PHYSICS FORMULA LIST
u
Speed of light
Planck constant
Gravitation constant
Boltzmann constant
Molar gas constant
Avogadro’s number
Charge of electron
Permeability of vacuum
Permitivity of vacuum
Coulomb constant
Faraday constant
Mass of electron
Mass of proton
Mass of neutron
Atomic mass unit
Atomic mass unit
Stefan-Boltzmann
constant
Rydberg constant
Bohr magneton
Bohr radius
Standard atmosphere
Wien
displacement
constant
c
h
hc
G
k
R
NA
e
µ0
3 × 108 m/s
6.63 × 10−34 J s
1242 eV-nm
6.67 × 10−11 m3 kg−1 s−2
1.38 × 10−23 J/K
8.314 J/(mol K)
6.023 × 1023 mol−1
1.602 × 10−19 C
4π × 10−7 N/A2
0
F
me
mp
mn
u
u
σ
8.85 × 10−12 F/m
9 × 109 N m2 /C2
96485 C/mol
9.1 × 10−31 kg
1.6726 × 10−27 kg
1.6749 × 10−27 kg
1.66 × 10−27 kg
931.49 MeV/c2
5.67 × 10−8 W/(m2 K4 )
R∞
µB
a0
atm
b
1.097 × 107 m−1
9.27 × 10−24 J/T
0.529 × 10−10 m
1.01325 × 105 Pa
2.9 × 10−3 m K
1
4π0
u sin θ
0.1: Physical Constants
Projectile Motion:
x
H
θ
u cos θ
O
R
y = ut sin θ − 21 gt2
g
y = x tan θ − 2
x2
2u cos2 θ
u2 sin 2θ
u2 sin2 θ
2u sin θ
, R=
, H=
T =
g
g
2g
x = ut cos θ,
1.3: Newton’s Laws and Friction
Linear momentum: p~ = m~v
Newton’s first law: inertial frame.
p
Newton’s second law: F~ = d~
dt ,
F~ = m~a
Newton’s third law: F~AB = −F~BA
Frictional force: fstatic, max = µs N,
fkinetic = µk N
2
2
µ+tan θ
Banking angle: vrg = tan θ, vrg = 1−µ
tan θ
--------------------------------------------------
Centripetal force: Fc = mv
r ,
MECHANICS
Pseudo force: F~pseudo = −m~a0 ,
2
2
ac = vr
Fcentrifugal = − mv
r
2
Minimum speed to complete vertical circle:
p
p
vmin, bottom = 5gl, vmin, top = gl
1.1: Vectors
Notation: ~a = ax ı̂ + ay ̂ + az k̂
q
Magnitude: a = |~a| = a2x + a2y + a2z
l
Conical pendulum: T = 2π
Dot product: ~a · ~b = ax bx + ay by + az bz = ab cos θ
Cross product:
y
~
a × ~b
l cos θ
g
~
a
k̂
θ
θ T
ı̂
~b
θ
q
mg
̂
1.4: Work, Power and Energy
~a ×~b = (ay bz − az by )ı̂ + (az bx − ax bz )̂ + (ax by − ay bx )k̂
|~a × ~b| = ab sin θ
R
~
F~ · dS
2
Potential energy: F = −∂U/∂x for conservative forces.
Average and Instantaneous Vel. and Accel.:
~vinst = d~r/dt
~ainst = d~v /dt
Motion in a straight line with constant a:
v = u + at,
W =
p
Kinetic energy: K = 12 mv 2 = 2m
1.2: Kinematics
~vav = ∆~r/∆t,
~aav = ∆~v /∆t
~ = F S cos θ,
Work: W = F~ · S
s = ut + 12 at2 ,
Relative Velocity: ~vA/B = ~vA − ~vB
v 2 − u2 = 2as
Ugravitational = mgh,
Uspring = 21 kx2
Work done by conservative forces is path independent
and depends only on initial and final points:
H
F~conservative · d~r = 0.
Work-energy theorem: W = ∆K
Mechanical energy: E = U + K. Conserved if forces are
conservative in nature.
