Double-Convex Lenses of One Radius

solved ThinLens

Q. Double-convex lenses are to be manufactured from a glass $n_{s} = 1$ of refractive index 1.55 $n = 1.55$ , with both faces of the same radius of curvature $R_{2} = - R_{1}$ . What is the radius of curvature required if the focal length is to be 20 cm $f = 20$ cm, in cm?

Inputs

index

n

1.55

focal length

f

20 cm

Outputs

radius

R_{1}

How it's solved

Solved with ThinLens

Given

$n = 1.55$
$f = 20$ cm
$n_{s} = 1$ from “manufactured from a glass”
$R_{2} = - R_{1}$ from “same radius of curvature”

Find $R_{1}$ .

Working

$f = 20$ cm $= 0.2$ m
lens makers formula(lens) $\frac{1}{f} = (\frac{n}{n_{s}} - 1) \cdot (\frac{1}{R_{1}} - \frac{1}{R_{2}})$
same radius of curvature $R_{2} = - R_{1}$
$R_{1} = 22$ cm
$R_{1} = 0.22$ m $= 22$ cm
$R_{1} \approx 22$ cmto 2 significant figures

Conclusion

The two faces bulge outward equally, so their radii are equal and opposite in sign and each bends the light by the same amount.

The web of concepts

given read from the words equation asked

The Ruby that solves it

The question and its solution are small Ruby files. The engine runs them.

question.rb

1require_relative "../physics"23question :double_convex_lenses_of_one_radius do4  kind :numerical5  describes "Double-convex lenses are to be manufactured from a glass of refractive index %{index}, with both " \6            "faces of the same radius of curvature. %{radius} if the focal length is to be %{focal_length}, in " \7            "cm?"8  given index: "1.55"9  given focal_length: "20 cm"10  asks radius: "What is the radius of curvature required"11end

solution.rb

1require_relative "../questions/double_convex_lenses_of_one_radius"2require_relative "../science/ray_optics_and_optical_instruments/thin_lens"34solve :double_convex_lenses_of_one_radius, using: ThinLens do5  objects :lens67  take index: lens.refractive_index_of_the_lens8  take focal_length: lens.focal_length910  deduce_from "same radius of curvature", lens.second_radius => -lens.first_radius11  deduce_from "manufactured from a glass", lens.index_of_the_surroundings => "1"1213  radius = lens.first_radius14  answer radius:, in: "cm", figures: 21516  conclusion "The two faces bulge outward equally, so their radii are equal and opposite in sign and each bends " \17             "the light by the same amount."18end

playground.rb

1require_relative "../solutions/double_convex_lenses_of_one_radius"23play_with :double_convex_lenses_of_one_radius do4  title "Double-Convex Lenses of One Radius"5end

The laws it drew on

quantity	written	units	within	from
refractive index of the lens	$n$	—	—	ThinLens
index of the surroundings	$n_{s}$	—	—	ThinLens
first radius	$R_{1}$	m	—	ThinLens
second radius	$R_{2}$	m	—	ThinLens
size of the second radius	$r_{2}$	m	—	ThinLens
focal length	$f$	m	—	ThinLens
size of the focal length	$f_{size}$	m	—	ThinLens
power of the lens	$P$	m⁻¹	—	ThinLens
distance in front	$d_{o}$	m	—	ThinLens
object distance	$u$	m	—	ThinLens
image distance	$v$	m	—	ThinLens
magnification	$m$	—	—	ThinLens
object height	$h_{o}$	m	—	ThinLens
image height	$h_{i}$	m	—	ThinLens
object distance of ground	$u_{g}$	m	—
distance in front of ground	$d_{o,g}$	m	—
object distance of lens	$u$	m	—
distance in front of lens	$d_{o}$	m	—
focal length of ground	$f_{g}$	m	—
refractive index of the lens of ground	$n_{g}$	—	—
index of the surroundings of ground	$n_{s,g}$	—	—
first radius of ground	$R_{1,g}$	m	—
second radius of ground	$R_{2,g}$	m	—
focal length of lens	$f$	m	—
refractive index of the lens of lens	$n$	—	—
index of the surroundings of lens	$n_{s}$	—	—
first radius of lens	$R_{1}$	m	—
second radius of lens	$R_{2}$	m	—
power of the lens of ground	$P_{g}$	m⁻¹	—
power of the lens of lens	$P$	m⁻¹	—
image distance of ground	$v_{g}$	m	—
image distance of lens	$v$	m	—
magnification of ground	$m_{g}$	—	—
magnification of lens	$m$	—	—
image height of ground	$h_{i,g}$	m	—
object height of ground	$h_{o,g}$	m	—
image height of lens	$h_{i}$	m	—
object height of lens	$h_{o}$	m	—
size of the second radius of lens	$r_{2}$	m	—
size of the focal length of lens	$f_{size}$	m	—

u_{l} = - d_{o,l}

an object in front is against the light

\frac{1}{f_{l}} = (\frac{n_{l}}{n_{s,l}} - 1) \cdot (\frac{1}{R_{1,l}} - \frac{1}{R_{2,l}})

lens makers formula

P_{l} = \frac{1}{f_{l}}

power of a lens

\frac{1}{v_{l}} - \frac{1}{u_{l}} = \frac{1}{f_{l}}

thin lens formula

m_{l} = \frac{v_{l}}{u_{l}}

magnification of a lens

m_{l} = \frac{h_{i,l}}{h_{o,l}}

magnification compares the heights

u_{g} = - d_{o,g}

an object in front is against the light(ground)

u = - d_{o}

an object in front is against the light(lens)

\frac{1}{f_{g}} = (\frac{n_{g}}{n_{s,g}} - 1) \cdot (\frac{1}{R_{1,g}} - \frac{1}{R_{2,g}})

lens makers formula(ground)

\frac{1}{f} = (\frac{n}{n_{s}} - 1) \cdot (\frac{1}{R_{1}} - \frac{1}{R_{2}})

lens makers formula(lens)

P_{g} = \frac{1}{f_{g}}

power of a lens(ground)

P = \frac{1}{f}

power of a lens(lens)

\frac{1}{v_{g}} - \frac{1}{u_{g}} = \frac{1}{f_{g}}

thin lens formula(ground)

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

thin lens formula(lens)

m_{g} = \frac{v_{g}}{u_{g}}

magnification of a lens(ground)

m = \frac{v}{u}

magnification of a lens(lens)

m_{g} = \frac{h_{i,g}}{h_{o,g}}

magnification compares the heights(ground)

m = \frac{h_{i}}{h_{o}}

magnification compares the heights(lens)

R_{2} = - R_{1}

same radius of curvature