Refraction by a Lens

When light passes through a lens, it undergoes refraction at both curved surfaces of the lens. The combined effect of refraction at the two surfaces produces image formation. For a thin lens, the first surface forms an intermediate image, and the second surface forms the final image.

A transparent refracting medium bounded by two surfaces, of which at least one is spherical, is called a lens.

The straight line passing through the optical centre and the centres of curvature of the lens surfaces is called the principal axis.

The point near the centre of a thin lens through which a ray of light passes without appreciable deviation is called the optical centre.

The point on the principal axis where rays parallel to the principal axis actually meet after refraction, or appear to diverge after refraction, is called the principal focus.

The distance between the optical centre and the principal focus is called the focal length.

The ratio of the height of the image to the height of the object is called magnification.

\[\frac{1}{f}=(\mu-1)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)\]

Where:

• f = focal length of the lens.
• μ = refractive index of the material of the lens with respect to air.
• R₁​ = radius of curvature of the first surface.
• R₂​ = radius of curvature of the second surface.

\[\frac{1}{v}-\frac{1}{u}=\frac{1}{f}\]

Where:

• u = object distance.
• v = image distance.
• f = focal length of the lens.

\[m=\frac{h_i}{h_o}=\frac{v}{u}\]

Where:

• m = magnification.
• h_i = height of image.
• h_o​ = height of object.
• v = image distance.
• u = object distance.

The new Cartesian sign convention is used for lenses.

• All distances are measured from the optical centre of the lens.
• Distances measured in the direction of incident light are taken as positive.
• Distances measured opposite to the direction of incident light are taken as negative.
• Heights measured above the principal axis are taken as positive.
• Heights measured below the principal axis are taken as negative.
• The focal length of a convex lens is positive.
• The focal length of a concave lens is negative.

• A ray passing through the optical centre of a thin lens passes without appreciable deviation.
• A ray parallel to the principal axis, after refraction through a convex lens, passes through the principal focus on the other side.
• A ray parallel to the principal axis, after refraction through a concave lens, appears to diverge from the principal focus on the same side.
• A ray passing through the principal focus of a convex lens emerges parallel to the principal axis.
• A ray directed towards the principal focus of a concave lens emerges parallel to the principal axis.

1. A convex lens has two refracting surfaces.

2. The first surface forms an intermediate image I₁:

• \[\frac{n_1}{OB}+\frac{n_2}{BI_1}=\frac{n_2-n_1}{BC_1}\]

3. The second surface takes I₁​ as a virtual object:

• \[-\frac{n_2}{DI_1}+\frac{n_1}{DI}=\frac{n_2-n_1}{DC_2}\]

4. For a thin lens, BI₁ = DI₁​. Adding the two equations cancels the intermediate image terms:

• \[\frac{n_1}{OB}+\frac{n_1}{DI}=(n_2-n_1)\left(\frac{1}{BC_1}+\frac{1}{DC_2}\right)\]

5. If the object is at infinity (OB → ∞, DI = f):

• \[\frac{1}{f}=\left(\frac{n_2}{n_1}-1\right)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)\]

This is the Lens Maker's Formula.

6. Using the sign convention (OB = −u, DI = +v):

• \[{\frac{1}{v}-\frac{1}{u}=\frac{1}{f}}\]

Magnification produced by a lens is the ratio of the height of the image to the height of the object.

For a thin lens, magnification is also given by:

Magnification is positive for an erect image and negative for an inverted image.

• The glass lens has a refractive index n = 1.47.
• For the lens to “disappear” in the liquid, light should not bend at the boundary between glass and liquid, so both must have the same refractive index.
• Therefore, the refractive index of the liquid must be equal to 1.47, so n₁ = n₂.
• If n₁ = n₂, the lens maker relation gives 1/f = 0, which means f → ∞ (infinite focal length).
• A lens with infinite focal length behaves like a simple plane sheet of glass and does not act like a converging or diverging lens.
• So, the liquid used is not water, because water has a different refractive index; instead, it could be glycerine, whose refractive index is close to that of glass (about 1.47).

A convex lens is used in magnifying glasses, cameras, microscopes, and the human eye because it can converge light and form clear images. A concave lens is used in spectacles for myopia because it diverges light before it enters the eye.