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Physics formulas from Mechanics, Waves, Optics, Heat and Thermodynamics, Electricity and Magnetism and Modern Physics. Also includes the value of Physical Constants. Helps in quick revision for CBSE, NEET, JEE Mains, and Advanced.
0.1: Physical Constants
Speed of light c 3 × 108 m/s Planck constant h 6. 63 × 10 −^34 J s hc 1242 eV-nm Gravitation constant G 6. 67 × 10 −^11 m^3 kg−^1 s−^2 Boltzmann constant k 1. 38 × 10 −^23 J/K Molar gas constant R 8 .314 J/(mol K) Avogadro’s number NA 6. 023 × 1023 mol−^1 Charge of electron e 1. 602 × 10 −^19 C Permeability of vac- uum
μ 0 4 π × 10 −^7 N/A^2
Permitivity of vacuum 0 8. 85 × 10 −^12 F/m Coulomb constant (^4) π^10 9 × 109 N m^2 /C^2 Faraday constant F 96485 C/mol Mass of electron me 9. 1 × 10 −^31 kg Mass of proton mp 1. 6726 × 10 −^27 kg Mass of neutron mn 1. 6749 × 10 −^27 kg Atomic mass unit u 1. 66 × 10 −^27 kg Atomic mass unit u 931 .49 MeV/c^2 Stefan-Boltzmann constant
σ 5. 67 × 10 −^8 W/(m^2 K^4 )
Rydberg constant R∞ 1. 097 × 107 m−^1 Bohr magneton μB 9. 27 × 10 −^24 J/T Bohr radius a 0 0. 529 × 10 −^10 m Standard atmosphere atm 1. 01325 × 105 Pa Wien displacement constant
b 2. 9 × 10 −^3 m K
1 MECHANICS
1.1: Vectors
Notation: ~a = ax ˆı + ay ˆ + az ˆk
Magnitude: a = |~a| =
a^2 x + a^2 y + a^2 z
Dot product: ~a ·~b = axbx + ay by + az bz = ab cos θ
Cross product: ~a
~a × ~b ~b θ
ˆı
ˆk ˆ
~a ×~b = (ay bz − az by )ˆı + (az bx − axbz )ˆ + (axby − ay bx)kˆ
|~a × ~b| = ab sin θ
1.2: Kinematics
Average and Instantaneous Vel. and Accel.:
~vav = ∆~r/∆t, ~vinst = d~r/dt ~aav = ∆~v/∆t ~ainst = d~v/dt
Motion in a straight line with constant a:
v = u + at, s = ut + 12 at^2 , v^2 − u^2 = 2as
Relative Velocity: ~vA/B = ~vA − ~vB
Projectile Motion:
x
y
O
u^ sin
θ
u cos θ
u
θ
R
H
x = ut cos θ, y = ut sin θ − 12 gt^2
y = x tan θ −
g 2 u^2 cos^2 θ x^2
T =
2 u sin θ g
, R =
u^2 sin 2θ g
, H =
u^2 sin^2 θ 2 g
1.3: Newton’s Laws and Friction
Linear momentum: ~p = m~v
Newton’s first law: inertial frame.
Newton’s second law: F~ = d d~pt , F~ = m~a
Newton’s third law: F~AB = − F~BA
Frictional force: fstatic, max = μsN, fkinetic = μkN
Banking angle: v
2 rg = tan^ θ,^
v^2 rg =^
μ+tan θ 1 −μ tan θ
Centripetal force: Fc = mv 2 r ,^ ac^ =^
v^2 r
Pseudo force: F~pseudo = −m~a 0 , Fcentrifugal = − mv
2 r Minimum speed to complete vertical circle:
vmin, bottom =
5 gl, vmin, top =
gl
Conical pendulum: T = 2π
l cos θ g
mg
T
l θ
θ
1.4: Work, Power and Energy
Work: W = F~ · S~ = F S cos θ, W =
F · dS~
Kinetic energy: K = 12 mv^2 = p
2 2 m Potential energy: F = −∂U/∂x for conservative forces.
