Convex and Concave Lenses Notes

Convex and Concave Lenses

Focus Question

• How are systems of lenses used to make optical devices?

New Vocabulary

• lens
• convex lens
• concave lens
• thin lens equation
• chromatic aberration
• achromatic lens
• nearsightedness
• farsightedness

Review Vocabulary

• Transparent: A property of a medium that allows it to transmit light and reflect a fraction of the light, allowing objects to be seen clearly through it.
• Index of Refraction: Determines the angle of refraction of light as it crosses the boundary between mediums. For a given medium, it is the ratio of the speed of light in a vacuum to the speed of light in that medium; represented by the symbol $n$.

Types of Lenses

• A lens is a piece of transparent material, such as glass or plastic, that is used to focus light and form an image.
• When light passes through a lens, refraction occurs at the two lens surfaces.
• A convex lens is thicker at the center than at the edges. A convex lens often is called a converging lens because, when surrounded by material with a lower index of refraction, it refracts parallel light rays so that the rays meet at a point.
• A concave lens is thinner in the middle than at the edges. A concave lens often is called a diverging lens because, when surrounded by material with a lower index of refraction, rays passing through it spread out.
• Using Snell’s law and geometry, you can predict the paths of rays passing through lenses.
• To simplify such problems, assume that all refraction occurs on a plane, called the principal plane, that passes through the center of the lens.
• This approximation, called the thin lens model, applies to all the lenses that you will learn about in this section.

Convex Lenses

• If the object is more than twice the focal length $(2F)$ from a concave lens, the image formed will be:
  • Located between $F$ and $2F$
  • Reduced
  • Inverted
  • Real
• If the object is at twice the focal length $(2F)$, the image will be the same size and located at $2F$.

• If the object is between the focal point $(F)$ and two focal lengths $(2F)$, the image formed will be:
  • Located beyond $2F$
  • Enlarged
  • Inverted
  • Real
• If the object is at the focal point $(F)$, no image will be formed. Instead the refracted rays will be parallel and form a beam.

• If the object is between the focal point $(F)$ and the lens, the image formed will be:
  • Located farther from the lens than the object
  • Enlarged
  • Upright
  • Virtual

Concave Lenses

• Regardless of where the object is placed, the image formed will be:
  • Located between the lens and $F$
  • Reduced
  • Upright
  • Virtual

Lens Equations

• The problems that you will solve involve spherical thin lenses, lenses that have faces with the same curvature as a sphere.
• Based on the thin lens model, as well as the other simplifications used in solving problems for spherical mirrors, equations have been developed that look exactly like the equations for spherical mirrors.
• The thin lens equation relates the focal length of a spherical thin lens to the object position and the image position.
• Thin Lens Equation: $\frac{1}{f} = \frac{1}{x<em>o} + \frac{1}{x</em>i}$
  • $f$ = focal length
  • $x_o$ = object position
  • $x_i$ = image position
• The magnification equation for spherical mirrors also can be used for spherical thin lenses.
• Magnification: $m = - \frac{x<em>i}{x</em>o}= \frac{h<em>i}{h</em>o}$
  • $h_i$ = image height
  • $h_o$ = object height

Lens Equations (Sign Conventions)

• It is important that you use the proper sign conventions when using these equations. The table summarizes these conventions.

Example Problem

A 5.0-cm-tall block is positioned 25.0 cm from a convex lens. The focal length of the lens is 14.0 cm. Predict the position, height, and orientation of the block’s image.

• Sketch and Analyze the Problem
  • Sketch the situation and draw a ray diagram.
  • List the knowns and unknowns.
    • KNOWN
      • x_o = 25.0 cm
      • h_o = 5.0 cm
      • f = 14.0 cm
    • UNKNOWN
      • x_i = ?
      • h_i = ?
• Solve for the Unknown
  • Use the thin lens equation to find the image location.
    \frac{1}{f} = \frac{1}{xo} + \frac{1}{xi}
    \frac{1}{14} = \frac{1}{25} + \frac{1}{xi} xi = 31.8 cm
  • Use the magnification equation to find the image height and orientation.
    m = -\frac{xi}{xo} = \frac{hi}{ho}
    hi = -\frac{xi * ho}{xo}
    h_i = - \frac{31.8 * 5}{25} = -6.4$$ cm
• The negative sign indicates that the image is inverted. Thus the image is 6.4 cm tall, inverted, and located 31.8 cm on the other side of the lens.
• Evaluate the Answer
  • For an object between 1 and 2 focal lengths from a convex lens, the image should be enlarged and inverted. This agrees with our answer.

