Physics · Units 9–11 · Quick reference
Physics 2
Grades 11–12+8 pages
Electric charge, fields and potential; capacitors; circuits with Ohm’s law, Kirchhoff’s rules and RC timing; magnetism and induction; waves and sound; mirrors, lenses and wave optics; photons, atoms, nuclear decay and half-life.
1Constants2Coulomb3Electric fields4Potential5Capacitors6Current & power7Kirchhoff8RC circuits9Magnetism10Induction & AC11Waves12Sound13Mirrors14Lenses15Wave optics16Photons & atoms17Nuclear18Fission & relativity
1Constants & Key Equationsvalues used on every page
| Constant | Value |
|---|---|
| k (Coulomb) | 8.99 × 10⁹ N·m²/C² |
| ε₀ (free space) | 8.85 × 10⁻¹² C²/(N·m²) |
| e (elementary charge) | 1.602 × 10⁻¹⁹ C |
| mₑ; mₚ | 9.11 × 10⁻³¹ kg; 1.67 × 10⁻²⁷ kg |
| μ₀ (magnetic) | 4π × 10⁻⁷ T·m/A |
| c (light) | 3.00 × 10⁸ m/s |
| h (Planck) | 6.626 × 10⁻³⁴ J·s |
| hc | 1240 eV·nm |
| 1 eV | 1.602 × 10⁻¹⁹ J |
| 1 u (mass unit) | 931.5 MeV of energy |
| I₀ (faintest sound) | 1.0 × 10⁻¹² W/m² |
Reading the symbols
- ℰ: emf, the voltage a battery or coil supplies. Φ: magnetic flux. μ: mass per meter of a string. φ: work function.
- Prefixes: μ = 10⁻⁶, n = 10⁻⁹, p = 10⁻¹²; k = 10³, M = 10⁶.
| Topic | Equations |
|---|---|
| Charge | F = kq₁q₂r²; E = Fq = kQr² |
| Potential | V = kQr; ΔU = qΔV; E = ΔVd |
| Capacitors | C = QV = ε₀Ad; U = ½CV² |
| Circuits | V = IR; P = IV; R = ρLA; τ = RC |
| Magnetism | F = qvB sin θ; r = mvqB; B = μ₀I2πr; B = μ₀nI |
| Induction | Φ = BA cos θ; ℰ = −NΔΦΔt; ℰ = BLv |
| Waves | v = fλ; v = √(FTμ); β = 10 log II₀ |
| Optics | 1f = 1do + 1di; m = −dido; d sin θ = mλ |
| Modern | E = hf = hcλ; KE = hf − φ; λ = hp; E = mc² |
Vector or scalar?
- Vectors (add with directions): F, E, B, v.
- Scalars (add with signs): q, V, U, I.
| Quantity | Unit | In base units |
|---|---|---|
| charge q | coulomb (C) | A·s |
| potential V | volt (V) | J/C |
| resistance R | ohm (Ω) | V/A |
| capacitance C | farad (F) | C/V |
| magnetic field B | tesla (T) | N/(A·m) |
| flux Φ | weber (Wb) | T·m² |
| energy | joule or eV | N·m |
Powers of ten on a calculator
- Enter 8.99 × 10⁹ as 8.99 EE 9 (or EXP), not 8.99 × 10 ^ 9 typed by hand.
- Wrap fraction bottoms in parentheses: kqr² is kq ÷ (r²).
Prefix first, then plug
Turn μC, nF, mm and kΩ into base units before you multiply; most wrong answers are off by a power of ten.
2Charge & Coulomb’s Lawforces between charges
Electric charge
- Two kinds, + and −; like charges repel, unlike attract.
- Charge is conserved: rubbing moves electrons, it never makes charge.
- Charge comes in whole multiples of e.
Strong for its size
- Between a proton and an electron, the electric pull is about 2 × 10³⁹ times their gravity.
- Two 1 μC charges 30 cm apart push with about the weight of 10 g.
Sizes in, signs out
Put only sizes into Coulomb’s law. Then draw the arrow: like charges push apart, unlike pull together.
Charging objects
- Friction: electrons move from one material to another.
- Contact: a charged object shares charge with a conductor it touches.
- Induction: bring a charge near, ground the far side, remove the ground; the object keeps the opposite charge.