Ugravitational = mgh, Uspring = 12 kx^2
Work done by conservative forces is path indepen- dent and depends only on initial and final points:∮ F~conservative · d~r = 0.
Work-energy theorem: W = ∆K
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Mechanical energy: E = U + K. Conserved if forces are conservative in nature.
Power Pav = ∆ ∆Wt , Pinst = F~ · ~v
1.5: Centre of Mass and Collision
Centre of mass: xcm =
∑ (^) x ∑ imi mi ,^ xcm^ =
∫ ∫ xdm dm
CM of few useful configurations:
- m 1 , m 2 separated by r:
m 1 m 2 C
r
m 2 r m 1 +m 2
m 1 r m 1 +m 2
- Triangle (CM ≡ Centroid) yc = h 3 C (^) h 3
h
- Semicircular ring: yc = (^2) πr C^2 r r π
- Semicircular disc: yc = (^34) πr C (^4) r r 3 π
- Hemispherical shell: yc = r 2 C r r 2
- Solid Hemisphere: yc = 38 r C r 38 r
- Cone: the height of CM from the base is h/4 for the solid cone and h/3 for the hollow cone.
Motion of the CM: M =
mi
~vcm =
mi~vi M
, ~pcm = M~vcm, ~acm =
F~ext M
Impulse: J~ =
F dt = ∆~p
Collision: (^) m 1 m 2
v 1 v 2
Before collision After collision m 1 m 2 v 1 ′ v 2 ′ Momentum conservation: m 1 v 1 +m 2 v 2 = m 1 v′ 1 +m 2 v 2 ′ Elastic Collision: 12 m 1 v 12 +^12 m 2 v 22 = 12 m 1 v 1 ′^2 +^12 m 2 v′ 22 Coefficient of restitution:
e =
−(v 1 ′ − v′ 2 ) v 1 − v 2
1 , completely elastic 0 , completely in-elastic
If v 2 = 0 and m 1 m 2 then v′ 1 = −v 1. If v 2 = 0 and m 1 m 2 then v′ 2 = 2v 1. Elastic collision with m 1 = m 2 : v′ 1 = v 2 and v 2 ′ = v 1.
1.6: Rigid Body Dynamics
Angular velocity: ωav = ∆ ∆θt , ω = d dθt , ~v = ~ω × ~r
Angular Accel.: αav = ∆ ∆ωt , α = d dωt , ~a = α~ × ~r
Rotation about an axis with constant α:
ω = ω 0 + αt, θ = ωt + 12 αt^2 , ω^2 − ω 02 = 2αθ
Moment of Inertia: I =
i miri
(^2) , I = ∫^ r (^2) dm
ring
mr^2
disk
(^12) mr 2
shell
(^23) mr 2
sphere
(^25) mr 2
rod
121 ml^2
hollow
mr^2
solid
(^12) mr 2
rectangle
m(a^2 +b^2 ) 12
a b
Theorem of Parallel Axes: I‖ = Icm + md^2 cm
I‖ d
Ic
Theorem of Perp. Axes: Iz = Ix + Iy x
z y
Radius of Gyration: k =
I/m
Angular Momentum: ~L = ~r × ~p, L~ = I~ω
Torque: ~τ = ~r × F ,~ ~τ = d dL~t , τ = Iα O x
y (^) P ~r
θ F ~
Conservation of L~: ~τext = 0 =⇒ L~ = const.
