Resource Lesson
Lensmaker Equation
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The following formula, called the
Lensmaker Equation
, is used to determine whether a lens will behave as a converging or diverging lens based on the curvature of its faces and the relative indices of the lens material [n
_{1}
] and the surrounding medium [n
_{2}
].
Usually the expression
is treated as a constant (K
_{shape}
) allowing us to work more often with the following second form of the equation:
Remember that K
_{shape}
represents the shape of the lens which remains constant regardless of the type of surrounding medium [n
_{2}
] into which the lens is used.
If this expression yields a negative value for 1/f, then the lens is diverging; a positive 1/f means that the lens is converging.
Refer to the following information for the next seven questions.
The following double convex lens (n
_{1}
= 1.52) has radii of curvatures equal to r
_{1}
= 15 cm and r
_{2}
= 10 cm.
As the light approaches the lens from the left, does the front surface have a positive or negative curvature?
As the light leaves the lens (having entered from the left), does the back surface of the lens have a positive or negative curvature?
State the values of r
_{1}
and r
_{2}
.
Calculate the value of K
_{shape}
.
What is the focal length of this lens in air (n
_{2}
= 1.00)?
What is the focal length of this lens if it were submerged in water (n
_{2}
= 1.333)?
What is the focal length of this lens if it were embedded in carbon disulfide (n
_{2}
= 1.628)?
It is the Lensmaker Equation that gives rise to our previous statements about the shapes of lenses and their functionality in air:
converging lenses
are lenses that are "thicker in the center" than on the edges (convex)
geometry
r
_{1}
> 0, r
_{2}
< 0 therefore K
_{shape}
> 0
Since
n
_{1}
> n
_{2}
and
K
_{shape}
> 0
1/f > 0 and these lenses will be converging.
r
_{1}
=
, r
_{2}
< 0 therefore K
_{shape}
> 0
diverging lenses
are lenses that are "thinner in the center" than on the edges (concave)
geometry
r
_{1}
< 0, r
_{2}
> 0 therefore K
_{shape}
< 0
Since
n
_{1}
> n
_{2}
and
K
_{shape}
< 0
1/f < 0 and these lenses will be diverging.
r
_{1}
=
, r
_{2}
> 0 therefore K
_{shape}
< 0
Power
To calculate the
power of a lens
, we use the relationship that
In this formula, the focal lengths are usually measured in meters resulting in the power of the lens being measured in a unit called a
Diopter
where 1 D = m
^{-1}
.
What is the power of the lens in the previous example while it is being used in air?
What is the power of the lens in the previous example while it is being used in water?
Lenses in close combination
When two or more lenses are nested or used in
close combination
, that is, with no space in between them, the equation to calculate the effective power of the combination is
At the end of an eye exam, a ophthalmologist has confirmed his patient's prescription with three thin lenses placed in close combination within his testing apparatus. Determine the power of the final lens his patient will need if the individual lenses have focal lengths of: 20 cm, -30 cm, and 25 cm.
image courtesy of
University of Illinois Eye Center
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