Forces add as vectors Worked
+3.0 μC at x = 0, −5.0 μC at x = 0.40 m; +2.0 μC at x = 0.20 m
from +3.0: (8.99 × 10⁹)(3.0 × 10⁻⁶)(2.0 × 10⁻⁶)0.20² = 1.35 N, pushed toward +xrepel
from −5.0: 2.25 N, pulled toward +xattract
net: 1.35 + 2.25 = 3.60 N toward the −5.0 μCsame direction, so add
Coulomb’s law
Coulomb measured it with a torsion balance (1785). Use sizes in the formula; get the direction from like or unlike.
F = kq₁q₂r²
Counting electrons Worked
a balloon picks up −8.0 nC
n = 8.0 × 10⁻⁹ C1.602 × 10⁻¹⁹ C = 4.99 × 10¹⁰ extra electrons
Sharing charge Worked
identical metal spheres: +6.0 nC and −2.0 nC touch, then part
total stays +4.0 nC, split evenly: +2.0 nC eachcharge is conserved
3Electric Fieldsforce per unit charge
Electric field
In N/C, the same as V/m. E points the way a + charge is pushed.
E = Fq = kQr²
E = Vd (parallel plates)
Field lines
- Start on +, end on −; they never cross.
- Closer lines mean a stronger field.
- A conductor at rest has E = 0 inside and its extra charge on the outside (a Faraday cage).
Watch out
- Electrons accelerate against the field arrows.
- Fields from several charges add as vectors; draw each before adding.
Follow the plus charge
To find a field’s direction, imagine a tiny + charge there and ask which way it would be pushed.
Field of a point charge Worked
+4.0 nC; point 0.30 m away
E = (8.99 × 10⁹)(4.0 × 10⁻⁹)0.30² = 400 N/C, pointing away
an electron there: F = eE = 6.41 × 10⁻¹⁷ N, toward the chargeopposite E
Floating an oil drop Worked
plates 1.0 cm apart at 245 V; drop mass 8.0 × 10⁻¹⁵ kg hangs still
E = 2450.010 = 2.45 × 10⁴ N/C
qE = mg → q = (8.0 × 10⁻¹⁵)(9.8)2.45 × 10⁴ = 3.2 × 10⁻¹⁸ C
= 3.2 × 10⁻¹⁸1.602 × 10⁻¹⁹ = 20 electron chargesMillikan’s idea (1909)
4Potential & Energyvolts are joules per coulomb
- Potential V
- energy per charge at a point, Uq (volts = J/C)
- Potential difference ΔV
- what a voltmeter reads; work per charge to move between two points
- Electron-volt (eV)
- energy one electron gains crossing 1 V = 1.602 × 10⁻¹⁹ J
- Equipotential
- surface of equal V; it crosses field lines at 90°, so moving along it takes no work
Uniform field Worked
9.0 V across plates 3.0 mm apart
E = 9.03.0 × 10⁻³ = 3.0 × 10³ V/mV/m = N/C
Potential and energy
V is a scalar: add the potentials of several charges with their signs.
V = kQr (point charge)
ΔU = qΔV, E = ΔVd
Electron gun Worked
an electron speeds up from rest across 500 V
KE = 500 eV = 8.01 × 10⁻¹⁷ JqΔV
v = √(2KEmₑ) = 1.33 × 10⁷ m/sabout 4% of c
Potential of two charges Worked
+6.0 nC at 0.20 m and −3.0 nC at 0.30 m from point P
V = k(6.0 × 10⁻⁹0.20 − 3.0 × 10⁻⁹0.30) = (8.99 × 10⁹)(3.0 × 10⁻⁸ − 1.0 × 10⁻⁸)
= 180 Vscalars, signs kept
bring +2.0 μC from far away: W = qV = 3.6 × 10⁻⁴ J
Charges roll downhill
A + charge set free moves from high V to low V, like a ball rolling downhill; an electron rolls “uphill” to higher V.
5Capacitorsstoring charge and energy
Capacitance
In farads (F). Two plates hold +Q and −Q. A dielectric (κ > 1) between the plates raises C.
C = QV, C = κε₀Ad
U = ½CV² = ½QV
Capacitors at work
- Touchscreens sense a finger by its capacitance; flashes and defibrillators dump stored charge fast.