Equilibrium condition:
F = ~ 0 ,
~τ = ~ 0
Kinetic Energy: Krot = 12 Iω^2
Dynamics:
~τcm = Icm ~α, F~ext = m~acm, p~cm = m~vcm K = 12 mvcm^2 + 12 Icmω^2 , ~L = Icm~ω + ~rcm × m~vcm
1.7: Gravitation
Gravitational force: F = G m^1 rm 22 m 1 F F m 2
r
Potential energy: U = − GM mr
Gravitational acceleration: g = GMR 2
Variation of g with depth: ginside ≈ g
1 − (^) Rh
Variation of g with height: goutside ≈ g
1 − (^2) Rh
Effect of non-spherical earth shape on g: gat pole > gat equator (∵ Re − Rp ≈ 21 km)
Effect of earth rotation on apparent weight:
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2 Waves
2.1: Waves Motion
General equation of wave: ∂
(^2) y ∂x^2 =^
1 v^2
∂^2 y ∂t^2. Notation: Amplitude A, Frequency ν, Wavelength λ, Pe- riod T , Angular Frequency ω, Wave Number k,
T =
ν
2 π ω
, v = νλ, k = 2 π λ
Progressive wave travelling with speed v:
y = f (t − x/v), +x; y = f (t + x/v), −x
Progressive sine wave:
λ 2 λ x
y A
y = A sin(kx − ωt) = A sin(2π (x/λ − t/T ))
2.2: Waves on a String
Speed of waves on a string with mass per unit length μ and tension T : v =
T /μ
Transmitted power: Pav = 2π^2 μvA^2 ν^2
Interference:
y 1 = A 1 sin(kx − ωt), y 2 = A 2 sin(kx − ωt + δ) y = y 1 + y 2 = A sin(kx − ωt + )
A =
A 12 + A 22 + 2A 1 A 2 cos δ
tan =
A 2 sin δ A 1 + A 2 cos δ
δ =
2 nπ, constructive; (2n + 1)π, destructive.
Standing Waves: 2 A
cos
kx A N A N A
x
λ/ 4
y 1 = A 1 sin(kx − ωt), y 2 = A 2 sin(kx + ωt)
y = y 1 + y 2 = (2A cos kx) sin ωt
x =
n + (^12)
) (^) λ 2 ,^ nodes;^ n^ = 0,^1 ,^2 ,... n λ 2 , antinodes. n = 0, 1 , 2 ,...
String fixed at both ends:
L
N A N A N
λ/ 2
- Boundary conditions: y = 0 at x = 0 and at x = L
- Allowed Freq.: L = n λ 2 , ν = 2 nL
T μ , n^ = 1,^2 ,^3 ,.. ..
- Fundamental/1st^ harmonics: ν 0 = (^21) L
T μ
- 1st^ overtone/2nd^ harmonics: ν 1 = (^22) L
T μ
- 2nd^ overtone/3rd^ harmonics: ν 2 = (^23) L
T μ
- All harmonics are present.
String fixed at one end:
L
N (^) A N A
λ/ 2
- Boundary conditions: y = 0 at x = 0
- Allowed Freq.: L = (2n + 1) λ 4 , ν = 2 n 4 +1L
T μ ,^ n^ = 0 , 1 , 2 ,.. ..
- Fundamental/1st^ harmonics: ν 0 = (^41) L
T μ
- 1st^ overtone/3rd^ harmonics: ν 1 = (^43) L
T μ
- 2nd^ overtone/5th^ harmonics: ν 2 = (^45) L
T μ
- Only odd harmonics are present.
Sonometer: ν ∝ (^) L^1 , ν ∝
T , ν ∝ √^1 μ. ν = 2 nL
T μ
2.3: Sound Waves
Displacement wave: s = s 0 sin ω(t − x/v)
Pressure wave: p = p 0 cos ω(t − x/v), p 0 = (Bω/v)s 0
Speed of sound waves:
vliquid =
B
ρ
, vsolid =
Y
ρ
, vgas =
γP ρ
Intensity: I = 2 π
(^2) B v s^0
(^2) ν (^2) = p 02 v 2 B =^
p 02 2 ρv
Standing longitudinal waves:
p 1 = p 0 sin ω(t − x/v), p 2 = p 0 sin ω(t + x/v) p = p 1 + p 2 = 2p 0 cos kx sin ωt
Closed organ pipe: L
- Boundary condition: y = 0 at x = 0
- Allowed freq.: L = (2n + 1) λ 4 , ν = (2n + 1) 4 vL , n = 0 , 1 , 2 ,...
