Image Formation by Convex and Concave Lenses - Tenth Grade Science
Lenses are transparent optical devices bounded by two curved surfaces, or by one curved and one plane surface, that refract light to converge or diverge rays and produce real or virtual images. Understanding how rays refract through convex and concave lenses provides the foundational principles behind eyeglasses, microscopes, telescopes, cameras, and the human visual apparatus.
Terminology of Spherical Lenses
Before analyzing ray trajectories, precise definitions of the structural and geometric components of spherical lenses must be established.
Convex and Concave Geometries
A convex lens (or converging lens) is thicker at the central optical axis than at the outer perimeter edges. It possesses two outwardly bulging spherical faces. A concave lens (or diverging lens) is thinner at the central optical axis and noticeably thicker along its peripheral rim, bounded by two inwardly curved spherical surfaces.
CONVEX LENS CONCAVE LENS
(Converging) (Diverging)
.-. \ /
.' '. \ /
/ | \ | |
| | | | |
| O | | O |
| | | | |
\ | / | |
'. .' / \
'-' / \
Thick center Thin center
Thin edges Thick edges
Optical Center, Principal Axis, and Radii of Curvature
Each curved face of a lens forms a segment of an imaginary sphere. The center of this imaginary sphere is designated as the Center of Curvature ($C$), and its radius is the Radius of Curvature ($R$). Because a typical lens has two curved surfaces, it possesses two centers of curvature, labeled $C_1$ and $C_2$. The imaginary straight line passing through both centers of curvature is the Principal Axis.
The Optical Center ($O$) is the central geometric point of the lens lying on the principal axis. An incident ray of light passing directly through the optical center emerges without experiencing any measurable lateral displacement or deviation. The effective functional diameter of the circular lens boundary exposed to incoming light is termed the Aperture.
ANATOMY OF A BIPHYSICAL LENS
Principal Focus Optical Center Principal Focus
F1 O F2
Principal Axis -----------*-----------------O----------------*-----------
( -f ) ( +f )
<-- Aperture -->
Principal Focus and Focal Length
When a beam of light rays parallel to the principal axis falls on a thin convex lens, the refracted rays converge to a single sharp point on the opposite side of the principal axis. This convergence point is known as the Principal Focus ($F_2$). For a concave lens, parallel incident rays diverge upon exiting the lens; when projected backward along their trajectories, they appear to emanate from a common virtual point on the incident side of the lens, known as its Principal Focus ($F_1$).
The distance between the optical center $O$ and either principal focus is the Focal Length ($f$). For a thin lens with symmetrical curvatures surrounded by identical media on both sides, the focal lengths on either side are equal in magnitude ($f_1 = f_2 = f$).
Rules for Ray Tracing in Lenses
To determine the position, size, and nature of an image formed by a lens, geometric ray diagrams are constructed using three standard principal rays.
Rule 1: Rays Parallel to the Principal Axis
An incident ray propagating parallel to the principal axis passes directly through the principal focus on the opposite side after refracting through a convex lens. In a concave lens, an incident ray parallel to the principal axis refracts outward such that its backward extension appears to diverge from the principal focus on the incident side.
CONVEX LENS (Parallel Ray) CONCAVE LENS (Parallel Ray)
\ / Refracted ray
-------+-------> F2 --------+< - - - -
Parallel / Parallel\ (Virtual trace to F1)
Rule 2: Rays Passing Through the Focus
An incident ray passing through (or directed toward) the principal focus emerges from the lens parallel to the principal axis after refraction. This behavior illustrates the principle of optical reversibility: reversing the direction of light preserves the exact geometric path.
Rule 3: Rays Directed Through the Optical Center
An incident ray passing directly through the optical center $O$ of a thin lens traverses the lens without any net change in angular direction. The ray continues straight along its original path.
RAY THROUGH OPTICAL CENTER (UNDEDEVIATED)
Lens
|
Ray |
--------------->O-----------------> Undeviated Ray
|
|
Image Formation by Convex Lenses
The character of an image generated by a convex lens depends entirely on the distance of the object from the optical center relative to the focal point $F$ and $2F$.
Object at Infinity
When an object is placed at infinity, incident rays strike the lens parallel to the principal axis. All refracted rays converge precisely in the focal plane at principal focus $F_2$. The resulting image is real, inverted, and diminished to a sharp point.
OBJECT AT INFINITY (CONVEX LENS)
Rays from \
distant object \
-------------------+-------------------
------------------/ \------------------
/ \
----------------+-----+----------------
/ \
\--> Focus F2 (Point image)
Object Beyond 2F1
When an object of finite height $AB$ stands on the principal axis beyond twice the focal length ($u > 2f$), the refracted rays intersect between $F_2$ and $2F_2$ on the opposite side. The image formed is real, inverted, and smaller than the object (diminished).
Object at 2F1
When the object is situated exactly at distance $2F_1$ ($u = 2f$), the refracted rays converge at position $2F_2$ on the opposite side. The formed image is real, inverted, and exactly equal in height to the object ($h' = -h$).
Object Between F1 and 2F1
As the object is brought closer, between $F_1$ and $2F_1$ ($f < u < 2f$), the refracted rays intersect at a location beyond $2F_2$. The resulting image is real, inverted, and magnified (larger than the object). This configuration serves as the projection mechanism in movie theaters and slide projectors.