Parallel plates Worked
A = 0.040 m², d = 1.0 mm, air, on 9.0 V
C = (8.85 × 10⁻¹²)(0.040)1.0 × 10⁻³ = 354 pF
Q = CV = 3.19 nC; U = ½CV² = 14.3 nJ
fill the gap with κ = 2.5: C = 885 pF
Capacitors flip the rules
Capacitors combine the opposite way from resistors: in parallel they simply add; in series use reciprocals.
| Parallel | Series | |
|---|---|---|
| Same on each | V | Q |
| Total | C₁ + C₂ + … | 1C = 1C₁ + 1C₂ + … |
| 6.0 μF with 3.0 μF | 9.0 μF | 2.0 μF |
Camera flash Worked
150 μF charged to 300 V
U = ½(150 × 10⁻⁶)(300²) = 6.75 Jdumped in a millisecond
6Current, Resistance & PowerOhm’s law at work
- Current I
- ΔqΔt in amperes; 1 A = 1 C/s
- Resistance R
- VI in ohms; R = ρLA
- Resistivity ρ
- a property of the material (Ω·m); in metals it rises with temperature
- emf ℰ
- energy per charge a source supplies
- Internal resistance r
- resistance inside the source: Vterminal = ℰ − Ir
Watts are volts times amps
P = IV. Swap in V = IR to get I²R or V²/R, then use the form with the two numbers you know.
Ohm’s law and power
Ohm (1827). Energy = Pt; utilities bill in kWh.
V = IR
P = IV = I²R = V²R
Resistance of a wire Worked
copper, ρ = 1.68 × 10⁻⁸ Ω·m; 20 m long, 1.0 mm across
R = (1.68 × 10⁻⁸)(20)π(0.50 × 10⁻³)² = 0.428 ΩA = πr²
Space heater Worked
1500 W on 120 V
I = 1500120 = 12.5 A; R = 12012.5 = 9.6 Ω
3.0 h a day for 30 days: 1.5 kW × 90 h = 135 kWh
at $0.15 per kWh: $20.25
A real battery Worked
ℰ = 9.0 V, r = 0.50 Ω, load 4.0 Ω
I = 9.04.0 + 0.50 = 2.0 A
terminal voltage = (2.0)(4.0) = 8.0 V; 1.0 V is lost inside
7Series-Parallel & Kirchhoffsolving any network
Combination circuit Worked
parallel part: (6)(12)6 + 12 = 4 Ω; total 4 + 4 = 8 Ωinnermost first
I = 128 = 1.5 A
4 Ω drops 6.0 V, so 6.0 V is left across the pair
branches: 6.06 = 1.0 A, 6.012 = 0.5 Asum 1.5 A
battery power: (12)(1.5) = 18 W
Kirchhoff’s rules (1845)
- Junction: current in = current out (charge is conserved).
- Loop: the voltage changes around any closed loop add to zero (energy is conserved).
- Across a resistor in the current’s direction: −IR. Across a battery from − to +: +ℰ.
Two-loop circuit Worked
junction: I₁ + I₃ = I₂
left loop: 12 − 2I₁ − 4I₂ = 0
right loop: 6 − 2I₃ − 4I₂ = 0
I₁ = 2.4 A, I₂ = 1.8 A, I₃ = −0.6 Asolve together
the minus sign: I₃ really flows down, charging the 6 V battery
Meters
- Ammeter: in series, nearly 0 Ω.
- Voltmeter: in parallel, very high resistance.
Junctions count, loops balance
At a junction count the amps in and out; around a loop the volts gained and lost balance to zero.
8RC Circuitscharging takes time
Discharging to half Worked
same 10 s time constant
12 = e−t/10 → t = 10 ln 2 = 6.93 sln 2 = 0.693
RC circuits
τ in seconds when R is in ohms and C in farads. The current falls as I₀e−t/RC in both cases.
τ = RC
charging: V = V₀(1 − e−t/RC)
discharging: V = V₀e−t/RC
Sixty-three up, thirty-seven down
After one time constant a charging capacitor is 63% full; a discharging one keeps 37%.
