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n 2 d o + n 1 d i = n 1 n 2 R 2 .

The image is real and on the opposite side from the object, so d i > 0 and d o > 0 . The second surface is convex away from the object, so R 2 < 0 . [link] can be simplified by noting that d o = | d i | + t , where we have taken the absolute value because d i is a negative number, whereas both d o and t are positive. We can dispense with the absolute value if we negate d i , which gives d o = d i + t . Inserting this into [link] gives

n 2 d i + t + n 1 d i = n 1 n 2 R 2 .

Summing [link] and [link] gives

n 1 d o + n 1 d i + n 2 d i + n 2 d i + t = ( n 2 n 1 ) ( 1 R 1 1 R 2 ) .

In the thin-lens approximation    , we assume that the lens is very thin compared to the first image distance, or t d i (or, equivalently, t R 1 and R 2 ). In this case, the third and fourth terms on the left-hand side of [link] cancel, leaving us with

n 1 d o + n 1 d i = ( n 2 n 1 ) ( 1 R 1 1 R 2 ) .

Dividing by n 1 gives us finally

1 d o + 1 d i = ( n 2 n 1 1 ) ( 1 R 1 1 R 2 ) .

The left-hand side looks suspiciously like the mirror equation that we derived above for spherical mirrors. As done for spherical mirrors, we can use ray tracing and geometry to show that, for a thin lens,

1 d o + 1 d i = 1 f

where f is the focal length of the thin lens (this derivation is left as an exercise). This is the thin-lens equation. The focal length of a thin lens is the same to the left and to the right of the lens. Combining [link] and [link] gives

1 f = ( n 2 n 1 1 ) ( 1 R 1 1 R 2 )

which is called the lens maker’s equation . It shows that the focal length of a thin lens depends only of the radii of curvature and the index of refraction of the lens and that of the surrounding medium. For a lens in air, n 1 = 1.0 and n 2 n , so the lens maker’s equation reduces to

1 f = ( n 1 ) ( 1 R 1 1 R 2 ) .

Sign conventions for lenses

To properly use the thin-lens equation, the following sign conventions must be obeyed:

  1. d i is positive if the image is on the side opposite the object (i.e., real image); otherwise, d i is negative (i.e., virtual image).
  2. f is positive for a converging lens and negative for a diverging lens.
  3. R is positive for a surface convex toward the object, and negative for a surface concave toward object.


By using a finite-size object on the optical axis and ray tracing, you can show that the magnification m of an image is

m h i h o = d i d o

(where the three lines mean “is defined as”). This is exactly the same equation as we obtained for mirrors (see [link] ). If m > 0 , then the image has the same vertical orientation as the object (called an “upright” image). If m < 0 , then the image has the opposite vertical orientation as the object (called an “inverted” image).

Using the thin-lens equation

The thin-lens equation and the lens maker’s equation are broadly applicable to situations involving thin lenses. We explore many features of image formation in the following examples.

Consider a thin converging lens. Where does the image form and what type of image is formed as the object approaches the lens from infinity? This may be seen by using the thin-lens equation for a given focal length to plot the image distance as a function of object distance. In other words, we plot

d i = ( 1 f 1 d o ) −1

for a given value of f . For f = 1 cm , the result is shown in part (a) of [link] .

Questions & Answers

The photoelectric effect is the emission of electrons when light shines on a material. 
Chris Reply
What is photoelectric effect
Amit Reply
it gives practical evidence of particke nature of light.
particle nature
photoelectric effect is the phenomenon of emission of electrons from a material(i.e Metal) when it is exposed to sunlight. Emitted electrons are called as photo electrons.
what are the applications of quantum mechanics to medicine?
application of quantum mechanics in medicine: 1) improved disease screening and treatment ; using a relatively new method known as BIO- BARCODE ASSAY we can detect disease-specific clues in our blood using gold nanoparticles. 2) in Genomic medicine 3) in protein folding 4) in radio theraphy(MRI)
Quantam physics ki basic concepts?
Laxmikanta Reply
why does not electron exits in nucleaus
Kabbo Reply
electrons have negative
Proton and meltdown has greater mass than electron. So it naturally electron will move around nucleus such as gases surrounded earth
.......proton and neutron....
excuse me yash what negative
coz, electron contained minus ion
negative sign rika shrestha ji
electron is the smallest negetive charge...An anaion i.e., negetive ion contains extra electrons. How ever an atom is neutral so it must contains proton and electron
yes yash ji
yes friends
koantam theory
yes prema
quantum theory tells us that both light and matter consists of tiny particles which have wave like propertise associated with them.
proton and nutron nuclear power is best than proton and electron kulamb force
what is de-broglie wave length?
plot a graph of MP against tan ( Angle/2) and determine the slope of the graph and find the error in it.
Ime Reply
expression for photon as wave
Are beta particle and eletron are same?
Amalesh Reply
how can you confirm?
If they are same then why they named differently?
because beta particles give the information that the electron is ejected from the nucleus with very high energy
what is meant by Z in nuclear physic
Shubhu Reply
atomic n.o
no of atoms present in nucleus
Note on spherical mirrors
Shamanth Reply
what is Draic equation? with explanation
M.D Reply
trpathy Reply
it's a subject
it's a branch in science which deals with the properties,uses and composition of matter
what is a Higgs Boson please?
god particles is know as higgs boson, when two proton are reacted than a particles came out which is used to make a bond between than materials
bro little abit getting confuse if i am wrong than please clarify me
the law of refraction of direct current lines at the boundary between two conducting media of
what is the black body of an ideal radiator
Areej Reply
uncertainty principles is applicable to
what is the cause of the expanding universe?
microscopic particles or gases
Astronomers theorize that the faster expansion rate is due to a mysterious, dark force that is pulling galaxies apart. One explanation for dark energy is that it is a property of space.
Thanks for your contribution Areej.
no problem
what is photoelectric equation
Practice Key Terms 5

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