Lens formula and power of a lens
A lens is an optical device made of transparent material, usually glass or plastic, bounded by two refracting surfaces, of which at least one is curved. Lenses are primarily used to converge or diverge light rays and form images of objects. The lens formula and power of a lens are two fundamental concepts in geometrical optics that describe the quantitative relationship between object distance, image distance, focal length, and the light-bending ability of a lens.
Types of Lenses
Lenses are broadly classified into two types based on the nature of refraction they produce:
- Convex lens (Converging lens) – Thicker at the centre and thinner at the edges, it converges parallel light rays to a point called the principal focus.
- Concave lens (Diverging lens) – Thinner at the centre and thicker at the edges, it diverges parallel rays outward, appearing to originate from a virtual focus.
These lenses obey well-defined laws of refraction and are used in various optical instruments such as cameras, microscopes, telescopes, and spectacles.
The Lens Formula
The lens formula gives a mathematical relationship between the object distance (u), image distance (v), and focal length (f) of a lens. It is expressed as:
1f=1v−1u\frac{1}{f} = \frac{1}{v} – \frac{1}{u}f1=v1−u1
This relation is valid for both convex and concave lenses, provided the sign convention is properly applied. It connects the three key parameters determining image formation.
Sign Convention (Cartesian Convention)
In optical problems, distances are measured from the optical centre of the lens. The following sign conventions are used:
- All distances measured in the direction of incident light are positive.
- All distances measured opposite to the direction of incident light are negative.
- Heights measured upward from the principal axis are positive; those measured downward are negative.
According to this convention:
- For a convex lens, the focal length (f) is positive.
- For a concave lens, the focal length (f) is negative.
Derivation of the Lens Formula
The lens formula can be derived using the refraction at spherical surfaces and applying the Gaussian formula for refraction.
For a spherical surface separating two media of refractive indices μ1\mu_1μ1 and μ2\mu_2μ2, the relationship between object distance (u), image distance (v), and radius of curvature (R) is:
μ2v−μ1u=μ2−μ1R\frac{\mu_2}{v} – \frac{\mu_1}{u} = \frac{\mu_2 – \mu_1}{R}vμ2−uμ1=Rμ2−μ1
Applying this relation twice—once for each refracting surface of the lens—and assuming the lens is thin (so that the two refractions occur very close to each other), we arrive at the lens maker’s formula:
1f=(μ−1)(1R1−1R2)\frac{1}{f} = (\mu – 1) \left( \frac{1}{R_1} – \frac{1}{R_2} \right)f1=(μ−1)(R11−R21)
Here,
- μ\muμ = refractive index of the lens material relative to the surrounding medium,
- R1R_1R1 and R2R_2R2 = radii of curvature of the two surfaces of the lens.
This equation gives the focal length of a lens in terms of its material and curvature. By combining this with the geometry of image formation, we obtain the lens formula:
1f=1v−1u\frac{1}{f} = \frac{1}{v} – \frac{1}{u}f1=v1−u1
Application of the Lens Formula
The lens formula is used to:
- Calculate image position and nature (real or virtual, magnified or diminished).
- Design optical instruments such as microscopes and telescopes.
- Determine the effective focal length when multiple lenses are combined.
Example
If an object is placed 30 cm in front of a convex lens of focal length 10 cm, the image distance (v) can be found using:
110=1v−1(−30)\frac{1}{10} = \frac{1}{v} – \frac{1}{(-30)}101=v1−(−30)1 ⇒1v=110−130=230\Rightarrow \frac{1}{v} = \frac{1}{10} – \frac{1}{30} = \frac{2}{30}⇒v1=101−301=302 ⇒v=15 cm\Rightarrow v = 15\, \text{cm}⇒v=15cm
Thus, the image is real and formed 15 cm on the opposite side of the lens.
Magnification Produced by a Lens
The magnification (m) produced by a lens is the ratio of the height of the image (hih_ihi) to the height of the object (hoh_oho), or equivalently, the ratio of image distance to object distance:
m=hiho=vum = \frac{h_i}{h_o} = \frac{v}{u}m=hohi=uv
- If m>1m > 1m>1: the image is magnified.
- If 0<m<10 < m < 10<m<1: the image is diminished.
- A positive magnification indicates a virtual and erect image.
- A negative magnification indicates a real and inverted image.
Power of a Lens
The power of a lens (P) is a measure of its ability to converge or diverge light rays. It is defined as the reciprocal of the focal length of the lens when the focal length is expressed in metres:
P=1fP = \frac{1}{f}P=f1
The unit of power is the dioptre (D), where
1 D=1 m−11\, \text{D} = 1\, \text{m}^{-1}1D=1m−1
Nature of Power
- A convex lens (converging) has a positive power.
- A concave lens (diverging) has a negative power.
Example
If a convex lens has a focal length of 0.5 m, then
P=10.5=+2 DP = \frac{1}{0.5} = +2\, \text{D}P=0.51=+2D
If a concave lens has a focal length of −0.25 m, then
P=1−0.25=−4 DP = \frac{1}{-0.25} = -4\, \text{D}P=−0.251=−4D
Combination of Lenses
When two or more thin lenses are placed in contact, the effective power (Pₑ) of the combination is the algebraic sum of their individual powers:
Pe=P1+P2+P3+…Pₑ = P_1 + P_2 + P_3 + \ldotsPe=P1+P2+P3+…
or equivalently,
1fe=1f1+1f2+1f3+…\frac{1}{fₑ} = \frac{1}{f_1} + \frac{1}{f_2} + \frac{1}{f_3} + \ldotsfe1=f11+f21+f31+…
This principle is used in optical instruments and in the manufacture of corrective lenses for spectacles.
Example
If a +2 D convex lens and a −1 D concave lens are placed together, the effective power is:
Pe=(+2)+(−1)=+1 DPₑ = (+2) + (-1) = +1\, \text{D}Pe=(+2)+(−1)=+1D
The system behaves as a convex lens of focal length 1 m.
Applications of Lens Power
- Spectacles and contact lenses: Correct myopia (short-sightedness) with concave lenses and hypermetropia (long-sightedness) with convex lenses.
- Optical instruments: Determine the focal length of components in cameras, microscopes, and telescopes.
- Ophthalmology: Measure the corrective power required for vision defects using lenses of known dioptre values.
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