Charging a capacitor Worked
50 kΩ and 200 μF on 9.0 V, starting empty
τ = (50 × 10³)(200 × 10⁻⁶) = 10 s
at 10 s: V = 9.0(1 − e−1) = 5.69 V63%
at 50 s: 8.94 V (99.3%), “full”5τ
current at the start: 9.050 × 10³ = 0.18 mAC acts like a wire
Two moments
- Just switched on: an empty capacitor passes current like a plain wire.
- Long after: a full capacitor blocks steady current like a gap.
- Uses: camera flashes, blinking lights, timers, filters.
9Magnetic Forces & Fieldsmoving charges and currents
Force on moving charge
F is perpendicular to both v and B, so it bends the path but does no work: the speed stays the same.
F = qvB sin θ
r = mvqB
Right-hand rule (Traditional)
Fingers along v, curl them toward B; your thumb gives F on a + charge. Flip it for an electron.
| Source | Field B |
|---|---|
| long straight wire | μ₀I2πr; circles the wire |
| solenoid (n turns per m) | μ₀nI inside, nearly uniform |
| wire in a field | F = BIL sin θ |
Solenoid Worked
2000 turns on 0.40 m; 1.5 A
n = 20000.40 = 5000 per m
B = (4π × 10⁻⁷)(5000)(1.5) = 9.42 mT
Proton in a magnet Worked
v = 2.0 × 10⁶ m/s, B = 0.50 T, at 90°
F = (1.602 × 10⁻¹⁹)(2.0 × 10⁶)(0.50) = 1.60 × 10⁻¹³ N
r = (1.67 × 10⁻²⁷)(2.0 × 10⁶)(1.602 × 10⁻¹⁹)(0.50) = 4.17 cmmass spectrometer idea
one lap: T = 2πmqB = 1.31 × 10⁻⁷ ssame for any speed
Motor coil Worked
50 turns, 0.010 m², 2.0 A, in 0.30 T, plane along B
τ = NIAB sin θ = (50)(2.0)(0.010)(0.30) = 0.30 N·mθ from the coil’s axis to B
the commutator flips the current each half turnDC motor
Parallel wires Worked
15 A and 10 A, same direction, 5.0 cm apart
FL = μ₀I₁I₂2πd = 6.0 × 10⁻⁴ N/m, attractingopposite currents repel
Field directions
- Wire: grip it with your right thumb along I; fingers curl the way B circles.
- Solenoid: curl right fingers with the current; the thumb points to the north end.
Velocity selector
- Crossed E and B fields let through only charges with v = EB: 3.0 × 10⁴ V/m with 0.10 T passes 3.0 × 10⁵ m/s.
- Thomson used crossed fields on cathode rays (1897).
10Induction & ACchanging flux makes voltage
Faraday’s law
Flux in webers (T·m²). Faraday found induction in 1831. The minus sign is Lenz’s law.
Φ = BA cos θ
ℰ = −NΔΦΔt, ℰ = BLv
Lenz’s law
- The induced current makes a field that opposes the change in flux.
- Push a magnet’s N pole into a coil: the coil’s near end becomes N and pushes back.
Sliding rod Worked
0.50 m rod at 4.0 m/s on rails, B = 0.80 T; circuit 2.0 Ω
ℰ = BLv = 1.6 V; I = 0.80 A
pushing force F = BIL = 0.32 Nagainst the motion
power: Fv = I²R = 1.28 Wenergy is conserved
Coil in a rising field Worked
200 turns, 0.010 m²; B rises 0 → 0.60 T in 0.20 s
ℰ = (200)(0.60)(0.010)0.20 = 6.0 V
Generator Worked
100 turns, 0.020 m², 0.40 T, 60 Hz
ω = 2π(60) = 377 rad/s
ℰmax = (100)(0.40)(0.020)(377) = 302 V
Why power lines use high voltage Worked
send 500 kW through 4.0 Ω of line
at 10 kV: I = 50 A, loss I²R = 10 kW (2.0%)
at 100 kV: I = 5.0 A, loss = 100 Wtransformers step up, then down
Change makes current
A steady field induces nothing. Move, turn, grow or shrink the flux, and a current appears that resists the change.
Generators and AC
- A coil turning in B gives ℰmax = NBAω, with ω = 2πf.
- Vrms = Vpeak√2: a U.S. outlet is 120 V rms, about 170 V peak.
Transformers
- VsVp = NsNp = IpIs (ideal): voltage up means current down.
- They work only with changing current (AC).
