Overview: Refraction of Light at Spherical Surfaces: Lenses

The rays starting from a fixed point on the principal axis of a lens, or appearing to go towards a fixed point on the axis, after refraction through the lens, become parallel to the principal axis. This point is called the 'first focus' of the lens.

The distance of the first focus from the optical centre of the lens is called the 'first focal-length' of the lens.

The rays travelling parallel to the axis of the lens, after refraction through the lens, either go towards a fixed point on the axis or appear to come from a point. This point is called the 'second focus' or the 'principal focus' of the lens.

The distance of the second focus from the optical centre of the lens is called the 'second focal length' or the 'principal focal length' of the lens.

The plane passing through the focus of a lens and perpendicular to the principal axis is called the 'focal plane'.

The linear magnification produced by a spherical (convex or concave) lens is the ratio of the size of the image formed by the lens to the size of the object, both measured perpendicular to the principal axis.

\[\frac{n}{v}-\frac{1}{u}=\frac{n-1}{R}\]

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

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

OR

m = \[\frac {f}{f + u}\]

1. Both the Lenses are Convex:
   \[\frac {1}{f}\] = \[\frac {1}{f_1}\] + \[\frac {1}{f_2}\]
2. One Lens is Convex and the Other is Concave:
   \[\frac {1}{f}\] = \[\frac {1}{f_1}\] - \[\frac {1}{f_2}\]
   
3. Combined Power: 
   P = P₁ + P₂

• The optical centre of the lens is taken as the origin; the principal axis is the X-axis and the perpendicular line through the optical centre is the Y-axis.
• Distances to the right of the optical centre are positive and to the left are negative; heights above the principal axis are positive and below are negative.

• Convexo-convex (Bi-convex): Both surfaces are convex; radii of curvature may be equal or different.
• Plano-convex: One surface is plane and the other is convex.
• Concavo-convex (Convex meniscus): One surface is concave and the other convex; thicker at the centre.
• Concavo-concave (Bi-concave): Both surfaces are concave; radii of curvature may be equal or different.
• Plano-concave: One surface is plane, and the other is concave.
• Convexo-concave (Concave meniscus): One surface is convex and the other concave; thinner at the centre.

• The focal length of a lens depends on its refractive index and the radii of curvature of its surfaces (lens maker’s formula).
• Changing the surrounding medium changes a lens's focal length; it increases in a denser medium and may even alter the lens's properties.

• For two coaxial lenses separated by distance d, the equivalent focal length and power depend on f₁, f₂, and d.
• A concave lens always forms a virtual image; therefore, its focal length is determined by combining it with a mirror.
• A convex mirror also always forms a virtual image, so it is combined with a convex lens to find its focal length.
• With a convex lens and a plane mirror, if the object and image coincide without parallax, the object position determines the lens's focal length.
• Focal lengths in lens–mirror combinations are calculated using the lens formula and non-parallax positions.