Hello, and welcome to today's physics lesson. We are going to explore Chapter Five: Refraction through a Lens. By the end of this session, you will understand what lenses are, how they bend light, and how images are formed. Let us begin.
A lens is a transparent refracting medium bounded by either two spherical surfaces, or one spherical surface and one plane surface. A plane surface can be treated as a spherical surface of infinite radius of curvature. There are two main types: convex lenses, which are thicker in the middle, and concave lenses, which are thinner in the middle. You encounter lenses every day — in spectacles, cameras, microscopes, and even your own eyes.
Lenses are of two fundamental types. First, the convex lens, also called a converging lens. It is thick in the middle and thin at the edges, bulging outward. When light passes through it, the rays bend inward and converge toward a point. Convex lenses come in three varieties: biconvex with both surfaces curved outward, plano-convex with one flat surface and one curved surface, and concavo-convex with one concave and one convex surface where the convex surface dominates, making it thicker in the middle and giving it converging action. Do not confuse concavo-convex with convexo-concave — the latter is a diverging lens.
Second, the concave lens, also called a diverging lens. It is thin in the middle and thick at the edges, curving inward. Light rays passing through it spread apart or diverge. Concave lenses also come in three types: biconcave with both surfaces curved inward, plano-concave with one flat and one concave surface, and convexo-concave where the concave surface dominates, making it thinner in the middle and giving it diverging action. Remember, a convexo-concave lens is thinner in the middle and diverges light, unlike a concavo-convex lens.
To understand how a lens works, imagine it as a combination of prisms. A convex lens can be thought of as having prisms at the top with their bases downward and at the bottom with their bases upward, while the center acts like a parallel-sided glass slab. When parallel rays strike a convex lens, the upper prism bends light downward toward its base, and the lower prism bends light upward toward its base. The central ray passes straight through. The result — all rays converge to a single point.
For a concave lens, the prism arrangement is reversed. The upper prism has its base upward, bending light upward toward its base, while the lower prism has its base downward, bending light downward toward its base. The rays appear to spread out from a point on the same side as the incoming light.
Now let us learn the technical terms associated with lenses. These definitions are essential for understanding image formation. Every lens has two centers of curvature, one for each spherical surface, marked C₁ and C₂. These are the centers of the spheres of which the lens surfaces form a part. The radii of curvature, R₁ and R₂, are the distances from these centers to the lens surfaces.
The principal axis is the straight line passing through both centers of curvature and the optical center of the lens. The optical center is a special point on the principal axis where a ray of light passes through undeviated — it continues in the same direction. For a thin lens, we consider this point to be at the center of the lens.
A lens has two principal foci, F₁ and F₂, one on each side. The first focal point F₁ is where rays must originate, or appear to originate, so that after refraction, they emerge parallel to the principal axis. The second focal point F₂ is where rays parallel to the principal axis converge after refraction for a convex lens, or appear to diverge from for a concave lens. For a convex lens, the second focus F₂ is real — rays actually pass through it. For a concave lens, the second focus F₂ is virtual — rays only appear to come from it.
The focal length is the distance from the optical center to a focus, denoted f. When the medium is the same on both sides of the lens, both focal lengths are equal in magnitude: f₁ equals f₂. By convention, we usually refer to the second focal length f₂ when speaking of the focal length of a lens.
Here is a precise definition you must remember. The focal length of a lens is the distance of the focus from the optical center of the lens. For a thin lens, the first focal length f₁ equals the distance from optical center O to first focus F₁, and the second focal length f₂ equals the distance from O to second focus F₂.
The focal length depends on two factors: the refractive index µ of the lens material relative to its surroundings, and the radii of curvature R₁ and R₂ of its surfaces. A thick lens has a shorter focal length than a thin lens of the same material. If you place a lens in water instead of air, its focal length increases because the difference between the refractive indices of the lens material and water is smaller than between the lens material and air, leading to less bending of light and a longer focal length. If you cover part of a lens, the focal length remains unchanged — only the intensity of the image decreases, while position, size and nature of the image stay the same.
Now we turn to image formation by lenses. To construct ray diagrams, we use three principal rays. First, a ray passing through the optical center travels undeviated. Second, a ray parallel to the principal axis, after refraction, passes through the second focus for a convex lens or appears to come from the second focus for a concave lens. Third, a ray passing through the first focus, or directed toward it, emerges parallel to the principal axis after refraction.
Where these refracted rays meet, or appear to meet, we find the image. A real image forms where rays actually converge — it can be projected on a screen and is always inverted. A virtual image forms where rays only appear to diverge from — it cannot be projected on a screen and is always erect.
Let us examine image formation by a convex lens for different object positions.
When the object is at infinity, parallel rays from the object converge at the second focus F₂ after refraction. It is real, inverted, and highly diminished — practically a point. This principle is used in burning glasses to focus sunlight.
When the object is beyond twice the focal length, that is beyond 2F₁, the image forms between the focus F₂ and twice the focal length 2F₂ on the other side of the lens. It is real, inverted, and diminished. Camera lenses work on this principle.