- More on motors and generators: the School Launch Magnetism and Induction sheet (Lesson 10.4).
Induction at work
- Generators, induction cooktops, wireless chargers, metal detectors and guitar pickups.
- Eddy currents in moving metal oppose the motion: magnetic brakes on trains and roller coasters.
11Waves & Standing Wavesstrings, pipes and beats
Wave basics
- Transverse: the medium moves across the wave (string, light). Longitudinal: along it (sound).
- The medium sets the speed; the source sets the frequency.
- Entering a new medium, f stays; v and λ change.
Waves on a string
FT = tension (N); μ = mass per length (kg/m). Tighter or lighter strings carry faster waves.
v = fλ, v = √(FTμ)
Tension and speed
- v grows as √FT: 4 times the tension makes waves 2 times as fast, raising every harmonic an octave.
Radio wavelength Worked
FM station at 101.5 MHz
λ = cf = 3.00 × 10⁸101.5 × 10⁶ = 2.96 mlight speed, not sound
| Holder | Allowed λ | Frequencies |
|---|---|---|
| string, or tube open at both ends | 2Ln | fn = nv2L, n = 1, 2, 3 … |
| tube closed at one end | 4Ln | fn = nv4L, n = 1, 3, 5 … |
Tuning a string Worked
0.80 m string, mass 4.0 g, tension 80 N
μ = 0.00400.80 = 0.0050 kg/m; v = √(800.0050) = 126 m/s
f₁ = 1262(0.80) = 79.1 Hz; f₃ = 237 Hz
Bottle-length tubes Worked
a 0.25 m tube, closed at one end
f₁ = 3434(0.25) = 343 Hz; next: 1029 Hzodd harmonics only
open at both ends: f₁ = 686 Hz
Interference and beats
- Path difference of mλ: waves add. Of (m + ½)λ: they cancel.
- Beats: fbeat = |f₁ − f₂|. 330 Hz with 333 Hz throbs 3 times a second.
- Chladni made 2-D standing waves visible with sand on a bowed plate (1787).
Resonance
- Driving a system at one of its natural frequencies builds large amplitudes: a pushed swing, a singer’s glass, an instrument’s body.
Closed ends are nodes
A fixed string end or a closed tube end cannot move: always a node. An open tube end swings freely: an antinode.
12Soundloudness, Doppler and booms
Intensity and decibels
I in W/m²; I₀ = 1.0 × 10⁻¹² W/m². Each +10 dB is 10 times the intensity.
I = P4πr²
β = 10 log II₀
Outdoor speaker Worked
2.0 W of sound, spreading evenly; listener 4.0 m away
I = 2.04π(4.0)² = 9.95 × 10⁻³ W/m²
β = 10 log 9.95 × 10⁻³10⁻¹² = 100 dB
at 8.0 m: ¼ the intensity, 94 dBabout 6 dB less
Closing in, pitch climbs
When source and listener get closer, every sign choice should raise f′; when they part, lower it. Check your answer that way.
| Level | Intensity | Like |
|---|---|---|
| 0 dB | 10⁻¹² W/m² | faintest sound heard |
| 30 dB | 10⁻⁹ W/m² | a whisper |
| 60 dB | 10⁻⁶ W/m² | conversation |
| 90 dB | 10⁻³ W/m² | a lawn mower up close |
| 120 dB | 1 W/m² | threshold of pain |
Two equal sources together: +3 dB, since 10 log 2 ≈ 3.
Shock waves
- Faster than sound, wave crests pile into a cone: a sonic boom. Mach number = v ÷ vsound.
- A jet at 515 m/s: Mach 1.50; cone half-angle sin⁻¹(11.50) = 41.8°.
Ten times means ten decibels
Every factor of 10 in intensity adds 10 dB; doubling your distance from a source drops about 6 dB.
Doppler effect
- f′ = f v ± vov ∓ vs, v = 343 m/s in air.
- Upper signs when source and observer move toward each other.
Siren and car Worked
700 Hz siren at 30 m/s; car at 20 m/s toward it
approaching: 700 × 343 + 20343 − 30 = 812 Hz
after passing: 700 × 343 − 20343 + 30 = 606 Hz
Echo Worked
a shout returns from a cliff after 1.2 s
d = (343)(1.2)2 = 206 mthere and back
Doppler with waves of light
- Radar guns and weather radar find speed from the frequency shift of reflected microwaves.
