The Glass of a Double Convex Lens

Q. The radii of curvature of the faces of a double convex lens are 10 cm and 15 cm. Its focal length is 12 cm. What is the refractive index of glass? expects 1.5

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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?

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Inputs

first

R_{1}

10 cm

second

r_{2}

15 cm

focal length

f

12 cm

Outputs

index

n

The solution

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 Ruby

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

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

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

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