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✦ בנוי על ידי פהד גאנם ✦

P (0, 0)  |  K (26, 0)  |  B (2, 0)  |  A (2, 4)
D (7, 0)  |  C (7, 2)  |  O (24, 6)
grid = 1 cm  ;  lens diameter = 12 cm  ;  part ו : 4 cm
2.0 cm
12.0 cm
12.0 cm
−10.0 cm
10.0 cm
−5.0 cm
0.50
2.0 cm
7.0 cm
3.18 cm
12.0 cm
6.35 cm
f = −10 cm u = 10 cm v = −5 cm m = 24 = 0.5 Dmin = 10817 = 6.35 cm

m = hiho = 24 = 0.5 < 1
converging, u < f : m = ff − u > 1
diverging : m = |f|u + |f| < 1
image CD : virtual  ·  upright  ·  m = 0.5 < 1  ⟹  diverging lens only

A (2, 4)  ·  C (7, 2)
slope = 2 − 47 − 2 = −25 = −0.4
x = 2 + 40.4 = 2 + 10 = 12 cm
optical centre : x = 12 cm  ·  lens plane ⊥ axis  ·  from (12, −6) to (12, +6)

hit point : (12, 4)
slope = 4 − 212 − 7 = 25 = 0.4
xF = 12 − 40.4 = 12 − 10 = 2 cm
|f| = 12 − 2 = 10 cm
4 − 0.2 · 10 = 2  ⟹  line y = 2 passes through C (7, 2) ✓
|f| = 10 cm  ·  F at x = 2 cm (coincides with B)  ·  F′ at x = 22 cm

u = 12 − 2 = 10 cm
v = −(12 − 7) = −5 cm
m = |v|u = 510 = 0.5 = hiho = 24 ✓
1f = 1u + 1v = 110 − 15 = −0.1 cm⁻¹ f = −10 cm  ⟹  |f| = 10 cm
(u − f)(v − f) = f²  →  (10 + 10)(−5 + 10) = 20 · 5 = 100 = (−10)² ✓
f = −10 cm  ·  |f| = 10 cm  ·  diverging lens

slopeCO = 6 − 224 − 7 = 417 ≈ 0.2353
y = 2 + (12 − 7) · 417 = 2 + 2017 = 5417 ≈ 3.18 cm
slopein = 54/17 − 412 − 2 = −785 ≈ −0.082
5417 − 5 · 417 = 3417 = 2 ✓
A (2, 4) → lens (12, 54/17 ≈ 3.18) → eye O (24, 6)  ·  back extension hits C (7, 2)

h = 5417 = 3.1765 cm
D = 4 cm  ⟹  D2 = 2 cm < 3.18 cm
Dmin = 2 · 5417 = 10817 ≈ 6.35 cm
ytop(24) = 2 + (2 − 2) · 175 = 2  <  6
the eye does NOT see the image  ·  needs D ≥ 108/17 ≈ 6.35 cm

d [cm]   | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.06
Δx [cm] | 0.61 | 0.29 | 0.20 | 0.17 | 0.12 | 0.10
L = 120 cm
0.015 cm
120 cm
0.402 cm
0.805 cm
0.404 cm
0.809 cm
503 nm
0.006037 cm²
0.0018 cm
0.997
503.1 nm
Δx = λ · Ld a = λ · L λ = aL

d · sin θ = n · λ
sin θ ≈ tan θ = xL
xn = n · λ · Ld
Δx = xn+1 − xn = (n+1) λ Ld − n λ Ld
Δx = λ · Ld

u = 1d  ⟹  Δx = (λ · L) · u
[u] = cm⁻¹  ;  [a] = cmcm⁻¹ = cm²
new variable : 1/d  ·  units cm⁻¹  ·  slope = λ·L in cm²

1/0.01 = 100  ;  1/0.02 = 50  ;  1/0.03 = 100/3 = 33.33 1/0.04 = 25  ;  1/0.05 = 20  ;  1/0.06 = 50/3 = 16.67
1/d [cm⁻¹] : 100  |  50  |  33.33  |  25  |  20  |  16.67 Δx [cm]    : 0.61  |  0.29  |  0.20  |  0.17  |  0.12  |  0.10
1/d [cm⁻¹] = 100 ; 50 ; 33.33 ; 25 ; 20 ; 16.67

