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 distance $d_{23}$from 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 $\left(NI\right)$, and (e) on the same side of lens 3 as object O or on the opposite side.

Short answer

1. The image distance due to the image produced by the lens 3 is $i_3=+7.5\text{cm}$.
2. The overall lateral magnification is $m=-0.75$.
3. The final image is real.
4. The final image is inverted.
5. The final image is on the opposite side of the object.

Step-by-step solution

Given data:

• The focal length of lens 1, $f_1=6.0\text{cm}$
• The focal length of lens 2, $f_2=3.0\mathrm{cm}$
• The focal length of lens 3, $f_3=3.0\mathrm{cm}$
• The object distance from the lens 1, $p_1=18.0\mathrm{cm}$
• The distance between lens 1 and 2, $d_1=15.0\mathrm{cm}$
• The distance between lens 2 and 3, $d_{23}=11.0\text{cm}$$d_{2}3=11.0\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, conclude 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:

The lens formula is,

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

The magnification is define by,

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

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

Determining the image distance  i1 due to the image produced by the lens 1.

All lenses are converging so that the focal length due to each lens is positive.

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

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

So,

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

Substituting known value in the above equation.

role="math" localid="1663154504917" $$\begin{gathered} i_1=\frac{18.0\text{cm}\times 6\text{cm}}{\left(18-6\right)\text{cm}} \\ =\frac{108{\text{cm}}^2}{12\text{cm}} \\ =9.0\text{cm} \end{gathered}$$

Determining the image distance i2  due to the image produced by the lens 2.

So, now find the$P_2$i.e. the object distance due to lens 2.

$p_2=d_{12}-i_1$

Putting the value of$i_1$ and $d_{12}$ in the above equation.

$$\begin{gathered} p_2=15.0\text{cm}-9.0\text{cm} \\ =6.0\text{cm} \end{gathered}$$

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

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

Rearrange the above equation for$i_2$.

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

Substituting the known value in the above equation.

role="math" localid="1663154845910" $$\begin{gathered} i_2=\frac{6.0\text{cm}\times 3\text{cm}}{\left(6-3\right)\text{cm}} \\ =\frac{18{\text{cm}}^2}{3\text{cm}} \\ =6.0\text{cm} \end{gathered}$$

(a) The image distance i3  due to the image produced by the lens 3.

Now, calculate the object distance$P_3$as,

$p_3=d_{23}-i_2$

Putting the value of$i_2$and $d_{23}$ in the above equation.

localid="1663155022457" $$\begin{gathered} p_3=11.0\text{cm}-6.0\text{cm} \\ =5.0\text{cm} \end{gathered}$$

Now, calculate the image distance due to lens 3 as

localid="1663155507111" $\frac{1}{i_3}=\frac{1}{f_3}-\frac{1}{p_3}$

localid="1663155513254" $i_3=\frac{f_3{p}_3}{p_3-f_3}$

Substituting the known value, and you get

$$\begin{gathered} i_3=\frac{5.0\text{cm}\times 3.0\text{cm}}{\left(5.0-3.0\right)\text{cm}} \\ =\frac{15.0{\text{cm}}^2}{2.0\text{cm}} \\ =7.5\text{cm} \end{gathered}$$

Hence, the image distance $i_3$ due to the image produced by lens 3 is $i_3=+7.5\text{cm}$.

(b) Determining the overall lateral magnification:

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

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

The magnification of lens 1 is,

$$\begin{gathered} m_1=-\frac{i_1}{p_1} \\ =-\frac{9.0\text{cm}}{18\text{cm}} \\ =-0.5 \end{gathered}$$

The magnification of lens 2 is,

$$\begin{gathered} m_2=-\frac{i_2}{p_2} \\ =-\frac{6.0\text{cm}}{6.0\text{cm}} \\ =-1 \end{gathered}$$

_{The magnification of lens 3 is,}

$$\begin{gathered} m_3=-\frac{i_3}{p_3} \\ =-\frac{7.5\text{cm}}{5\text{cm}} \\ =-1.5 \end{gathered}$$

So, the total magnification is,

$$\begin{gathered} m=m_1{m}_2{m}_3 \\ =\left(-0.5\right)\times \left(-1\right)\times \left(-1.5\right) \\ =-0.75 \end{gathered}$$

Therefore, the overall lateral magnification is $m=-0.75$.

(c) Determining final image is real or virtual.

If the final image distance from the last lens is positive, then the image is real and if the image distance from the last lens is negative, then the image is virtual.

So in this case, the $i_3$is positive so that the image is real.

(d) Determining 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.

Therefore, the final image is inverted.

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

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

Result (c) conclude that the image is on the opposite side of the lens 3 from the object.

Hence, the final image is on the opposite side of the object.