Power Pav = ∆W
∆t ,
Pinst = F~ · ~v
1.5: Centre of Mass and Collision
Centre of mass: xcm =
mr 2
R
P
Pxi mi ,
mi
2
1
2 mr
2
2
3 mr
ring
disk
shell
C
m2 r
m1 +m2
C
Ik
Theorem of Parallel Axes: Ik = Icm + md
C
2r
π
Theorem of Perp. Axes: Iz = Ix + Iy
Radius of Gyration: k =
r
Hemispherical shell: yc = 2r
r
6. Solid Hemisphere: yc = 3r
8
r
x
~ = I~
L
ω
4r
3π
C
r
2
C
3r
8
~
~τ = ddtL ,
y
P θ ~
F
~
r x
τ = Iα
O
7. Cone: the height of CM from the base is h/4 for
the solid cone and h/3 for the hollow cone.
P
Motion of the CM: M = mi
y
p
I/m
~ = ~r × p~,
Angular Momentum: L
C
Ic
d
cm
Torque: ~τ = ~r × F~ ,
R
solid rectangle
h
3
r
p~cm = M~vcm ,
hollow
z
h
4r
4. Semicircular disc: yc = 3π
Impulse: J~ =
rod
2
m1 r
m1 +m2
2. Triangle (CM ≡ Centroid) yc = h3
~vcm =
sphere
m2
r
3. Semicircular ring: yc = 2r
π
b
a
1. m1 , m2 separated by r:
mi~vi
,
M
2
1
2
2
2 mr m(a +b )
12
mr 2
xdm
m1
P
2
1
12 ml
xcm = R dm
CM of few useful configurations:
5.
2
2
5 mr
~ ~τext = 0 =⇒ L
~ = const.
Conservation of L:
P~
P
Equilibrium condition:
F = ~0,
~τ = ~0
Kinetic Energy: Krot = 12 Iω 2
Dynamics:
F~ext = m~acm ,
p~cm = m~vcm
2
2
1
1
~ = Icm ω
K = mvcm + Icm ω , L
~ + ~rcm × m~vcm
~τcm = Icm α
~,
F~ext
~acm =
M
2
F~ dt = ∆~
p
2
1.7: Gravitation
Before collision After collision
Collision:
m1
m2
v1
m1
m2
v10
v2
m1
2
Gravitational force: F = G mr1 m
2
F F
v20
Momentum conservation: m1 v1 +m2 v2 = m1 v10 +m2 v20
2
2
Elastic Collision: 12 m1 v1 2+ 12 m2 v2 2 = 12 m1 v10 + 12 m2 v20
Coefficient of restitution:
−(v10 − v20 )
1, completely elastic
e=
=
0, completely in-elastic
v1 − v2
If v2 = 0 and m1 m2 then v10 = −v1 .
If v2 = 0 and m1 m2 then v20 = 2v1 .
Elastic collision with m1 = m2 : v10 = v2 and v20 = v1 .
m2
r
Potential energy: U = − GMr m
Gravitational acceleration: g = GM
R2
Variation of g with depth: ginside ≈ g 1 − 2h
R
h
Variation of g with height: goutside ≈ g 1 − R
Effect of non-spherical earth shape on g:
gat pole > gat equator (∵ Re − Rp ≈ 21 km)
Effect of earth rotation on apparent weight:
ω
~
1.6: Rigid Body Dynamics
Angular velocity: ωav = ∆θ
∆t ,
ω = dθ
dt ,
~v = ω
~ × ~r
Angular Accel.: αav = ∆ω
∆t ,
α = dω
dt ,
~a = α
~ × ~r
mg
mgθ0 = mg − mω 2 R cos2 θ
θ
R
Rotation about an axis with constant α:
ω = ω0 + αt,
θ = ωt + 21 αt2 ,
Moment of Inertia: I =
P
i mi ri
2
,
ω 2 − ω0 2 = 2αθ
I=
R
Orbital velocity of satellite: vo =
r2 dm
Escape velocity: ve =
q
2GM
R
q
GM
R
mω 2 R cos θ
vo
Kepler’s laws:
a
1.9: Properties of Matter
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).