(100, 0.61) (50, 0.29) (33.33, 0.20) (25, 0.17) (20, 0.12) (16.67, 0.10)
Δx = 0.006037 · (1/d) + 0.0018
scatter of 6 points  ·  one straight trend line  ·  passes very close to the origin

a = λ · L
(16.67 , 0.1024)  ;  (100 , 0.6055) a = 0.6055 − 0.1024100 − 16.67 = 0.503183.33 = 0.006037 cm²
λ = aL = 0.006037120 = 5.031 · 10⁻⁵ cm
5.031 · 10⁻⁵ cm = 5.031 · 10⁻⁷ m = 503 nm
λ ≈ 5.0 · 10⁻⁷ m = 503 nm  ·  green light

x2 − x0 = 2 λ Ld − 0 = 2 · Δx
distance (n = 0 → n = 2) = 2 · Δx

1/d = 10.015 = 2003 = 66.67 cm⁻¹
Δx = 0.006037 · 66.67 + 0.0018 = 0.4025 + 0.0018 = 0.404 cm
distance = 2 · 0.404 = 0.809 ≈ 0.81 cm = 8.1 · 10⁻³ m
Δx = 5.0·10⁻⁵ · 1200.015 = 0.0060.015 = 0.400  ⟹  0.80 cm
Δx = 0.404 cm  ·  distance = 2Δx ≈ 0.81 cm = 8.1 · 10⁻³ m

Ephoton = 2.75 eV = 4.400·10⁻¹⁹ J   ·   λ₀ = 551 nm
h = 6.626·10⁻³⁴ J·s   ·   c = 3·10⁸ m/s   ·   1 eV = 1.6·10⁻¹⁹ J
h·c = 1.9878·10⁻²⁵ J·m = 1242.375 eV·nm   ·   me = 9.109·10⁻³¹ kg

1.20 V
451.77 nm
100 %
1.00
2.750 eV
4.400·10⁻¹⁹ J
6.640·10¹⁴ Hz
451.77 nm
2.255 eV
5.445·10¹⁴ Hz
0.4951 eV
7.924·10⁻²⁰ J
0.495 V
4.171·10⁵ m/s
100 %
1.000 mA
0.992 mA
—
E = h·f f = Eh E = W + Ek(max) W = h·cλ₀ Vstop = Ek(max)e Isat ∝ 1d²

E = h·f f = Eh
E = 2.75 · 1.6·10⁻¹⁹ = 4.400·10⁻¹⁹ J
f = 4.400·10⁻¹⁹6.626·10⁻³⁴ = 6.6405·10¹⁴ Hz
λ = h·cE = 1.9878·10⁻²⁵4.400·10⁻¹⁹ = 4.518·10⁻⁷ m

Ephoton = W + Ek(max)
W = h·cλ₀ = 1.9878·10⁻²⁵5.51·10⁻⁷ = 3.6076·10⁻¹⁹ J W = 3.6076·10⁻¹⁹1.6·10⁻¹⁹ = 2.255 eV
Ek(max) = 4.400·10⁻¹⁹ − 3.6076·10⁻¹⁹ = 7.924·10⁻²⁰ J Ek(max) = 7.924·10⁻²⁰1.6·10⁻¹⁹ = 0.495 eV
v = √2·Ek(max)me = 4.171·10⁵ m/s

e·Vstop = Ek(max) Vstop = Ek(max)e = 7.924·10⁻²⁰1.6·10⁻¹⁹ = 0.495 V

I = N·et
IA = IC = Isat

Ephoton = W + Ek(max)
λ = h·cW + Ek(max)

Isat ∝ 1d²
Ek(max) = h·f − W

n = ∞  →  0 eV   ·   n = 6  →  −0.37 eV   ·   n = 5  →  −0.54 eV
n = 4  →  −0.85 eV   ·   n = 3  →  ???   ·   n = 2  →  −3.4 eV   ·   n = 1  →  −13.6 eV
h·c = 1.9878·10⁻²⁵ J·m = 1242.375 eV·nm   ·   1 eV = 1.6·10⁻¹⁹ J

12.500 eV
−1.5111 eV
−2.418·10⁻¹⁹ J
12.0889 eV
12.7500 eV
12.500 eV
n = 1
—
—
3.400 eV
5.440·10⁻¹⁹ J
En = − 13.6n² eV λ = h·cΔE Eion = 0 − En

En = − 13.6n² eV
E3 = − 13.63² = − 13.69 = −1.5111 eV E3 = −1.5111 · 1.6·10⁻¹⁹ = −2.418·10⁻¹⁹ J
n = 2 : −13.64 = −3.40  ·  n = 4 : −13.616 = −0.85 n = 5 : −13.625 = −0.544 ≈ −0.54  ·  n = 6 : −13.636 = −0.378 ≈ −0.37

ΔE = E3 − E1 = (−1.5111) − (−13.6) = 12.0889 eV
12.5 ≠ 12.0889
Ephoton = Efinal − Einitial
E4 − E1 = (−0.85) − (−13.6) = 12.75 eV

