- Transparent medium bounded by two surfaces, at least one of which is curved.
- Every lens is part of some sphere.
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Types of Lenses
Convex Lens (Converging Lens):
- Thicker at the middle, thinner at the edges.
- Positive focal length.
Concave Lens (Diverging Lens):
- Thinner at the middle, thicker at the edges.
- Negative focal length.
Sr. No. Convex Lens Concave Lens 1 Double convex Double concave 2 Plano convex Plano concave 3 Concavo convex Convexo concave -
Important Definitions
- Centre of Curvature ($C_1, C_2$): Centre of sphere from which lens surface is obtained. Every lens has two centers of curvature.
- Radius of Curvature ($R_1, R_2$): Radius of sphere from which lens surface is obtained. Every lens has two radii of curvature.
- Principal Axis: Line joining the two centers of curvature.
- Principal Focus (F):
- For convex lens: Point of convergence of refracted light ray (real point).
- For concave lens: Point from which refracted light ray appears to diverge (imaginary point).
- Every lens has two foci, one on each side.
- Optical Centre (O): Point inside the body of lens through which light rays pass undeviated.
- Focal Length (f): Distance between principal focus and optical center. Positive for convex, negative for concave.
- Aperture: Size of diameter of lens.
- Power of a Lens (P): Ability to deviate light ray from its original path.
- $P = \frac{1}{f}$ (where $f$ is focal length in meters).
- If $f$ is in cm, $P = \frac{100}{f}$.
- SI unit: Dioptre (D). $1~D = 1~m^{-1}$.
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Combination of Lenses
- General rule for combination of 'n' lenses: $P = P_1 + P_2 + ... + P_n$.
- $\frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2} + ... + \frac{1}{f_n}$.
- For two double convex lenses: $P = P_1 + P_2$, $\frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2} = \frac{f_1+f_2}{f_1f_2} \implies f = \frac{f_1f_2}{f_1+f_2}$. (Behaves as a convex lens).
- For two concave lenses: $P = P_1 + P_2$, $\frac{1}{f} = \frac{1}{-f_1} + \frac{1}{-f_2} = -\frac{(f_1+f_2)}{f_1f_2} \implies f = -\frac{f_1f_2}{f_1+f_2}$. (Behaves as a concave lens).
- For one convex and one concave lens: $\frac{1}{f} = \frac{1}{f_1} + \frac{1}{-f_2} = \frac{f_2-f_1}{f_1f_2} \implies f = \frac{f_1f_2}{f_2-f_1}$.
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Lens Formula
- $\frac{1}{p} + \frac{1}{q} = \frac{1}{f}$.
- $p$: object distance (positive for real object, negative for virtual object).
- $q$: image distance (positive for real image, negative for virtual image).
- $f$: focal length (positive for convex, negative for concave).
- Positive sign is used where rays actually intersect. If not, negative sign.
Position and Nature of Image for Convex Lens:
Object Position of Image Nature of Image Beyond 2F Between F and 2F Real, inverted, small At 2F At 2F Real, inverted, equal Between F and 2F Beyond 2F Real, inverted, enlarged At F At infinity Real, inverted, enlarged Inside F On same side further away from lens than the object Virtual, erect, enlarged -
Magnification of a Lens (M)
- Linear magnification: $M = \frac{q}{p} = \frac{h_i}{h_o}$ (size of image / size of object).
- Angular magnification: $M_a = \frac{\text{Angle formed at aided eye}}{\text{Angle formed at naked eye}}$.
- For real image: $M = \frac{q}{p}$. For virtual image: $M = -\frac{q}{p}$.
- For very small angles, linear and angular magnifications can be equated.
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Visual Angle
- Angle made by object at observer's eye.
- Apparent size of object varies with visual angle.
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Resolving Power
- Ability of an instrument to reveal the minor details of the object under examination.
- Angle of Resolution ($\alpha_{min}$): Minimum angle that allows two point sources to appear distinctly separated.
- $\alpha_{min} = 1.22 \frac{\lambda}{D}$.
- Rayleigh showed that resolving power of a lens of aperture $D$, under a light source of wavelength $\lambda$ is $R = \frac{D}{1.22\lambda}$.
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Lens Aberrations
- Spherical Aberration: Blurred image due to size of lens (outer rays undergo more refraction). Remedy: Use stops or combine convex and concave lenses.
- Chromatic Aberration: Inability of a lens to bring all light rays (all colors) at single focus. Remedy: Use achromatic lenses.
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Least Distance of Distinct Vision
- Minimum distance at which a normal eye can see an object clearly.
- $d = 25~cm$.