- Fundamental/1st^ harmonics: ν 0 = 4 vL
- 1st^ overtone/3rd^ harmonics: ν 1 = 3ν 0 = 34 vL
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- 2nd^ overtone/5th^ harmonics: ν 2 = 5ν 0 = 54 vL
- Only odd harmonics are present.
Open organ pipe: (^) L
A
N
A
N
A
- Boundary condition: y = 0 at x = 0 Allowed freq.: L = n λ 2 , ν = n 4 vL , n = 1, 2 ,...
- Fundamental/1st^ harmonics: ν 0 = 2 vL
- 1st^ overtone/2nd^ harmonics: ν 1 = 2ν 0 = (^22) Lv
- 2nd^ overtone/3rd^ harmonics: ν 2 = 3ν 0 = 32 vL
- All harmonics are present.
Resonance column:
l^1
d
l^2
d
l 1 + d = λ 2 , l 2 + d = 34 λ , v = 2(l 2 − l 1 )ν
Beats: two waves of almost equal frequencies ω 1 ≈ ω 2
p 1 = p 0 sin ω 1 (t − x/v), p 2 = p 0 sin ω 2 (t − x/v) p = p 1 + p 2 = 2p 0 cos ∆ω(t − x/v) sin ω(t − x/v) ω = (ω 1 + ω 2 )/ 2 , ∆ω = ω 1 − ω 2 (beats freq.)
Doppler Effect:
ν =
v + uo v − us
ν 0
where, v is the speed of sound in the medium, u 0 is the speed of the observer w.r.t. the medium, consid- ered positive when it moves towards the source and negative when it moves away from the source, and us is the speed of the source w.r.t. the medium, consid- ered positive when it moves towards the observer and negative when it moves away from the observer.
2.4: Light Waves
Plane Wave: E = E 0 sin ω(t − xv ), I = I 0
Spherical Wave: E = aE r 0 sin ω(t − rv ), I = I r^02
Young’s double slit experiment
Path difference: ∆x = dyD
S 1 P
S 2
d
y
D
θ
Phase difference: δ = (^2) λπ ∆x
Interference Conditions: for integer n,
δ =
2 nπ, constructive; (2n + 1)π, destructive,
∆x =
nλ,( constructive; n + (^12)
λ, destructive
Intensity:
I = I 1 + I 2 + 2
I 1 I 2 cos δ,
Imax =
I 1 +
I 2
, Imin =
I 1 −
I 2
I 1 = I 2 : I = 4I 0 cos^2 δ 2 , Imax = 4I 0 , Imin = 0
Fringe width: w = λDd
Optical path: ∆x′^ = μ∆x
Interference of waves transmitted through thin film:
∆x = 2μd =
nλ,( constructive; n + (^12)
λ, destructive.
Diffraction from a single slit: (^) b θ
y y D For Minima: nλ = b sin θ ≈ b(y/D)
Resolution: sin θ = 1.^22 bλ
Law of Malus: I = I 0 cos^2 θ (^) I 0 I
θ
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4 Heat and Thermodynamics
4.1: Heat and Temperature
Temp. scales: F = 32 + 95 C, K = C + 273. 16
Ideal gas equation: pV = nRT , n : number of moles
van der Waals equation:
p + (^) Va 2
(V − b) = nRT
Thermal expansion: L = L 0 (1 + α∆T ), A = A 0 (1 + β∆T ), V = V 0 (1 + γ∆T ), γ = 2β = 3α
Thermal stress of a material: FA = Y ∆ll
4.2: Kinetic Theory of Gases
General: M = mNA, k = R/NA
Maxwell distribution of speed:
v
n
vp ¯v vrms
RMS speed: vrms =
3 kT m =
3 RT M
Average speed: ¯v =
8 kT πm =
8 RT πM
Most probable speed: vp =
2 kT m
Pressure: p = 13 ρv^2 rms
Equipartition of energy: K = 12 kT for each degree of freedom. Thus, K = f 2 kT for molecule having f de- grees of freedoms.
Internal energy of n moles of an ideal gas is U = f 2 nRT.