- Light from a receding star shifts toward red (redshift).
13Mirrorsray diagrams and images
Mirror equation
The same equation as a thin lens.
f = R2
1f = 1do + 1di, m = −dido
| Sign | + | − |
|---|---|---|
| f | concave | convex |
| di | real, in front | virtual, behind |
| m | upright | inverted |
| Ray in | Ray out (concave) |
|---|---|
| parallel to the axis | through F |
| through F | parallel to the axis |
| through C (center) | straight back on itself |
| to the center of the mirror | at an equal angle below the axis |
Concave mirror Worked
R = 20 cm, so f = 10 cm; object at 15 cm
1di = 110 − 115 → di = 30 cm
m = −3015 = −2.0: real, inverted, twice the sizematches the figure
Store security mirror Worked
convex, f = −20 cm; shopper 30 cm away
di = −12 cm; m = 0.40virtual, upright, smaller
convex mirrors always shrink: wide view
Makeup mirror Worked
concave, f = 20 cm; face at 10 cm
di = −20 cm; m = 2.0: upright, magnifiedobject inside F
| Object (concave mirror) | Image |
|---|---|
| beyond C | real, inverted, smaller |
| at C | real, inverted, same size |
| between C and F | real, inverted, larger |
| at F | none: rays leave parallel |
| inside F | virtual, upright, larger |
| any (plane mirror) | virtual, upright, same size, as far behind |
Telescopes
- A spherical mirror blurs rays far from the axis; a parabolic mirror brings all parallel rays to one focus.
- Newton’s reflecting telescope used a concave mirror instead of a lens, avoiding color fringes.
Concave caves in (Traditional)
A concave mirror curves inward like a cave and gathers light to a focus; a convex one bulges out and spreads it.
14Refraction & Lensesbending light to form images
Snell’s law and lenses
Converging (convex) lens: f > 0. Diverging (concave) lens: f < 0, always a virtual, upright, smaller image.
n₁ sin θ₁ = n₂ sin θ₂
1f = 1do + 1di
Diverging lens Worked
f = −10 cm, object at 15 cm
1di = −110 − 115 → di = −6.0 cmvirtual
m = −−6.015 = 0.40upright, smaller
Into a glass block Worked
light hits glass (n = 1.52) at 40° from the normal
sin θ₂ = sin 40°1.52 → θ₂ = 25.0°bends toward the normal
it leaves the far side at 40° again, shifted sidewaysparallel faces
Lens power and eyes
- Power in diopters: P = 1f (f in meters): f = 0.50 m is +2.0 D.
- Farsighted eyes need converging (+) lenses; nearsighted, diverging (−): a −2.5 D lens has f = −40 cm.
Two lenses Worked
f₁ = 10 cm, object at 15 cm: image 30 cm past lens 1, m₁ = −2first image
lens 2 (f = 5 cm) sits 40 cm from lens 1: object 10 cm away → di = 10 cm, m₂ = −1
total m = (−2)(−1) = +2: upright, twice the sizemultiply the m’s
Magnifying glass Worked
f = 5.0 cm; stamp at 4.0 cminside F
1di = 15.0 − 14.0 → di = −20 cm
m = −−204.0 = 5.0upright, 5 times larger
Water tricks Worked
pool 2.0 m deep, seen from straight above
looks 2.01.33 = 1.50 m deepapparent depth
from under water, light past sin⁻¹(11.33) = 48.8° reflects totallycritical angle
Dispersion
- n depends on color: violet bends most, red least, so a prism spreads white light into a spectrum.
Negative means virtual
A negative di is a virtual image on the object’s side; a negative f is a diverging lens or a convex mirror.
15Wave Optics & Lightinterference, diffraction, polarization
Light is an electromagnetic wave
- Maxwell (1865) found c = 1√(μ₀ε₀) = 3.00 × 10⁸ m/s; Hertz made radio waves (1887).
- “And God said, Let there be light: and there was light” (Genesis 1:3, KJV).
