Oy EYx NO6 UBFnb 0 VUFB5 e
Oy EYx NO6 UBFnb 0 VUFB5 e
Oy EYx NO6 UBFnb 0 VUFB5 e
u sin θ
x
H
Speed of light c 3 × 108 m/s Projectile Motion:
θ
Planck constant h 6.63 × 10−34 J s O u cos θ
hc 1242 eV-nm
R
Gravitation constant G 6.67 × 10−11 m3 kg−1 s−2
Boltzmann constant k 1.38 × 10−23 J/K
x = ut cos θ, y = ut sin θ − 12 gt2
Molar gas constant R 8.314 J/(mol K)
6.023 × 1023 mol−1 g
Avogadro’s number NA y = x tan θ − 2 x2
Charge of electron e 1.602 × 10−19 C 2u cos2 θ
Permeability of vac- µ0 4π × 10−7 N/A2 2u sin θ u2 sin 2θ u2 sin2 θ
uum T = , R= , H=
g g 2g
Permitivity of vacuum 0 8.85 × 10−12 F/m
Coulomb constant 1
4π0
9 × 109 N m2 /C2
Faraday constant F 96485 C/mol 1.3: Newton’s Laws and Friction
Mass of electron me 9.1 × 10−31 kg
Mass of proton mp 1.6726 × 10−27 kg Linear momentum: p~ = m~v
Mass of neutron mn 1.6749 × 10−27 kg
Atomic mass unit u 1.66 × 10−27 kg Newton’s first law: inertial frame.
Atomic mass unit u 931.49 MeV/c2
Stefan-Boltzmann σ 5.67 × 10−8 W/(m2 K4 ) Newton’s second law: F~ = d~
p
dt , F~ = m~a
constant
Rydberg constant R∞ 1.097 × 107 m−1 Newton’s third law: F~AB = −F~BA
Bohr magneton µB 9.27 × 10−24 J/T
Bohr radius a0 0.529 × 10−10 m Frictional force: fstatic, max = µs N, fkinetic = µk N
Standard atmosphere atm 1.01325 × 105 Pa
2.9 × 10−3 m K v2 v2 µ+tan θ
Wien displacement b Banking angle: rg = tan θ, rg = 1−µ tan θ
constant
mv 2 v2
Centripetal force: Fc = r , ac = r
2
ı̂ mg
a × ~b
~ ~b
Cross product:
θ k̂ ̂
~
a
1.4: Work, Power and Energy
~a ×~b = (ay bz − az by )ı̂ + (az bx − ax bz )̂ + (ax by − ay bx )k̂
Work: W = F~ · S
~ = F S cos θ, F~ · dS
~
R
W =
|~a × ~b| = ab sin θ
p2
Kinetic energy: K = 21 mv 2 = 2m
Average and Instantaneous Vel. and Accel.: Ugravitational = mgh, Uspring = 21 kx2
~vav = ∆~r/∆t, ~vinst = d~r/dt Work done by conservative forces is path indepen-
~aav = ∆~v /∆t ~ainst = d~v /dt dent and depends only on initial and final points:
F~conservative · d~r = 0.
H
Work-energy theorem: W = ∆K
Motion in a straight line with constant a:
Mechanical energy: E = U + K. Conserved if forces are
v = u + at, s = ut + 21 at2 , v 2 − u2 = 2as
conservative in nature.
h z y
2. Triangle (CM ≡ Centroid) yc = 3 h Theorem of Perp. Axes: Iz = Ix + Iy
C x
h
3
2r p
3. Semicircular ring: yc = π
C
2r Radius of Gyration: k = I/m
r π
~ = ~r × p~,
Angular Momentum: L ~ = I~
L ω
4r
4. Semicircular disc: yc = 3π C 4r
r 3π y
~ P θ ~
Torque: ~τ = ~r × F~ , ~τ = dL
dt , τ = Iα F
r ~
r x
5. Hemispherical shell: yc = 2 C r O
r 2
~ ~τext = 0 =⇒ L
Conservation of L: ~ = const.