4.3: Specific Heat
Specific heat: s = (^) mQ∆T
Latent heat: L = Q/m
Specific heat at constant volume: Cv = (^) n∆∆QT
V
Specific heat at constant pressure: Cp = (^) n∆∆QT
p Relation between Cp and Cv : Cp − Cv = R
Ratio of specific heats: γ = Cp/Cv
Relation between U and Cv : ∆U = nCv ∆T
Specific heat of gas mixture:
Cv =
n 1 Cv 1 + n 2 Cv 2 n 1 + n 2
, γ =
n 1 Cp 1 + n 2 Cp 2 n 1 Cv 1 + n 2 Cv 2
Molar internal energy of an ideal gas: U = f 2 RT , f = 3 for monatomic and f = 5 for diatomic gas.
4.4: Theromodynamic Processes
First law of thermodynamics: ∆Q = ∆U + ∆W
Work done by the gas:
∆W = p∆V, W =
∫ V 2
V 1
pdV
Wisothermal = nRT ln
V 2
V 1
Wisobaric = p(V 2 − V 1 )
Wadiabatic =
p 1 V 1 − p 2 V 2 γ − 1 Wisochoric = 0
Efficiency of the heat engine:
T 1
T 2
Q 1
Q 2
W
η =
work done by the engine heat supplied to it
Q 1 − Q 2
Q 1
ηcarnot = 1 −
Q 2
Q 1
T 2
T 1
Coeff. of performance of refrigerator:
T 1
T 2
Q 1
Q 2
W
COP = Q W^2 = Q 1 Q−^2 Q 2
Entropy: ∆S = ∆TQ , Sf − Si =
∫ (^) f i
∆Q T
Const. T : ∆S = QT , Varying T : ∆S = ms ln Tf Ti
Adiabatic process: ∆Q = 0, pV γ^ = constant
4.5: Heat Transfer
Conduction: ∆ ∆Qt = −KA ∆xT
Thermal resistance: R = (^) KAx
Rseries = R 1 + R 2 = (^) A^1
x 1 K 1 +^
x 2 K 2
x 1
A x 2
K 1 K 2
1 Rparallel =^
1 R 1 +^
1 R 2 =^
1 x (K^1 A^1 +^ K^2 A^2 )^ K^1
K 2
x
A 1
A 2
Kirchhoff ’s Law: (^) absorptive poweremissive power = E abodybody = Eblackbody
Wien’s displacement law: λmT = b λ
Eλ
λm
Stefan-Boltzmann law: ∆ ∆Qt = σeAT 4
Newton’s law of cooling: d dTt = −bA(T − T 0 )
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5 Electricity and Magnetism
5.1: Electrostatics
Coulomb’s law: F~ = (^4) π^10 q^1 rq 22 rˆ (^) q 1 r q 2
Electric field: E~(~r) = (^4) π^10 r^ q 2 ˆr ~r q E~
Electrostatic energy: U = − (^4) π^10 q^1 rq^2
Electrostatic potential: V = (^4) π^10 qr
dV = − E~ · ~r, V (~r) = −
∫ (^) ~r
∞
E^ ~ · d~r
Electric dipole moment: ~p = q d~ (^) −q ~p +q d
Potential of a dipole: V = (^4) π^10 p^ cosr 2 θ ~p
r
V (r) θ
Field of a dipole: ~p
r
Er
Eθ
θ
Er = (^4) π^102 p^ rcos 3 θ, Eθ = (^4) π^10 p^ sinr 3 θ
Torque on a dipole placed in E~: ~τ = ~p × E~
Pot. energy of a dipole placed in E~: U = −~p · E~
5.2: Gauss’s Law and its Applications
Electric flux: φ =
E~ · dS~
Gauss’s law:
E · dS~ = qin/ 0
Field of a uniformly charged ring on its axis:
EP = (^4) π^10 (a (^2) +^ qxx (^2) ) 3 / 2 a q x (^) P E~
E and V of a uniformly charged sphere:
E =
1 4 π 0
Qr R^3 ,^ for^ r < R 1 4 π 0
Q r^2 ,^ for^ r^ ≥^ R^ O (^) R r
E
V =
Q 8 π 0 R
3 − r 2 R^2
, for r < R 1 4 π 0
Q r ,^ for^ r^ ≥^ R^ O (^) R r