Coated lens Worked
MgF₂ coating, n = 1.38, to cancel 550 nm reflections
t = λ4n = 5504(1.38) = 99.6 nmthin film
Two polarizers Worked
800 W/m² of unpolarized light
first filter: half passes, 400 W/m²
second at 60°: I = I₀cos²θ = 400(0.5)² = 100 W/m²Malus’s law
Whole waves shine bright
Bright spots come where paths differ by a whole number of wavelengths; dark ones at a half-wave extra.
Gratings and slits
d = slit spacing; a = slit width; m = 1, 2, 3 … The central bright band of a single slit is twice as wide as the others.
grating, bright: d sin θ = mλ
single slit, dark: a sin θ = mλ
Diffraction grating Worked
600 lines/mm; green laser, 532 nm
d = 1 mm600 = 1.67 × 10⁻⁶ m
m = 1: 18.6°; m = 2: 39.7°; m = 3: 73.3°sin θ = mλd
m = 4 would need sin θ > 1: none
Soap film Worked
film n = 1.33 in air; brightest reflection of 600 nm
one reflection flips phase, so 2nt = λ2: t = 6004(1.33) = 113 nmthinnest film
Single slit Worked
633 nm through a 0.10 mm slit; screen 1.5 m away
central band width ≈ 2λLa = 19.0 mmsmall angles
| Band | Wavelength | Use |
|---|---|---|
| radio | longer than 1 m | broadcasting |
| microwave | 1 mm to 1 m | radar, ovens, Wi-Fi |
| infrared | 700 nm to 1 mm | heat lamps, remotes |
| visible | 400 to 700 nm | sight: 3.10 to 1.77 eV |
| ultraviolet | 10 to 400 nm | sunburn, sterilizing |
| X-ray | 0.01 to 10 nm | medical images |
| gamma | under 0.01 nm | nuclear decay, cancer care |
Polarized light at work
- Polarized sunglasses block glare reflected from water and roads; LCD screens use crossed polarizers.
16Photons & Atomslight packets and energy levels
Photons
Planck (1900): light energy comes in packets. Einstein (1905): each photon acts alone; Millikan confirmed it (1916).
E = hf = hcλ
E (eV) = 1240λ (nm)
Light on a metal Worked
φ = 2.30 eV; violet light, 400 nm
E = 1240400 = 3.10 eV; KEmax = 3.10 − 2.30 = 0.80 eVstopping voltage 0.80 V
red, 700 nm: 1.77 eV < 2.30 eV: no electrons, however bright
cutoff: λ = 12402.30 = 539 nm
One number both ways
E in eV = 1240 ÷ λ in nm, and λ = 1240 ÷ E. Short waves carry big packets.
Photoelectric effect
- KEmax = hf − φ, where φ (work function) is the least energy that frees an electron.
- Brighter light frees more electrons, not faster ones; below a cutoff frequency, none at all.
Hydrogen’s blue-green line Worked
electron drops from n = 4 to n = 2
E = −0.85 − (−3.40) = 2.55 eV
λ = 12402.55 = 486 nmblue-green
n = 2 → 1: 10.2 eV, 122 nmultraviolet
Photons from a laser pointer Worked
5.0 mW, 532 nm
E = 1240532 = 2.33 eV = 3.73 × 10⁻¹⁹ J
photons per second: 5.0 × 10⁻³3.73 × 10⁻¹⁹ = 1.34 × 10¹⁶
Bohr model of hydrogen
- En = −13.6 eVn²: n = 1, 2, 3, 4 give −13.6, −3.40, −1.51, −0.85 eV.
- A drop between levels emits one photon carrying the difference; freeing the electron from n = 1 takes 13.6 eV.
- Bohr (1913) explained the lines Balmer fit in 1885.
Matter waves
- de Broglie (1920s): λ = hp = hmv.
- An electron after 500 V: λ = h√(2mₑeV) = 5.48 × 10⁻¹¹ m, about the spacing of atoms, so electrons diffract in crystals.
- A 0.145 kg ball at 40 m/s: 1.14 × 10⁻³⁴ m, far too small to notice.
17Nucleus, Decay & Half-Lifeinside the nucleus
The nucleus
- Rutherford found the tiny, dense nucleus (1911); Chadwick found the neutron (1932).
- AZX: A = protons + neutrons, Z = protons. Isotopes share Z but differ in A.
- The strong force holds the nucleus together against the protons’ repulsion.
Half-life
T = half-life. Decay is random for one nucleus but steady for billions.