When the object is at twice the focal length, at 2F₁, the image forms at twice the focal length 2F₂ on the other side of the lens. It is real, inverted, and the same size as the object.
When the object is between the focus F₁ and twice the focal length 2F₁, the image forms beyond twice the focal length 2F₂ on the other side of the lens. It is real, inverted, and magnified. This is how projectors create enlarged images on screens.
When the object is at the focus F₁, the refracted rays become parallel and the image forms at infinity. It is real, inverted, and highly magnified.
Finally, when the object is between the lens and the focus F₁, that is when the object distance is less than the focal length, something remarkable happens. The image forms on the same side of the lens as the object. It is virtual, erect, and magnified. This is how a magnifying glass works.
For a concave lens, the situation is simpler and follows a consistent pattern. Regardless of where the object is placed, the image is always virtual, erect, diminished, and located between the optical center and the focus on the same side of the lens as the object. As the object moves from infinity toward the lens, the image shifts from the focus toward the optical center and gradually increases in size, but it always remains smaller than the object.
Now we introduce the sign convention and the lens formula. We use the Cartesian sign convention. The optical center is the origin. The object is always placed to the left of the lens. Distances measured in the direction of incident light are positive; opposite to it are negative. Distances above the principal axis are positive; below are negative.
For a convex lens, the focal length is positive. For a concave lens, it is negative. The object distance is always negative. The image distance is positive for real images and negative for virtual images.
Here is the lens formula you must know. One over the image distance v minus one over the object distance u equals one over the focal length f. Symbolically: 1/v − 1/u = 1/f. Here, v is the image distance from the optical center, u is the object distance from the optical center, and f is the focal length.
Linear magnification tells us how large the image is compared to the object. It equals the image distance v divided by the object distance u, or equivalently the image height h₂ divided by the object height h₁. Symbolically: m = v/u or m = h₂/h₁. Here, m is the linear magnification, v and u are image and object distances, and h₂ and h₁ are image and object heights. For real images, magnification is negative, indicating inversion. For virtual images, it is positive, indicating an erect image. A concave lens always produces a magnification between zero and one — the image is always diminished.
Let us work through an example. Imagine you place a candle 24 centimeters in front of a convex lens with focal length 8 centimeters. By the sign convention, u equals minus 24 centimeters and f equals plus 8 centimeters. Rearranging the lens formula: one over v equals one over f plus one over u, which becomes one over v equals one over 8 plus one over minus 24. This equals 3 minus 1 over 24, which equals 2 over 24, or one over 12. So v equals 12 centimeters. The image forms 12 centimeters behind the lens. The magnification m equals v over u, so 12 divided by minus 24, giving minus one-half. The image is real, inverted, and half the size of the object.
Now we define the power of a lens. The power of a lens is defined as the reciprocal of its focal length expressed in meters. Its unit is the dioptre, symbol D.
Here is the precise statement. The power P of a lens in dioptres equals the reciprocal of its focal length f in meters. Symbolically: P = 1/f, where f is in meters.
A convex lens has positive power; a concave lens has negative power. A lens with focal length 20 centimeters, or 0.2 meters, has power plus 5 dioptres if convex, or minus 5 dioptres if concave. Thus, thicker lenses with shorter focal lengths have greater power. When two thin lenses are in contact, their powers add algebraically.
A magnifying glass, or simple microscope, is simply a convex lens of short focal length. To use it, place the object within the focal length. The lens produces a virtual, erect, and magnified image at the least distance of distinct vision — about 25 centimeters for a normal eye. The magnifying power is given by 1 + D/f, where D is 25 centimeters and f is the focal length. Shorter focal lengths give greater magnification.
Lenses have countless applications. Convex lenses serve as camera objectives, projector lenses, and the lenses in our eyes. They correct long-sightedness in spectacles. Concave lenses correct short-sightedness and serve as eyepieces in Galilean telescopes. Combinations of lenses overcome defects like chromatic aberration.
Let us recap the key takeaways from this lesson.
First, convex lenses converge light and can form real or virtual images depending on object position, while concave lenses always diverge light and form only virtual, diminished images.
Second, the lens formula 1/v − 1/u = 1/f relates object distance u, image distance v, and focal length f for any thin lens.
Third, linear magnification m equals v over u, with negative values indicating real inverted images and positive values indicating virtual erect images.
Fourth, lens power P in dioptres equals one over focal length f in meters, with convex lenses having positive power and concave lenses having negative power.
Fifth, a magnifying glass or simple microscope uses a convex lens with the object placed within its focal length to produce an enlarged virtual image at the least distance of distinct vision.
Sixth, the Cartesian sign convention consistently assigns positive and negative values to distances based on direction from the optical center.
That brings us to the end of our exploration of refraction through lenses. Remember to practice drawing ray diagrams using principal rays and solving numerical problems using the lens formula. The more you work with these concepts, the clearer they become. Keep curious, keep questioning, and I look forward to seeing you in the next lesson.