The Glass of a Double Convex Lens

solved ThinLens

Q. The radii of curvature $n_{s} = 1$ of the faces of a double convex $R_{2} = - r_{2}$ lens are 10 cm $R_{1} = 10$ cm and 15 cm $r_{2} = 15$ cm. Its focal length is 12 cm $f = 12$ cm. What is the refractive index of glass?

Inputs

first

R_{1}

10 cm

second

r_{2}

15 cm

focal length

f

12 cm

Outputs

index

n

How it's solved

Solved with ThinLens

Given

$R_{1} = 10$ cm
$r_{2} = 15$ cm
$f = 12$ cm
$n_{s} = 1$ from “radii of curvature”
$R_{2} = - r_{2}$ from “double convex”

Find $n$ .

Working

$R_{1} = 10$ cm $= 0.1$ m
$r_{2} = 15$ cm $= 0.15$ m
$f = 12$ cm $= 0.12$ m
double convex $R_{2} = - r_{2}$
$R_{2} = - 0.15$
$R_{2} = −0.15$ m
lens makers formula(lens) $\frac{1}{f} = (\frac{n}{n_{s}} - 1) \cdot (\frac{1}{R_{1}} - \frac{1}{R_{2}})$
$\frac{1}{0.12} = (\frac{n}{1} - 1) \cdot (\frac{1}{0.1} - \frac{1}{(−0.15)})$
$n = 1.5$
$n \approx 1.5$ to 2 significant figures

Conclusion

Both faces bulge outward, so the second face's radius counts negative and the two faces add their bending; the lens-maker's formula then gives the glass.

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 :the_glass_of_a_double_convex_lens do4  kind :numerical5  describes "The radii of curvature of the faces of a double convex lens are %{first} and %{second}. Its focal length " \6            "is %{focal_length}. %{index}?"7  given first: "10 cm"8  given second: "15 cm"9  given focal_length: "12 cm"10  asks index: "What is the refractive index of glass"11end

solution.rb

1require_relative "../questions/the_glass_of_a_double_convex_lens"2require_relative "../science/ray_optics_and_optical_instruments/thin_lens"34solve :the_glass_of_a_double_convex_lens, using: ThinLens do5  objects :lens67  take first: lens.first_radius8  take second: lens.size_of_the_second_radius9  take focal_length: lens.focal_length1011  deduce_from "double convex", lens.second_radius => -lens.size_of_the_second_radius12  deduce_from "radii of curvature", lens.index_of_the_surroundings => "1"1314  index = lens.refractive_index_of_the_lens15  answer index:, figures: 21617  conclusion "Both faces bulge outward, so the second face's radius counts negative and the two faces add " \18             "their bending; the lens-maker's formula then gives the glass."19end

playground.rb

1require_relative "../solutions/the_glass_of_a_double_convex_lens"23play_with :the_glass_of_a_double_convex_lens do4  title "The Glass of a Double Convex Lens"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_{2}

double convex