Defects of Spherical Lenses

• Spherical lenses exhibit a spherical aberration, just as spherical mirrors do. To avoid this, slightly nonspherical lenses or a system of several lenses can be used.
• In addition, a lens is like a prism, so different wavelengths of light are refracted at slightly different angles, which causes the light that passes through a lens, especially near the edges, to be slightly dispersed.
• An object viewed through a lens appears to be ringed with color, an effect called chromatic aberration.
• Chromatic aberration is always present when a single lens is used, but it can be greatly reduced by an achromatic lens, which is a system of two or more lenses, such as a convex lens with a concave lens, that have different indices of refraction.

Lenses in Eyes

• Light enters the eye through the cornea. Light entering the eye is primarily focused by the cornea because the air-cornea surface has the greatest difference in indices of refraction.
• The light then passes through the lens and focuses onto the retina that is at the back of the eye.
• The lens is responsible for the fine focus that allows you to clearly see both distant and nearby objects. Using a process called accommodation, muscles surrounding the lens can contract or relax, thereby changing the shape and focal length of the lens.
• Specialized cells on the retina absorb this light and send information about the image along the optic nerve to the brain.
• The eyes of many people do not focus sharp images on the retina. Instead, images are focused either in front of the retina or behind it.
• External lenses, in the form of eyeglasses or contact lenses, are needed to adjust the focal length and move images to the retina.
• Nearsightedness (Myopia): The focal length of the eye is too short to focus light on the retina and images are formed in front of the retina. Concave lenses correct this by diverging light, thereby increasing images’ distances from the lens and forming images on the retina.
• Farsightedness (Hyperopia): The condition in which the focal length of the eye is too long and images are formed past the retina. Convex lenses produce virtual images farther from the eye than the associated objects. The image from the lens becomes the object for the eye, thereby correcting the defect.

Refracting Telescopes

• An astronomical refracting telescope uses lenses to magnify distant objects.
• Parallel light rays from distant stars enter the objective convex lens and are focused as a real, inverted image at the focal point of the objective lens.
• This image then becomes the object for the convex lens of the eyepiece and is located between the eyepiece lens and its focal point.
• This means that a virtual image is produced that is upright and larger than the first image.
• However, because the first image was already inverted, the final image is still inverted.

Cameras

• As light enters the camera, it passes through an achromatic lens.
• This lens system refracts the light much like a single convex lens would, forming an image that is inverted on the reflex mirror.
• The image is reflected upward to a prism that redirects the light to the viewfinder.
• When the person holding the camera takes a photograph, he or she presses the shutter- release button, which briefly raises the mirror.
• The light, instead of being diverted upward to the prism, then travels along a straight path to focus on the film.

Microscopes

• Microscopes are used to view small objects.
• In a simple compound microscope, the object is located between one and two focal lengths from the objective lens.
• The objective lens produces a real image that is inverted and larger than the object.
• This image then becomes the object for the eyepiece and is located between the eyepiece and its focal point.
• A virtual image is produced that is upright and larger than the image of the objective lens.
• Thus, the viewer sees an image that is inverted and greatly larger than the original object.

Binoculars

• Binoculars, like telescopes, produce magnified images of faraway objects. Each side of the binoculars is like a small telescope.
• Light enters a convex objective lens, which inverts the image.
• The light then travels through two prisms that use total internal reflection to invert the image again.
• The viewer sees an image that is upright compared to the object.
• Binoculars provide a three- dimensional view of a distant object.

Quiz Questions and Answers

1. Which is a piece of transparent material, such as glass or plastic, that is used to focus light and form an image?
   • A. a lens (CORRECT)
2. Which type of lens is thicker at the center than at the edges?
   • D. a convex lens (CORRECT)
3. Which type of lens is thicker at the center than at the edges?
   • D. a convex lens (CORRECT)
4. Which is the effect when an object viewed through a lens appears to be ringed with color?
   • B. chromatic aberration (CORRECT)
5. Which is a system of two or more lenses, such as a convex lens with a concave lens, that have different indices of refraction?
   • C. achromatic lens (CORRECT)