V
E and V of a uniformly charged spherical shell:
E =
0 , for r < R 1 4 π 0
Q r^2 ,^ for^ r^ ≥^ R^ O (^) R r
E
V =
1 4 π 0
Q R ,^ for^ r < R 1 4 π 0
Q r ,^ for^ r^ ≥^ R^ O R
r
V
Field of a line charge: E = (^2) πλ 0 r
Field of an infinite sheet: E = 2 σ 0
Field in the vicinity of conducting surface: E = (^) σ 0
5.3: Capacitors
Capacitance: C = q/V
Parallel plate capacitor: C = 0 A/d A
−q A
+q
d
Spherical capacitor: C = 4 π r 20 −rr^11 r 2 r 1
r 2
−q +q
Cylindrical capacitor: C = (^) ln(^2 rπ 2 /r^0 l 1 ) l r 1
r 2
Capacitors in parallel: Ceq = C 1 + C 2 C 2 A B
C 1
Capacitors in series: (^) C^1 eq = (^) C^11 + (^) C^12 C^1 C^2 A B
Force between plates of a parallel plate capacitor: F = Q
2 2 A 0
Energy stored in capacitor: U = 12 CV 2 = Q
2 2 C =^
1 2 QV Energy density in electric field E: U/V = 12 0 E^2
Capacitor with dielectric: C = ^0 KAd
5.4: Current electricity
Current density: j = i/A = σE
Drift speed: vd = 12 eEm τ = (^) neAi
Resistance of a wire: R = ρl/A, where ρ = 1/σ
Temp. dependence of resistance: R = R 0 (1 + α∆T )
Ohm’s law: V = iR
Kirchhoff ’s Laws: (i) The Junction Law: The algebraic sum of all the currents directed towards a node is zero i.e., Σnode Ii = 0. (ii)The Loop Law: The algebraic sum of all the potential differences along a closed loop in a circuit is zero i.e., Σloop∆ Vi = 0.
Resistors in parallel: (^) R^1 eq = (^) R^11 + (^) R^12 R 2 A B
R 1
Resistors in series: Req = R 1 + R 2 R 1 R 2 A B
Wheatstone bridge:
R 1 R 2
R 3 R 4 V
↑ (^) G
Balanced if R 1 /R 2 = R 3 /R 4.
Electric Power: P = V 2 /R = I^2 R = IV
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5.7: Electromagnetic Induction
Magnetic flux: φ =
B · dS~
Faraday’s law: e = − d dφt
Lenz’s Law: Induced current create a B-field that op- poses the change in magnetic flux.
Motional emf: e = Blv
−
l ~v ⊗^ B~
Self inductance: φ = Li, e = −L d dit
Self inductance of a solenoid: L = μ 0 n^2 (πr^2 l)
Growth of current in LR circuit: i = (^) Re
[
1 − e−^ L/Rt^ ]
e
L R
S^ i^ t
i
LR
- (^63) Re
Decay of current in LR circuit: i = i 0 e−^ L/Rt L R
S^ i^ t
i i 0
L R
- 37 i 0
Time constant of LR circuit: τ = L/R
Energy stored in an inductor: U = 12 Li^2
Energy density of B field: u = UV = B
2 2 μ 0
Mutual inductance: φ = M i, e = −M d dit
EMF induced in a rotating coil: e = N ABω sin ωt
Alternating current: (^) t
i T i = i 0 sin(ωt + φ), T = 2π/ω
Average current in AC: ¯i = (^) T^1
∫ T
0 i^ dt^ = 0
RMS current: irms =
[
1 T
∫ T
0 i
(^2) dt
] 1 / 2
= √i^02 t
i^2
T
Energy: E = irms^2 RT
Capacitive reactance: Xc = (^) ωC^1
