# Thin Lenses

"An intellectual is someone whose mind watches itself"
Albert Camus

• Lenses form images by refraction of incident light.  Most commonly they are constructed of glass, although other transparent materials are also used.  Similar to spherical mirrors, there are two basic types of lens, converging and diverging

• Converging Lens.  Parallel light incident on one side of a converging lens passes through a single point on the other side called the "focal point" (F).  This is achieved due to refraction at both surfaces of the lens because light travels more slowly in glass than air.

• Diverging Lens.  Parallel light incident on one side of a diverging lens is seen to diverge, as if the light came from a single point on the same side of the lens as the incident light.  This point is known as a " virtual focal point", since after passage through the lens, light does not pass through it. As in the case of the converging lens light is refracted at both surfaces passing more slowly through glass than air.

• All lenses have 2 focal points, one on each side of the lens, but only one focal length.

• Lens Makers Formula.  The focal length of a lens depends on the refractive index of the material of the lens (n L ), the refractive index of the medium in which it is immersed (n m ) and the radii of curvature of the two faces of the lens (R 1 and R 2 ) according to the   formula (valid for thin lenses),

• Sign Conventions:

 SIGN + - Focal length - f Converging Lens Diverging Lens Radii of curvature - R Convex Surface Concave Surface

Whether a surface is convex or concave is determined by light striking the lens before passing through the lens.  For example, the first surface struck by the light in the diagram of the diverging lens above is concave.  If this convention is adhered to it does not matter which surface is labeled R1 or R 2.

• Image Formation:  Similar to the spherical mirror case there are three characteristic rays, any two of which enable the location of the image to be determined.
• Rays passing through the centre of a lens are transmitted without deflection.
• Rays parallel to the axis are refracted such that they pass through the focal point of the lens (converging) or appear to have come from the focal point (diverging).
• Rays passing through the focal point are then refracted parallel to the axis of the lens (converging) or rays which are headed towards the focal point on the other side of the lens are refracted parallel to the exis (diverging).

• Converging Lens:  Examples of image formation using the characteristic rays are shown below.
•

• Diverging Lens:  Example of image formation using the characteristic rays is shown below (animation).

• Lens equation:  For objects close to the axis of a thin lens the focal length (f), object distance (p) and image distance (q) are related by

• Magnification.  The magnification of a thin lens (m) is defined by

• Sign Conventions: The following sign conventions must be followed,

 SIGN + - f - focal length Converging lens Diverging lens p - object distance Real Virtual q - image distance Real Virtual m - magnification Upright image Inverted image

• Image Properties:  Exactly as in the case of the spherical mirror, there are four basic properties of  images, described in the table below.

 Lens Object location Image location Type Orientation Relative size Application CONVERGING Infinity At F Real Inverted Smaller Telescope CONVERGING Beyond 2F Between F and 2F Real Inverted Smaller Camera CONVERGING At 2F At 2F Real Inverted Same size Copier CONVERGING Between F and 2F Beyond 2F Real Inverted Larger Projector CONVERGING At F At infinity No image No image No image Lighthouse CONVERGING Closer than F Same side of lens Virtual Upright Larger Magnifying glass DIVERGING Anywhere Same side of lens Virtual Upright Smaller

• Note that the above analysis is applicable for THIN lenses - where the height of the lens is much larger than the maximum thickness of the lens.  A more complex analysis must be considered for lenses which do not satisfy this condition - THICK lenses.  Lenses in high-end optical instruments are typically thick lenses.
• The diopter is a measure of the focal length of a lens used by opticians.  Its is equal to the reciprocal of the focal length of the lens (measured in metres).  For example a lens with a focal length of 50 cm has a diopter value of 2.

"There are 1011 stars in the galaxy. That used to be a huge number.But it's only a hundred billion. It's less than the national
deficit! We used to call them astronomical numbers. Now we should call them economical numbers."
Richard Feynman

Dr. C. L. Davis
Physics Department
University of Louisville
email: c.l.davis@louisville.edu