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 p1. Lens 2 is mounted within the farther boxed region, at distance 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 i2 for the image produced by lens 2 (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 (R) or virtual (V), (d) inverted (I) from object O or non-inverted (NI), and (e) on the same side of lens 2 as object O or on the opposite side.

Short Answer

Expert verified
  1. Image distance for the image produced by lens 2,i2=-3.4cm .
  2. Overall lateral magnification, including sign,M=-1.1.
  3. Virtual (V).
  4. Inverted (I).
  5. On the same side as the object.

Step by step solution

01

Step 1: Given data

  • The object stands on the common central axis of two thin symmetric lenses.
  • Distance between object and lens 1,p1=+12cm .
  • Distance between lenses 1 and 2,d=30cm .
  • Lens 1 is converging, focal length,f1=8cm .
  • Lens 2 is diverging, focal length, f2=-8cm.
02

Determining the concept

By using the relation between focal length, image distance, and object distance, find image distance i2. Then by using the formula for overall magnification, find the same.

From parts a and b, answer parts c, d, and e.

Formulae are as follows:

  • The formula for focal length,1f=1p+1i.
  • Overall magnification,M=m1m2.
  • Magnification,m=-ip.

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

03

(a) Determining the image distance for the image produced by lens 2, i2

For lens 1, focal lengthf1, object distancep1:

Using the expression for focal length,

1f1=1p1+1i11i1=1f1-1p11i1=p1-f1f1p11f1=1p1+1i11i1=1f1-1p11i1=p1-f1f1p1

Solving further as,

i1=f1p1p1-f1

Substitute the values in the above expression, and we get,

i1=8×1212-8=24cm

This serves as an object for lens 2, which is diverging:

p2=d-i1=30-24=6cm
It is given that f2=-8cm.

Modifying equation 1 for lens 2,

i2=f2p2p2-f2=-8×66--8=-3.4cm

Therefore, the image produced by lens 2 is at -3.4 cm.

04

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

To find the overall magnification use the formula,

M=m1m2

Magnification,

m=-ip

M=-i1p1×-i2p2=-2412×--3.46=-1.1

Therefore, the overall magnification for the given lens system is –1.1.
05

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

Since lens 2 is diverging, the object for lens 2 is inside the focal point. The final image distance is negative.

Hence the image formed by this lens system is virtual.

06

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

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

Hence, the image is inverted.

07

(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 negative, which is on the same side of the object relative to lens 2, which is diverging.

Hence, the image is diverging.

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

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Most popular questions from this chapter

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 mθ of the eyepiece, and (e) the overall magnification M of the microscope?


Isaac Newton, having convinced himself (erroneously as it turned out) that chromatic aberration is an inherent property of refracting telescopes, invented the reflecting telescope, shown schematically in Fig. 34-59. He presented his second model of this telescope, with a magnifying power of 38, to the Royal Society (of London), which still has it. In Fig. 34-59, incident light falls, closely parallel to the telescope axis, on the objective mirror. After reflection from the small mirror (the figure is not to scale), the rays form a real, inverted image in the focal plane (the plane perpendicular to the line of sight, at focal point F). This image is then viewed through an eyepiece. (a) Show that the angular magnification for the device is given by Eq. 34-15:

mθ=fob/fey

fob

the focal length of the objective is a mirror and

feyis that of the eyepiece.

(b) The 200 in. mirror in the reflecting telescope at Mt. Palomar in California has a focal length of 16.8 m. Estimate the size of the image formed by this mirror when the object is a meter stick 2.0 km away. Assume parallel incident rays. (c) The mirror of a different reflecting astronomical telescope has an effective radius of curvature of 10 m (“effective” because such mirrors are ground to a parabolic rather than a spherical shape, to eliminate spherical aberration defects). To give an angular magnification of 200, what must be the focal length of the eyepiece?

Two thin lenses of focal lengths f1andf2 are in contact and share the same central axis. Show that, in image formation, they are equivalent to a single thin lens for which the focal length is f=f1f2(f1+f2).

58 through 67 61 59 Lenses with given radii. Object stands in front of a thin lens, on the central axis. For this situation, each problem in Table 34-7 gives object distance , index of refraction n of the lens, radius of the nearer lens surface, and radius of the farther lens surface. (All distances are in centimetres.) Find (a) the image distance and (b) the lateral magnification m of the object, including signs. Also, determine whether the image is (c) real (R) or virtual , (d) inverted from object or non-inverted (NI), and (e) on the same side of the lens as object or on the opposite side

Light travels from point A to B point via reflection at point O on the surface of a mirror. Without using calculus, show that length AOB is a minimum when the angle of incidence θis equal to the angle of reflection ϕ.

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