Inductive reactance: XL = ωL
Imepedance: Z = e 0 /i 0
RC circuit: (^) i
C (^) R
e 0 sin˜ ωt R
ωC^1 Z φ
Z =
R^2 + (1/ωC)^2 , tan φ = (^) ωCR^1
LR circuit: i
L (^) R
e 0 sin˜ ωt
R ωL Z
φ
Z =
R^2 + ω^2 L^2 , tan φ = ωLR
LCR Circuit: (^) i
L C^ R
e 0 sin˜ ωt R
ωC^1
ωL
Z (^1) φ^ ωC^ −^ ωL
Z =
R^2 +
ωC −^ ωL
, tan φ = ωC^1 −ωL R νresonance = (^21) π
1 LC
Power factor: P = ermsirms cos φ
Transformer: N N^12 = e e^12 , e 1 i 1 = e 2 i 2 i 1
N 1 i 2
e 1 N 2
e 2
Speed of the EM waves in vacuum: c = 1/
μ 0 0
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6 Modern Physics
6.1: Photo-electric effect
Photon’s energy: E = hν = hc/λ
Photon’s momentum: p = h/λ = E/c
Max. KE of ejected photo-electron: Kmax = hν − φ
Threshold freq. in photo-electric effect: ν 0 = φ/h
Stopping potential: Vo = hce
λ
− φe (^1) λ
V 0
φ hc
hce
− φe
de Broglie wavelength: λ = h/p
6.2: The Atom
Energy in nth Bohr’s orbit:
En = −
mZ^2 e^4 8 02 h^2 n^2
, En = −
13. 6 Z^2
n^2
eV
Radius of the nth Bohr’s orbit:
rn =
0 h^2 n^2 πmZe^2
, rn =
n^2 a 0 Z
, a 0 = 0.529 ˚A
Quantization of the angular momentum: l = nh 2 π
Photon energy in state transition: E 2 − E 1 = hν
E 1
E 2 hν Emission E 1
E 2 hν Absorption
Wavelength of emitted radiation: for a transition from nth to mth state: 1 λ
= RZ^2
[
n^2
m^2
]
X-ray spectrum: λmin = (^) eVhc
λ
I
λmin λα
Kα Kβ
Moseley’s law:
ν = a(Z − b)
X-ray diffraction: 2 d sin θ = nλ
Heisenberg uncertainity principle: ∆p∆x ≥ h/(2π), ∆E∆t ≥ h/(2π)
6.3: The Nucleus
Nuclear radius: R = R 0 A^1 /^3 , R 0 ≈ 1. 1 × 10 −^15 m
Decay rate: d dNt = −λN
Population at time t: N = N 0 e−λt
O t
N 0
N
N 0 2 t 1 / 2
Half life: t 1 / 2 = 0. 693 /λ
Average life: tav = 1/λ
Population after n half lives: N = N 0 / 2 n.
Mass defect: ∆m = [Zmp + (A − Z)mn] − M
Binding energy: B = [Zmp + (A − Z)mn − M ] c^2
Q-value: Q = Ui − Uf
Energy released in nuclear reaction: ∆E = ∆mc^2 where ∆m = mreactants − mproducts.
6.4: Vacuum tubes and Semiconductors
Half Wave Rectifier:
D
R Output
Full Wave Rectifier:
˜ Output
Triode Valve: Filament Plate
Grid Cathode
Plate resistance of a triode: rp = ∆ ∆Vipp
∆Vg =
Transconductance of a triode: gm = (^) ∆∆Vipg
∆Vp=
Amplification by a triode: μ = − ∆ ∆VVpg
∆ip=
Relation between rp, μ, and gm: μ = rp × gm
Current in a transistor: Ie = Ib + Ic
Ic
Ib
Ie
α and β parameters of a transistor: α = I Ice , β = Ic Ib , β^ =^
α 1 −α
Transconductance: gm = (^) ∆∆VIbec
Logic Gates: AND OR NAND NOR XOR A B AB A+B AB A + B A B + ¯ AB¯ 0 0 0 0 1 1 0 0 1 0 1 1 0 1 1 0 0 1 1 0 1 1 1 1 1 0 0 0
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