Question

80 through 87 80, 87 SSM WWW 83 Two-lens systems. In Fig. 34-45, stick figure O (the object) stands on the common central axis of two thin, symmetric lenses, which are mounted in the boxed regions. Lens 1 is mounted within the boxed region closer to O, which is at object distance ${\mathit{p}}_{\mathbf{1}}$. Lens 2 is mounted within the farther boxed region, at distance $\mathit{d}$. Each problem in Table 34-9 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 its focal points (the proper sign of the focal distance is not indicated). Find (a) the image distance ${\mathit{i}}_{\mathbf{2}}$for the image produced by lens 2 (the final image produced by the system) and (b) the overall lateral magnification $\mathit{M}$for the system, including signs.Also, determine whether the final image is (c) real(R) or virtual (V), (d) inverted (I)from object O or non-inverted (NI), and (e) on the same side of lens 2 as the object O or on the opposite side.

Short answer

a. The image distance for the image produced by lens 2, $i_2=+9.8\text{cm}$.

b. The overall lateral magnification, including sign,$M=-0.27$.

The final image is,

c. Real (R)

d. Inverted (I)

e. On the opposite side of the object.

Step-by-step solution

Step 1: Given data

• The object stands on the common central axis of two thin symmetric lenses.
• Distance between object and lenses 1,$p_1=+8\text{cm}$
• Distance between lens 1 and 2,$d=12\text{cm}$
• Lens 1 is diverging, focal length$f_1=-6cm$
• Lens 2 is converging, focal length$f_2=6\text{cm}$

Determining the concept

Using the relation between focal length, the image distance, and the object distance, find the image distance$i_2$. Then using the formula for the overall magnification find its value.

From the solution of part a and b answer part c, d and e.

Formulae are as follows:

The formula for focal length,$\frac{1}{f}=\frac{1}{p}+\frac{1}{i}$

Overall magnification,$M=m_1 m_2$

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

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

(a) Determining the image distance for the image produced by lens 2

For lens 1 which is diverging focal length,$f_1=-6cm$, object distance$p_1$.

Using the expression for focal length,

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

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

$\frac{1}{i_1}=\frac{p_1-f_1}{f_1{p}_1}$

$i_1=\frac{f_1{p}_1}{p_1-f_1}........................\left(i\right)$

$\begin{array}{rcl}{i}_1& =& \frac{-6\times 8}{8-\left(-6\right)}\\ & =& \frac{-48}{14}\\ & =& -3.4\text{cm}\\ & & \end{array}$

This serves as an object for lens 2,

$\begin{array}{rcl}{p}_2& =& d-i_1\\ & =& 12-(-3.42)\\ & =& 15.42cm\end{array}$

It is given that$f_2=6cm$

Modifying equation (1) for lens 2,

$\begin{array}{rcl}{i}_2& =& \frac{f_2{p}_2}{p_2-f_2}\\ & =& \frac{6\times 15.42}{15.42-6}\\ & =& +9.8\text{cm}\end{array}$

Therefore, the image produced by lens 2 is at +9.8 cm.

(b) Determine the overall lateral magnification, including sign, M

To find overall magnification use the formula

$M=m_1{m}_2$

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

$\therefore M=-\frac{i_1}{p_1}\times -\frac{i_2}{p_2}$

$M=\left(-\frac{\left(-3.42\right)}{8}\right)\times \left(-\frac{9.8}{15.42}\right)$

$M=-0.27$

Therefore, the overall magnification for the given lens system is $-0.27$.

(c) Determining whether the final image is real (R) or virtual (V)

Since the lens 2 is converging, the object for lens 2 is outside the focal point. The final image distance is positive.

Hence, the image formed by this lens system is real.

(d) Determining whether the final image is inverted (I) or non-inverted (NI)

The overall magnification for this lens system is negative which shows that image and the object have the opposite orientation.

Hence, the image is inverted.

(e) Determine whether the final image is on the same side of lens 2 as object O or on the opposite side.

The final image distance is positive, which shows that it is at positive side of the lens 2, that is on the opposite side of the object.

Hence, the image ison the opposite side from the object.

The focal length and overall magnification of two-lens system can be found using corresponding formulae. The nature of the image can be predicted from the characteristics of image formed due to given two-lens system.