Object at F1
When the object is positioned exactly at the primary focal point $F_1$ ($u = f$), the refracted rays emerge completely parallel to one another. Because parallel lines intersect only at infinity, the image is formed at infinity. The image is real, inverted, and magnified to immense proportions. This arrangement powers searchlights, lighthouse beacons, and stage spotlights.
Object Between F1 and Optical Center O
When the object is brought inside the focal length ($u < f$), the refracted rays diverge upon exiting the opposite side of the lens. To an observer viewing from the transmission side, these diverging rays appear to emanate from a virtual intersection point behind the object on the same side of the lens. The resulting image is virtual, erect, and magnified. This optical configuration is the operating principle of a simple handheld magnifying glass.
MAGNIFYING GLASS CONFIGURATION (OBJECT INSIDE F)
Virtual Image (A'B') Object (AB)
^ ^
| | Lens
| | |
A' A |
| | | / Refracted ray
------------+-----------------------+------O-----+----------------
B' B| F1 | /
| | | /
: : | /
: - - - - - - - - - - - - - - -+ / (Backward extensions meet)
: |/
SUMMARY TABLE: CONVEX LENS IMAGING
Object Position Image Position Nature Relative Size
----------------------------------------------------------------------
Infinity At Focus F2 Real, Inverted Point-sized
Beyond 2F1 Between F2 & 2F2 Real, Inverted Diminished
At 2F1 At 2F2 Real, Inverted Same size
Between F1 & 2F1 Beyond 2F2 Real, Inverted Magnified
At Focus F1 At Infinity Real, Inverted Highly magnified
Between F1 & O Same side (virtual)Virtual, Erect Magnified
Image Formation by Concave Lenses
Unlike convex lenses, concave lenses produce diverging rays for all real object positions. Consequently, they form only virtual, erect, and diminished images regardless of object distance.
Object at Infinity
When incident rays originate from an infinitely distant source, they arrive parallel to the principal axis. Upon refraction, the rays diverge outward away from the optical axis. When traced backward, these diverging trajectories converge at the focal point $F_1$ on the same side as the incoming light. The image formed is virtual, erect, and reduced to a dimensionless point.
Object at Finite Distance
For an object placed at any finite distance from the optical center, one ray traveling parallel to the principal axis diverges outward from $F_1$, while a second ray directed toward the optical center $O$ continues undeflected. The backward extension of the first ray intersects the undeflected ray between the optical center $O$ and the focal point $F_1$. The resulting image is consistently virtual, erect, and diminished.
CONCAVE LENS RAY TRACING (ANY FINITE DISTANCE)
Object (AB)
^ Lens
| | / Diverging ray
A | /
| a' | /
-----------------------+----*----+----O-----------------------
B b'F1 |
| |
|<--f---->|
Virtual, erect, diminished image (a'b') formed between F1 and O
SUMMARY TABLE: CONCAVE LENS IMAGING
Object Position Image Position Nature Relative Size
----------------------------------------------------------------------
Infinity At Focus F1 Virtual, Erect Point-sized
Finite Distance Between F1 and O Virtual, Erect Diminished
Chapter Exercises and Solutions
Exercise 1: An object is positioned 30 cm in front of a symmetrical convex lens having a focal length of 15 cm. Describe the exact nature, position, and size of the image produced without executing complex algebraic derivations.
Solution 1: The focal length of the convex lens is $f = 15 \text{ cm}$. Twice the focal length is $2f = 2 \times 15 \text{ cm} = 30 \text{ cm}$. The object is located at an object distance of 30 cm, which corresponds exactly to the $2F_1$ position. According to the geometric ray rules for a convex lens, whenever an object is situated precisely at $2F_1$, the refracted rays converge at $2F_2$ on the opposite side of the lens at an identical distance of 30 cm from the optical center. The resulting image is real, inverted, located 30 cm behind the lens, and has a height exactly equal to that of the original object.
Exercise 2: Contrast the optical behavior and image characteristics produced when a reading card is placed 5 cm away from a convex lens of focal length 10 cm versus a concave lens of focal length 10 cm.
Solution 2: For the convex lens of focal length 10 cm, an object distance of 5 cm places the reading card between the optical center and the principal focus ($u < f$). Under this condition, the convex lens functions as a simple magnifier, refracting the rays such that their backward virtual extensions intersect behind the card on the same side. The observer sees an image that is virtual, upright (erect), and significantly magnified. For the concave lens of focal length 10 cm, placing the card at 5 cm results in diverging refraction that forms a virtual image between the card and the optical center. The image seen through the concave lens is virtual, upright (erect), but noticeably reduced in size (diminished).
Exercise 3: Explain why concave lenses are uniquely suited for wide-angle peepholes installed in building entrance doors, referencing ray convergence principles.
Solution 3: A concave lens always diverges incoming light rays, causing its refracted rays to appear as though they originate from a compact region situated between the optical center and the focal point. Because it compresses light entering from very wide exterior angles into a small visual cone, it produces a virtual, erect, and diminished view covering an expansive field of view. A resident looking through the peephole from the interior can observe an entire hallway and recognize visitors standing anywhere in front of the door. A convex lens could not perform this role because objects placed beyond its focal length would produce inverted, disorienting images, while objects closer than the focal point would be magnified beyond the small viewport.