed region, at distancefrom lens 2. Each problem in Table 34-10 refers to a different combination of lenses and different values for distances, which are given in centimeters. The type of lens is indicated by C for converging and D for diverging; the number after C or D is the distance between a lens and either of the focal points (the proper sign of the focal distance is not indicated). Find (a) the image distance $i_3$for the (final) image produced by lens 3 (the final image produced by the system) and (b) the overall lateral magnification M for the system, including signs. Also, determine whether the final image is (c) real $\left(\mathrm{R}\right)$or virtual $\left(\mathrm{V}\right)$, (d) inverted $\left(\mathrm{I}\right)$from object O or non-inverted $\left(\mathrm{Nl}\right)$, and (e) on the same side of lens 3 as object O or on the opposite side.

Short answer

1. The image distancedue to the image produced by the lens 3 is$i_3=-4.0\mathrm{cm}$ .
2. The overall lateral magnification is$m=-1.2$.
3. The final image is virtual.
4. The final image is inverted.
5. The final image is on the same side as the object.

Step-by-step solution

Given data

• Focal length

$f_1=6.0\mathrm{cm}$

$f_2=6.0\mathrm{cm}$

$f_3=4.0\mathrm{cm}$

• Objects distance;$p_1=4.0\mathrm{cm}$
• Distance between lens 1 and 2 is$d_1=9.6\mathrm{cm}$

Distance between lens 2 and 3 is$d_{2}3=14.0\mathrm{cm}$

Determining the concept

First, find out the image distance for each lens using the lens formula. After that, find the magnification of each lens using the corresponding formula. Using this, find the total magnification. From the sign of the total magnification, whether the image is inverted or non-inverted. Also, from the sign of the final image distance for the last lens, conclude whether the image is real or virtual and on which side of the object the final image is present.

Formulae are as follows:

$\frac{1}{f}=\left(\frac{1}{p}+\frac{1}{i}\right)$

$m=-\frac{i}{p}$

Here,$m$ is the magnification, $p$is the pole, $f$is the focal length, and $i$is the image distance.

(a) Determining the image distancei3 due to the image produced by the lens 3.

According to equation 34-4 first, find the image distance $i_1$as$\frac{1}{f}=\left(\frac{1}{p}+\frac{1}{i}\right)$

Lens 1 is diverging so that the focal length is negative.

$\frac{1}{i_1}=\frac{1}{f_1}-\frac{1}{p_1}$

So,

$i_1=\frac{f_1{p}_1}{p_1-f_1}$

Substituting the value,

$$\begin{gathered} i_1=\frac{4.0\times \left(-6\right)}{4+6} \\ =\frac{-24}{10} \\ =-2.4\text{cm} \end{gathered}$$

So, now find thei.e. the object distance due to lens 2.

$p_2=d_{12}-i_1$

Putting the value of$i_1{p}_2=9.6+2.4$

$p_2=12\mathrm{cm}$

From$P_2$ , calculate the image distance$i_2$of the image produced by the lens 2 is,

$\frac{1}{f_2}=\frac{1}{p_2}+\frac{1}{i_2}$

Lens 2 is converging so that the focal length due to lens 2 is positive.

$\frac{1}{i_2}=\frac{1}{f_2}-\frac{1}{p_2}$

So,

$i_2=\frac{f_2{p}_2}{p_2-f_2}$

Substituting the value,

$i_2=\frac{12\times 6}{12-6}$

$$\begin{gathered} i_2=\frac{72}{6} \\ =12\text{cm} \end{gathered}$$

Now, calculate the object distance role="math" localid="1663151409496" $P_3$as,

$p_3=d_{23}-i_2$

Putting the value of$i_2$ ,

$$\begin{gathered} p_3=14-\left(12\right) \\ =2\text{cm} \end{gathered}$$

Now, calculate the image distance due to lens 3 as$\frac{1}{f_3}=\frac{1}{p_3}+\frac{1}{i_3}$

The lens 3 is converging so that the focal length due to lens 3 is positive.

$\frac{1}{i_3}=\frac{1}{f_3}-\frac{1}{p_3}$

So,

$i_3=\frac{f_3{p}_3}{p_3-f_3}$

Substituting the value,

$$\begin{gathered} i_3=\frac{2\times 4}{2-4} \\ =-\frac{8}{2} \\ =-4.0\text{cm} \end{gathered}$$

Therefore, the image distance $i_3$due to the image produced by lens 3 is$i_3=-4.0\mathrm{cm}$

(b) Determine the overall lateral magnification.

Now, calculate the lateral magnification. The total magnification is the product of the magnification due to each lens.

$m=-\frac{i}{p}$

Magnification due to lens 1,

$m_1=-\frac{i_1}{p_1}$

Substituting the values,

$$\begin{gathered} m_1=-\frac{\left(-2.4\right)}{4} \\ =0.6 \end{gathered}$$

Magnification due to lens 2,

$m_2=-\frac{i_2}{p_2}$

Substituting the values,

$$\begin{gathered} m_2=-\frac{\left(12\right)}{12} \\ =-1 \end{gathered}$$

Magnification due to lens 2,

role="math" $m_3=-\frac{i_3}{p_3}$

Substituting the values,

$$\begin{gathered} m_3=-\frac{\left(-4\right)}{2} \\ =2 \end{gathered}$$

So, the total magnification is,

$m=m_1{m}_2{m}_3$

Substitute the above calculated values of the magnification,

$$\begin{gathered} m=-0.6\times 2\times \left(1\right) \\ =-1.2 \end{gathered}$$

Therefore, the overall lateral magnification is$m=-1.2$ .

(c) Determine whether the final image is real or virtual.

If the image distance due to the last lens is positive, then the image is real and if the image distance due to the last lens is positive, then the image is virtual.

So, in our case,$i_3$is negative, so that the image is virtual.

Therefore, the final image is virtual.

(d) Determine whether the final image is inverted or non-inverted.

If the total magnification is positive, then the image is non-inverted and if the total magnification is negative, then the image is inverted.

In our case, the total magnification is negative. So, the image is inverted.

Therefore, the final image is inverted.

(e) Determine the final image on the same side or opposite side of the object

From the result, (c) and conclude that the image is virtual i.e. the image forms on the same side of the lens 3 as the object.

Therefore, the final image is on the same side as the object.

The image distance for any lens and overall magnification of a three-lens system can be found using the corresponding formulae. The nature of the image can be predicted from the characteristics of the image formed due to the given three-lens system.