Concave lens = diverging lens. It has a negative focal length (f < 0). It always forms virtual, erect, diminished images on the same side as the object.
Rule for ray parallel to principal axis through a concave lens: After refraction, the ray diverges. When the diverged ray is extended backwards (on the same side as the incident ray), it appears to come from the first principal focus F₁ — which is on the same side as the object (left side of the lens).
Why not other options? Option 1 (parallel) — that's only if no refraction occurs. Option 3 (2F) — that's for a convex lens when object is at infinity, image at 2F. Option 4 (second focus) — a convex lens converges a parallel ray to the second principal focus F₂; a concave lens diverges it.
Concave lens: parallel ray → diverges → appears to come from F₁ (first focus, same side as object)
Convex lens: parallel ray → converges → passes through F₂ (second focus, opposite side)
To locate an image formed by a lens, we use three standard rays. Any two of these are sufficient to find the image.
📌 Ray 1 — Parallel to principal axis: Convex: converges through F₂. Concave: diverges, appears to come from F₁.
📌 Ray 2 — Through optical centre: Passes straight through without bending (for both convex and concave).
📌 Ray 3 — Through first focus F₁: Convex: emerges parallel to principal axis. Concave: directed toward F₁, emerges parallel to principal axis.
1/v − 1/u = 1/f (Lens formula)
Magnification m = v/u = h'/h
Sign convention: distances measured from optical centre; rightward positive, leftward negative
For a concave lens: f is always negative. Object is always placed on the left (u is negative). Image is always virtual (v is negative, on same side as object), erect (m positive), and diminished (|m| < 1). No matter where the object is placed in front of a concave lens, the image is always virtual, erect, and smaller — between F₁ and the optical centre.
Power P = 1/f (in metres), unit = dioptre (D). Convex lens: positive power (converging). Concave lens: negative power (diverging). When lenses are placed in contact, powers add: P_total = P₁ + P₂ + P₃ + .... This is the principle used in spectacle lenses — a combination of lenses can give any desired focal length. Example: P₁ = +3D, P₂ = −1D → P_total = +2D → f = 50 cm converging lens.
📌 Object anywhere (∞ to optical centre): Image is always virtual, erect, diminished
📌 Object at ∞: Image at F₁ (virtual, point-sized)
📌 Object at 2F₁: Image between F₁ and optical centre (virtual, erect, size = object/3)
📌 Object at F₁: Image between F₁ and optical centre (virtual, erect, size = object/2)
📌 Object between F₁ and lens: Image between F₁ and optical centre (virtual, erect, slightly less diminished)
📌 Concave lens NEVER forms real image — unlike convex lens which can
📌 Object at ∞: Real image at F₂ (used in cameras, telescopes)
📌 Object beyond 2F: Real, inverted, diminished image between F₂ and 2F₂
📌 Object at 2F: Real, inverted, same-size image at 2F₂
📌 Object between F and 2F: Real, inverted, magnified image beyond 2F₂
📌 Object at F: Image at infinity (parallel rays — used in projectors)
📌 Object between F and lens: Virtual, erect, magnified image on same side (magnifying glass)
The focal length of a lens depends on its refractive index and radii of curvature: 1/f = (μ − 1)(1/R₁ − 1/R₂), where μ is the refractive index of the lens material relative to the surrounding medium, R₁ and R₂ are the radii of curvature of the two surfaces (with sign convention). This formula is used by opticians to grind lenses with specific focal lengths. If the lens is immersed in a medium with refractive index close to its own, the focal length becomes very large (lens loses its converging/diverging power).
When light travels from a denser medium (μ₂) to a rarer medium (μ₁), and the angle of incidence exceeds the critical angle θ_c, all light is reflected back — no refraction. Critical angle: sin θ_c = μ₁/μ₂ = 1/μ (if μ₁ = 1 for air). Applications: optical fibres (light trapped by TIR — used in internet cables, endoscopes), diamonds (cut to exploit TIR for brilliance), prisms (45°−45°−90° prism deflects light by 90° or 180° using TIR).
For refraction at a spherical surface: μ₂/v − μ₁/u = (μ₂ − μ₁)/R. This is the basis for deriving the lens formula — a lens has two spherical surfaces, and applying this formula to each surface and combining gives the lens maker's formula. This formula also applies to the human eye (cornea is a spherical refracting surface) and fish-eye view — objects appear at different distances when seen from water into air due to this refraction.