N = N₀(½)t/T
Medical tracer Worked
technetium-99m, T = 6.0 h; 80 mg delivered
246.0 = 4 half-lives: 80 → 40 → 20 → 10 → 5.0 mg
| Decay | Emits | Change | Stopped by |
|---|---|---|---|
| alpha α | ⁴₂He nucleus | A − 4, Z − 2 | paper or skin |
| beta β⁻ | electron | A same, Z + 1 | a few mm of aluminum |
| gamma γ | photon | no change in A or Z | thick lead or concrete |
E = mc²: binding energy Worked
2 protons + 2 neutrons: 2(1.00728) + 2(1.00866) u
helium-4 nucleus: 4.00151 u; missing mass 0.0304 umass defect
E = 0.0304 × 931.5 = 28.3 MeVbinding energy
Half-life and age
- Half-lives are measured in labs today and hold steady under ordinary conditions.
- Turning one into an age also needs the starting amounts and a closed sample, which no one observed. Christians agree decay is real and differ on the ages; see the course (Lesson 11.3) and High School Geology.
Balancing decays Worked
²¹⁰₈₄Po → ²⁰⁶₈₂Pb + ⁴₂He210 = 206 + 4; 84 = 82 + 2
¹⁴₆C → ¹⁴₇N + ⁰₋₁echarge: 6 = 7 − 1
⁶⁰₂₇Co → ⁶⁰₂₈Ni + ⁰₋₁e + γsterilizes supplies
Not a whole number Worked
cobalt-60, T = 5.27 years; 10.0 g for 8.0 years
N = 10.0(½)8.0/5.27 = 3.49 guse the yx key
Activity
- Decays per second, in becquerels (1 Bq = 1 decay/s). Activity halves every half-life, just as N does.
Radiation safety
- Time: less exposure, less dose. Distance: dose falls off like 1/r². Shielding: match the barrier to the radiation.
- Alpha sources are dangerous mainly if swallowed or breathed in.
Alpha sheds four and two
Alpha: A drops 4 and Z drops 2. Beta-minus: Z rises 1, A stays. Gamma: only energy leaves.
18Fission, Fusion & RelativityE = mc² at work
Why both release energy
- Binding energy per nucleon peaks near iron-56.
- Splitting heavy nuclei (fission) or joining light ones (fusion) both move toward iron and free energy: E = Δm × 931.5 MeV per u.
Splitting uranium-235 Worked
²³⁵₉₂U + ¹₀n → ¹⁴¹₅₆Ba + ⁹²₃₆Kr + 3 ¹₀n236 = 141 + 92 + 3; 92 = 56 + 36
about 200 MeV per split, millions of times a chemical reaction
the 3 new neutrons can split more nuclei: a chain reactionFermi’s pile, Chicago, 1942
Fast particle Worked
lives 2.2 μs at rest; moves at 0.980c
γ = 1√(1 − 0.980²) = 5.03
lab lifetime 2.2 × 5.03 = 11.1 μs: travels 3.25 km, not 0.65 kmwhy cosmic-ray muons reach the ground
Fusing deuterium Worked
²H + ²H → ³He + ¹₀n
Δm = 2(2.01355) − 3.01493 − 1.00866 = 0.00351 u
E = 0.00351 × 931.5 = 3.27 MeVD + T gives 17.6 MeV
| Fission | Fusion | |
|---|---|---|
| Nuclei | heavy split (U-235) | light join (H isotopes) |
| Needs | slow neutrons | enormous temperature |
| Where | power plants (Shippingport, 1957) | the Sun (Bethe, 1939); research reactors |
| Waste | radioactive fragments | mostly helium |
Moving clocks run slow
A clock passing you ticks slow by a factor γ; a ruler passing you is short by γ.
Special relativity (1905)
- Postulates: the laws of physics are the same at every steady velocity, and light’s speed c is the same for every observer.
- Michelson and Morley (1887) found no change in c with Earth’s motion.
Lorentz factor
Moving clocks run slow; moving lengths shrink along the motion. Both matter only near c.
γ = 1√(1 − v²/c²)
t = γt₀, L = L₀γ
Clocks in orbit
- GPS satellite clocks run about 38 μs a day fast (speed and weaker gravity together); light goes 11 km in that time, so engineers correct them.