Waves and Light: Convex and Concave Lenses

Learning Targets

Define a lens and differentiate between convex and concave lenses based on their shape.
Explain how lenses use refraction to focus light and form images.
Identify the principal plane and understand the thin lens model as a simplification of lens behavior.
Predict the characteristics (location, size, orientation, and type) of images formed by convex and concave lenses based on the object's position.
Explain how lenses are used in various optical devices, including the human eye, telescopes, cameras, microscopes, and binoculars.

Focus Question

How are systems of lenses used to make optical devices?

New Vocabulary

lens: A piece of transparent material used to focus light and form an image.
convex lens: A lens that is thicker at the center than at the edges, often called a converging lens.
concave lens: A lens that is thinner in the middle than at the edges, often called a diverging lens.
thin lens equation: Relates the focal length of a thin lens to the object and image positions.
chromatic aberration: An effect where an object viewed through a lens appears to be ringed with color due to different wavelengths of light being refracted at slightly different angles.
achromatic lens: A system of two or more lenses with different indices of refraction used to reduce chromatic aberration.
nearsightedness: Also known as myopia, a condition where the focal length of the eye is too short, causing images to form in front of the retina.
farsightedness: Also known as hyperopia, a condition in which the focal length of the eye is too long, causing images to form behind the retina.

Review Vocabulary

transparent: A property of a medium that allows light to transmit through it with minimal reflection, enabling clear visibility of objects.
index of refraction: Represented by the symbol $n$ , it determines the angle of refraction as light crosses the boundary between mediums, defined as the ratio of the speed of light in a vacuum to its speed in the medium.

Types of Lenses

A lens is a piece of transparent material, such as glass or plastic, 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; it's a converging lens because it refracts parallel light rays to meet at a point when surrounded by a material with a lower index of refraction.
A concave lens is thinner in the middle than at the edges; it's a diverging lens because rays passing through it spread out when surrounded by a material with a lower index of refraction.
Snell’s law and geometry can predict the paths of rays passing through lenses.
The thin lens model simplifies problems by assuming all refraction occurs on a plane, called the principal plane, passing through the center of the lens.

Convex Lenses

If the object is more than twice the focal length ( $2F$ ) from a convex lens:
- Image is located between $F$ and $2F$ .
- Image is reduced.
- Image is inverted.
- Image is real.
If the object is at twice the focal length ( $2F$ ):
- Image is the same size.
- Image is located at $2F$ .
If the object is between the focal point ( $F$ ) and two focal lengths ( $2F$ ):
- Image is located beyond $2F$ .
- Image is enlarged.
- Image is inverted.
- Image is real.
If the object is at the focal point ( $F$ ):
- No image will be formed; the refracted rays will be parallel and form a beam.
If the object is between the focal point ( $F$ ) and the lens:
- Image is located farther from the lens than the object.
- Image is enlarged.
- Image is upright.
- Image is virtual.

Quantity	Sign (+/-)
f	+
xo	+
xi	+
m	−

Concave Lenses

Regardless of where the object is placed:
- Image is located between the lens and $F$ .
- Image is reduced.
- Image is upright.
- Image is virtual.

Quantity	Sign (+/-)
f	−
xo	+
xi	−
m	+

Lens Equations

The problems involve spherical thin lenses, which have faces with the same curvature as a sphere.
Based on the thin lens model and simplifications used for spherical mirrors, similar equations have been developed.
The thin lens equation relates the focal length ( $f$ ) of a spherical thin lens to the object position ( $xo$ ) and the image position ( $xi$ ):
- $\frac{1}{f} = \frac{1}{xo} + \frac{1}{xi}$
The magnification equation for spherical mirrors can also be used for spherical thin lenses:
- $m = -\frac{xi}{xo} = \frac{hi}{ho}$
 - Where $hi$ is the image height and $ho$ is the object height.

Sign Conventions

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

Lens Type	f	xo	xi	m	Image
Convex	+	xo > 2f	2f > xi > f	reduced	reduced, inverted
			xi > 2f	enlarged	enlarged,inverted
		f > xo > 0		xi	> xo (negative)
Concave	-	xo > 0		f	>

Example Problem

A 5.0-cm-tall block is positioned 25.0 cm from a convex lens with a focal length of 14.0 cm. Predict the position, height, and orientation of the block’s image.
- Known:
 - $x_o = 25.0 \text{ cm}$
 - $h_o = 5.0 \text{ cm}$
 - $f = 14.0 \text{ cm}$
- Unknown:
 - $x_i = ?$
 - $h_i = ?$
- Solution:
 - Use the thin lens equation to find the image location:
 - $\frac{1}{14} = \frac{1}{25} + \frac{1}{x_i}$
 - $x_i = 31.8 \text{ cm}$
 - Use the magnification equation to find the image height and orientation:
 - $m = -\frac{31.8}{25} = -1.27$
 - $hi = m \cdot ho = -1.27 \cdot 5.0 = -6.4 \text{ cm}$
 - 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.
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 spherical aberration, similar to spherical mirrors. To avoid this, slightly nonspherical lenses or a system of several lenses can be used.
A lens is like a prism, so different wavelengths of light are refracted at slightly different angles, causing light passing through a lens, especially near the edges, to be slightly dispersed.
Chromatic aberration occurs when an object viewed through a lens appears ringed with color.
Chromatic aberration is always present when a single lens is used but can be greatly reduced by an achromatic lens, which is a system of two or more lenses with different indices of refraction (e.g., a convex lens with a concave lens).

Lenses in Eyes

Light enters the eye through the cornea. The cornea provides most of the focusing because the air-cornea surface has the greatest difference in indices of refraction.
Light then passes through the lens and focuses onto the retina 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.
Accommodation is the process where muscles surrounding the lens contract or relax, 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.
Many people's eyes do not focus sharp images on the retina, with images focused either in front of or behind the retina.
External lenses, like eyeglasses or contact lenses, adjust the focal length and move images to the retina.
Nearsightedness (Myopia):
- The focal length of the eye is too short, and images are formed in front of the retina.
- Concave lenses correct this by diverging light, increasing the image distance and forming images on the retina.
Farsightedness (Hyperopia):
- 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.
- The image from the lens becomes the object for the eye, 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.
A virtual image is produced that is upright and larger than the first image.
The final image is still inverted because the first image was already 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 takes a photograph, the shutter-release button raises the mirror.
The light travels along a straight path to focus on the film instead of being diverted to the prism.

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.
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.

Quizzes

Which is a piece of transparent material, such as glass or plastic, that is used to focus light and form an image?
- A lens
Which type of lens is thicker at the center than at the edges?
- A convex lens
Which type of lens is thicker at the center than at the edges?
- A convex lens
Which is the effect when an object viewed through a lens appears to be ringed with color?
- Chromatic aberration
Which is a system of two or more lenses, such as a convex lens with a concave lens, that have different indices of refraction?
- Achromatic lens