λ = h·cΔE
ΔE = E3 − E2 = (−1.5111) − (−3.4) = 1.8889 eV = 3.0222·10⁻¹⁹ J λ = 1.9878·10⁻²⁵3.0222·10⁻¹⁹ = 6.577·10⁻⁷ m = 657.7 nm
4 → 2 : ΔE = 2.55 eV → λ = 1242.3752.55 = 487.2 nm 5 → 2 : ΔE = 2.86 eV → λ = 1242.3752.86 = 434.4 nm 6 → 2 : ΔE = 3.03 eV → λ = 1242.3753.03 = 410.0 nm
λlimit = 1242.3753.4 = 365.4 nm

Ek = e·V = 12.5 eV = 2.0·10⁻¹⁸ J
1 → 2 : 10.2000 eV ≤ 12.5 ✓  ·  עודף 2.300 eV 1 → 3 : 12.0889 eV ≤ 12.5 ✓  ·  עודף 0.411 eV 1 → 4 : 12.7500 eV ✗
3 → 2 : ΔE = 1.8889 eV → λ = 1.9878·10⁻²⁵3.0222·10⁻¹⁹ = 657.7 nm 2 → 1 : ΔE = 10.2 eV → λ = 1.9878·10⁻²⁵1.632·10⁻¹⁸ = 121.8 nm 3 → 1 : ΔE = 12.0889 eV → λ = 1.9878·10⁻²⁵1.9342·10⁻¹⁸ = 102.8 nm
1102.77 = 1657.73 + 1121.80

Eion = E∞ − E2 = 0 − (−3.4) = 3.4 eV Eion = 3.4 · 1.6·10⁻¹⁹ = 5.44·10⁻¹⁹ J

¹⁴₇N + ¹₀n  ⟶  ¹⁴₆C + ¹₁p
¹⁴₆C  ⟶  ¹⁴₇N + ⁰₋₁e
R(0) = 57 Bq  ;  T½ = 5730 years  ;  R(t₁) = 40 Bq  ;  ΔN = 4.63·10¹²
0 y
57 Bq
5730 y
—
—
—
—
—
—
—
—
—
—
—
λ = ln2T½ = — R = λ · N = —

¹⁴₇N + ¹₀n  ⟶  ¹⁴₆C + ¹₁p
Z : 7 + 0 = Z + 1  ⟹  Z = 6
A : 14 + 1 = A + 1  ⟹  A = 14
N = A − Z = 14 − 6 = 8

¹⁴₆C  ⟶  ¹⁴₇N + X
A : 14 = 14 + A(X)  ⟹  A(X) = 0
Z : 6 = 7 + Z(X)  ⟹  Z(X) = −1
¹⁴₆C  ⟶  ¹⁴₇N + ⁰₋₁e   (β⁻)
¹₀n  ⟶  ¹₁p + ⁰₋₁e

6 = 7 + (−1)  ✓
14 = 14 + 0  ✓

R(0) = 57 Bq  ;  R(t₁) = 40 Bq
t₁ ≈ 2900 … 3000 years
R(t₁)R(0) = N(t₁)N(0) = 4057 = 0.702
R(t) = R₀ · 2−t/T½  ⟹  t = T½ · log₂ R₀R t₁ = 5730 · log₂ 5740 = 5730 · 0.511 = 2928 years

λ = ln2T½  ;  λ · T½ = ln2 = 0.693
1 year = 365 · 24 · 3600 = 3.1536·107 s T½ = 5730 · 3.1536·107 = 1.807·1011 s
λ = 0.6931.807·1011 = 3.84·10−12 1/s
3.84·10−12 · 1.807·1011 = 0.693  ✓

R = λ · N  ;  R₀ = λ · N(0)  ;  R(t₁) = λ · N(t₁)
N(t₁) = N(0) − ΔN R(t₁) = λ·(N(0) − ΔN) = λ·N(0) − λ·ΔN = R₀ − λ·ΔN
λ·ΔN = 3.84·10−12 · 4.63·1012 = 17.76 Bq
R(t₁) = 57 − 17.76 = 39.24 ≈ 39.2 Bq
N(0) = R₀λ = 573.84·10−12 = 1.486·1013 N(t₁) = 1.486·1013 − 0.463·1013 = 1.023·1013 R(t₁) = 3.84·10−12 · 1.023·1013 = 39.2 Bq  ✓

t₁ ≈ 3050 … 3100 years
t₁ = 5730 · log₂ 5739.24 = 5730 · 0.5386 = 3086 years
Δt = 3086 − 2928 = 159 years
ΔtT½ = log₂ 4039.24 = 0.0277  ⟹  Δt = 159 years
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