3r
6. Solid Hemisphere: yc = 8 C 3r
r 8 P~
F = ~0, ~τ = ~0
P
Equilibrium condition:
7. Cone: the height of CM from the base is h/4 for
Kinetic Energy: Krot = 21 Iω 2
the solid cone and h/3 for the hollow cone.
P Dynamics:
Motion of the CM: M = mi
~τcm = Icm α
~, F~ext = m~acm , p~cm = m~vcm
F~ext
P
mi~vi
~vcm = , p~cm = M~vcm , ~acm = 1 2 1 2
K = 2 mvcm + 2 Icm ω , L ~ = Icm ω
~ + ~rcm × m~vcm
M M
Impulse: J~ = F~ dt = ∆~
R
p
1.7: Gravitation
Before collision After collision
Collision: m1 F F m2
m1 m2 m1 m2 Gravitational force: F = G mr1 m
2
2
v1 v2 v10 v20 r
Momentum conservation: m1 v1 +m2 v2 = m1 v10 +m2 v20
2
Elastic Collision: 21 m1 v1 2+ 21 m2 v2 2 = 12 m1 v10 + 12 m2 v20
2 Potential energy: U = − GMr m
Coefficient of restitution: GM
Gravitational acceleration: g = R2
−(v10 − v20 )
1, completely elastic
e= = 2h
v1 − v2 0, completely in-elastic Variation of g with depth: ginside ≈ g 1 − R
h
Variation of g with height: goutside ≈ g 1 − R
If v2 = 0 and m1 m2 then = −v1 . v10
If v2 = 0 and m1 m2 then = 2v1 . v20 Effect of non-spherical earth shape on g:
Elastic collision with m1 = m2 : v10 = v2 and v20 = v1 . gat pole > gat equator (∵ Re − Rp ≈ 21 km)
∆θ dθ mω 2 R cos θ
Angular velocity: ωav = ∆t , ω= dt , ~ × ~r
~v = ω mg
mgθ0 = mg − mω 2 R cos2 θ
∆ω dω θ
Angular Accel.: αav = ∆t , α= dt , ~ × ~r
~a = α R
mi ri 2 , r2 dm
P R
Moment of Inertia: I = i I= q
2GM
Escape velocity: ve = R
vo
1.9: Properties of Matter
Kepler’s laws:
a F/A ∆P F
Modulus of rigidity: Y = ∆l/l , B = −V ∆V , η= Aθ
First: Elliptical orbit with sun at one of the focus.
~
Second: Areal velocity is constant. (∵ dL/dt = 0). Compressibility: K = 1
= − V1 dV
B dP
2 3 2 4π 2 3
Third: T ∝ a . In circular orbit T = GM a . ∆D/D
lateral strain
Poisson’s ratio: σ = longitudinal strain = ∆l/l
1
1.8: Simple Harmonic Motion Elastic energy: U = 2 stress × strain × volume
Total energy: E = U + K = 21 mω 2 A2
Bernoulli’s equation: p + 12 ρv 2 + ρgh = constant
√
Torricelli’s theorem: vefflux = 2gh
q
l dv
Simple pendulum: T = 2π g l Viscous force: F = −ηA dx
F
q
I
Stoke’s law: F = 6πηrv
Physical Pendulum: T = 2π mgl
v
1 1 1
Springs in series: keq = k1 + k2
k1 k2
~
A
~2
A
Superposition of two SHM’s: δ
~1
A
1 2π 2π
T = = , v = νλ, k= String fixed at one end: N A
ν ω λ A N
λ/2
Progressive wave travelling with speed v:
1. Boundary conditions: y = 0 at x = 0
y = f (t − x/v), +x; y = f (t + x/v), −x q
2. Allowed Freq.: L = (2n + 1) λ4 , ν = 2n+1
4L
T
µ, n =
y 0, 1, 2, . . ..
A q
x 1 T
λ λ 3. Fundamental/1st harmonics: ν0 = 4L µ
Progressive sine wave: 2
q
3 T
4. 1st overtone/3rd harmonics: ν1 = 4L µ
y = A sin(kx − ωt) = A sin(2π (x/λ − t/T ))
q
5 T
5. 2nd overtone/5th harmonics: ν2 = 4L µ
x
Standing Waves: A N A N A
Standing longitudinal waves:
λ/4
p1 = p0 sin ω(t − x/v), p2 = p0 sin ω(t + x/v)
y1 = A1 sin(kx − ωt), y2 = A2 sin(kx + ωt) p = p1 + p2 = 2p0 cos kx sin ωt
y = y1 + y2 = (2A cos kx) sin ωt
n + 21 λ2 , nodes; n = 0, 1, 2, . . .
x=
n λ2 , antinodes. n = 0, 1, 2, . . .
L
Closed organ pipe:
L
2π
Phase difference: δ = λ ∆x
A
1. Boundary condition: y = 0 at x = 0
nλ, constructive;
Allowed freq.: L = n λ2 , ν = n 4L
v
, n = 1, 2, . . . ∆x =
n + 21 λ, destructive
v
2. Fundamental/1st harmonics: ν0 = 2L
2v Intensity:
3. 1st overtone/2nd harmonics: ν1 = 2ν0 = 2L
p
4. 2nd overtone/3rd harmonics: ν2 = 3ν0 = 3v I = I1 + I2 + 2 I1 I2 cos δ,
2L p p 2 p p 2
5. All harmonics are present. Imax = I1 + I2 , Imin = I1 − I2
I1 = I2 : I = 4I0 cos2 2δ , Imax = 4I0 , Imin = 0
l1 + d
λD
Fringe width: w = d
l2 + d
aE0 I0
Spherical Wave: E = r sin ω(t − vr ), I = r2
I
Spherical Mirror: O
f 3.3: Optical Instruments
v
u
Simple microscope: m = D/f in normal adjustment.
1. Focal length f = R/2 Objective Eyepiece
1 1 1
2. Mirror equation: v + u = f
O ∞
3. Magnification: m = − uv Compound microscope:
u v fe
Astronomical telescope:
real depth d d0
Apparent depth: µ = apparent depth = d0 d I
O
A 3.4: Dispersion
δ
Deviation by a prism: i0 Cauchy’s equation: µ = µ0 + A
λ2 , A>0
i r r0
µ2 µ1 µ2 − µ1 µ1 v
− = , m=
v u R µ2 u
4 Heat and Thermodynamics 4.4: Theromodynamic Processes
x
Ratio of specific heats: γ = Cp /Cv
emissive power Ebody
Relation between U and Cv : ∆U = nCv ∆T Kirchhoff ’s Law: absorptive power = abody = Eblackbody
∆Q
Molar internal energy of an ideal gas: U = f2 RT , Stefan-Boltzmann law: ∆t = σeAT 4
f = 3 for monatomic and f = 5 for diatomic gas. dT
Newton’s law of cooling: dt = −bA(T − T0 )
5 Electricity and Magnetism 5.3: Capacitors
−q +q
Coulomb’s law: F~ = 1 q1 q2
4π0 r 2 r̂ q1 r q2 Parallel plate capacitor: C = 0 A/d
A A
d
~ r) =
Electric field: E(~ 1 q
4π0 r 2 r̂
~
E
q ~
r
r2
1 q1 q2
Electrostatic energy: U = − 4π 0 r Spherical capacitor: C = 4π0 r1 r2
−q +q
r2 −r1
r1
1 q
Electrostatic potential: V = 4π0 r
Z ~
r
~ · ~r,
dV = −E V (~r) = − ~ · d~r
E 2π0 l r2
∞ Cylindrical capacitor: C = ln(r2 /r1 ) l
r1
p
~
Electric dipole moment: p~ = q d~ −q +q
d A
Capacitors in parallel: Ceq = C1 + C2 C1 C2
B
1 p cos θ V (r)
Potential of a dipole: V = 4π0 r 2
θ r
1 1 1
p
~ Capacitors in series: Ceq = C1 + C2
C1 C2
A B
~ ~τ = p~ × E
Torque on a dipole placed in E: ~ Energy density in electric field E: U/V = 12 0 E 2
0 KA
~ U = −~
Pot. energy of a dipole placed in E: ~
p·E Capacitor with dielectric: C = d
Field of a uniformly charged ring on its axis: Resistance of a wire: R = ρl/A, where ρ = 1/σ
1 qx a
EP = 4π0 (a2 +x2 )3/2 q ~
E Temp. dependence of resistance: R = R0 (1 + α∆T )
x P
Ohm’s law: V = iR
E and V (of a uniformly charged sphere:
1 Qr Kirchhoff ’s Laws: (i) The Junction Law: The algebraic
4π0 R3 , for r < R E
E= 1 Q sum of all the currents directed towards a node is zero
4π0 r 2 , for r ≥ R O
r
( R i.e., Σnode Ii = 0. (ii)The Loop Law: The algebraic
2
1 Qr sum of all the potential differences along a closed loop
V = 4π0 R3 , for r < R V
1 Q
for r ≥ R in a circuit is zero i.e., Σloop ∆ Vi = 0.
4π0 r ,
r
O R
1 1 1 A
E and V of a uniformly charged spherical shell: Resistors in parallel: Req = R1 + R2 R1 R2
B
0, for r < R E
E= 1 Q
4π0 r 2 , for r≥R O
r Resistors in series: Req = R1 + R2 R1 R2
R A B
(
1 Q
V = 4π0 R , for r < R V R1 R2
1 Q
4π0 r , for r ≥ R r ↑ G
O R Wheatstone bridge:
R3 R4
λ
Field of a line charge: E = 2π0 r V
σ Balanced if R1 /R2 = R3 /R4 .
Field of an infinite sheet: E = 20
σ Electric Power: P = V 2 /R = I 2 R = IV
Field in the vicinity of conducting surface: E = 0
i ig G i ~
Energy of a magnetic dipole placed in B:
Galvanometer as an Ammeter: i − ig U = −~µ·B~
S
~
ig G = (i − ig )S Bi l B
Hall effect: Vw = ned w
y
x
R G i d z
Galvanometer as a Voltmeter: ↑
A ig B
VAB = ig (R + G)
5.6: Magnetic Field due to Current
R C
i ~
⊗B
Charging of capacitors: ~ = µ0 i d~l×~
r
Biot-Savart law: dB 4π r 3 θ
~
r
V d~l
h t
i
q(t) = CV 1 − e− RC
θ2
C
t Field due to a straight conductor: i
d ~
Discharging of capacitors: q(t) = q0 e− RC q(t)
⊗B
θ1
R
µ0 i
B= 4πd (cos θ1 − cos θ2 )
Time constant in RC circuit: τ = RC
µ0 i
Field due to an infinite straight wire: B = 2πd
∆H Peltier heat
Peltier effect: emf e = ∆Q = charge transferred . dF µ0 i1 i2 i1 i2
Force between parallel wires: dl = 2πd
e d
Seeback effect: T
T0 Tn Ti
a
P
1. Thermo-emf: e = aT + 21 bT 2 Field on the axis of a ring: i ~
B
d
2. Thermoelectric power: de/dt = a + bT .
µ0 ia2
3. Neutral temp.: Tn = −a/b. BP = 2(a2 +d2 )3/2
4. Inversion temp.: Ti = −2a/b.
a
µ0 iθ
∆H Thomson heat Field at the centre of an arc: B = ~ θ i
Thomson effect: emf e = ∆Q = charge transferred = σ∆T . 4πa B
a
Faraday’s law of electrolysis: The mass deposited is
µ0 i
Field at the centre of a ring: B = 2a
1
m = Zit = F Eit
~ · d~l = µ0 Iin
H
Ampere’s law: B
where i is current, t is time, Z is electrochemical equiv-
alent, E is chemical equivalent, and F = 96485 C/g is Field inside a solenoid: B = µ0 ni, n = N
l
Faraday constant. l
µ0 N i
5.5: Magnetism Field inside a toroid: B = 2πr r
~
B Horizontal Bh
Angle of dip: Bh = B cos δ δ
Force on a current carrying wire: ~l
Bv B
~
F i
F~ = i ~l × B
~
Tangent galvanometer: Bh tan θ = µ0 ni
2r , i = K tan θ
Magnetic moment of a current loop (dipole): k
Moving coil galvanometer: niAB = kθ, i= nAB θ
µ ~
~ A
µ ~
~ = iA q
i I
Time period of magnetometer: T = 2π M Bh
~ ~τ = µ
Torque on a magnetic dipole placed in B: ~
~ ×B
~ = µH
Permeability: B ~
C R
5.7: Electromagnetic Induction 1
Z
RC circuit: i ωC
φ
~ · dS
~
H
Magnetic flux: φ = B
p ˜
e0 sin ωt
1
R
+ √ ωL
˜
e0 sin ωt
Z
Z= R2 + ω 2 L2 , tan φ = R
Motional emf: e = Blv l ~
v ⊗B
~
− L C R 1
ωC Z 1
LCR Circuit: i φ ωC − ωL
di
Self inductance: φ = Li, e= −L dt
q ˜
e0 sin ωt
ωL
1
R
i
L R Power factor: P = erms irms cos φ
e
e 0.63 R
t N1 e1 e1 N1 N2 e2
Transformer: = e2 , e1 i1 = e2 i2
S i L
R
N2
˜ i1 i2
˜
t
− L/R √
Decay of current in LR circuit: i = i0 e Speed of the EM waves in vacuum: c = 1/ µ0 0
L R i
i0
0.37i0
t
S i L
R
di
Mutual inductance: φ = M i, e = −M dt
h R i1/2 i2
1 T i0
RMS current: irms = T 0
i2 dt = √
2
t
Energy: E = irms 2 RT
1
Capacitive reactance: Xc = ωC
Inductive reactance: XL = ωL
Imepedance: Z = e0 /i0
6 Modern Physics N0
N
O t1/2 t
Photon’s energy: E = hν = hc/λ
Threshold freq. in photo-electric effect: ν0 = φ/h Population after n half lives: N = N0 /2n .
mZ 2 e4 13.6Z 2 D
En = − , En = − eV Half Wave Rectifier:
80 2 h2 n2 n2 R Output
nh
Quantization of the angular momentum: l = 2π
Grid
E2 E2
hν hν
∆Vp
E1
Emission
E1 Plate resistance of a triode: rp = ∆ip
Absorption ∆Vg =0
Wavelength of emitted radiation: for a transition Transconductance of a triode: gm =
∆ip
∆Vg
from nth to mth state: ∆Vp =0
1 2 1 1 ∆V
Amplification by a triode: µ = − ∆Vgp
= RZ − 2 ∆ip =0
λ n2 m
Relation between rp , µ, and gm : µ = rp × gm
I Kα
Kβ
hc
X-ray spectrum: λmin = eV
Ie Ic
Current in a transistor: Ie = Ib + Ic
λmin λα λ
Ib
√
Moseley’s law: ν = a(Z − b)
Ic
α and β parameters of a transistor: α = Ie , β =
X-ray diffraction: 2d sin θ = nλ Ic α
Ib , β = 1−α