4π 2 3
2
3
2
Third: T ∝ a . In circular orbit T = GM a .
Compressibility: K = B1 = − V1 dV
dP
∆D/D
lateral strain
Poisson’s ratio: σ = longitudinal
strain = ∆l/l
Elastic energy: U = 12 stress × strain × volume
1.8: Simple Harmonic Motion
Hooke’s law: F = −kx (for small elongation x.)
Surface tension: S = F/l
2
k
x = −ω 2 x
Acceleration: a = ddt2x = − m
pm
Time period: T = 2π
ω = 2π
k
Surface energy: U = SA
Excess pressure in bubble:
Displacement: x = A sin(ωt + φ)
∆pair = 2S/R,
√
Velocity: v = Aω cos(ωt + φ) = ±ω A2 − x2
∆psoap = 4S/R
cos θ
Capillary rise: h = 2Srρg
U
Potential energy: U = 12 kx2
−A
Kinetic energy K = 12 mv 2
0
x
A
Buoyant force: FB = ρV g = Weight of displaced liquid
K
−A
0
x
A
Total energy: E = U + K = 12 mω 2 A2
Simple pendulum: T = 2π
q
Hydrostatic pressure: p = ρgh
Equation of continuity: A1 v1 = A2 v2
v2
v1
Bernoulli’s equation: p + 21 ρv 2 + ρgh = constant
√
Torricelli’s theorem: vefflux = 2gh
l
g
l
dv
Viscous force: F = −ηA dx
F
Physical Pendulum: T = 2π
q
Stoke’s law: F = 6πηrv
I
mgl
v
r
4
Torsional Pendulum T = 2π
q
flow
Poiseuilli’s equation: Volume
= πpr
time
8ηl
I
k
2
Terminal velocity: vt = 2r (ρ−σ)g
9η
Springs in series: k1eq = k11 + k12
k1
k2
k2
k1
Springs in parallel: keq = k1 + k2
~
A
Superposition of two SHM’s:
~1
A
x1 = A1 sin ωt,
x2 = A2 sin(ωt + δ)
x = x1 + x2 = A sin(ωt + )
q
A = A1 2 + A2 2 + 2A1 A2 cos δ
tan =
A2 sin δ
A1 + A2 cos δ
~2
A
δ
l
Waves
2
4. 1st overtone/2nd harmonics: ν1 = 2L
2.1: Waves Motion
q
q
3
5. 2nd overtone/3rd harmonics: ν2 = 2L
2
2
∂ y
1 ∂ y
General equation of wave: ∂x
2 = v 2 ∂t2 .
T
µ
6. All harmonics are present.
Notation: Amplitude A, Frequency ν, Wavelength λ, Period T , Angular Frequency ω, Wave Number k,
1
2π
T = =
,
ν
ω
2π
k=
λ
v = νλ,
L
String fixed at one end:
N
A
+x;
1. Boundary conditions: y = 0 at x = 0
−x
y = f (t + x/v),
y
A
x
λ
λ
2
Progressive sine wave:
y = A sin(kx − ωt) = A sin(2π (x/λ − t/T ))
2.2: Waves on a String
2. Allowed Freq.: L = (2n + 1) λ4 , ν = 2n+1
4L
0, 1, 2, . . ..
q
T
1
3. Fundamental/1st harmonics: ν0 = 4L
µ
q
3
T
4. 1st overtone/3rd harmonics: ν1 = 4L
µ
q
5
T
5. 2nd overtone/5th harmonics: ν2 = 4L
µ
q
T
µ,
n =
6. Only odd harmonics are present.
Speed of waves on a string
with mass per unit length µ
p
and tension T : v = T /µ
Sonometer: ν ∝ L1 , ν ∝
Transmitted power: Pav = 2π 2 µvA2 ν 2
Interference:
y1 = A1 sin(kx − ωt),
A
N
λ/2
Progressive wave travelling with speed v:
y = f (t − x/v),
T
µ
√
n
T , ν ∝ √1µ . ν = 2L
q
T
µ
2.3: Sound Waves
y2 = A2 sin(kx − ωt + δ)
y = y1 + y2 = A sin(kx − ωt + )
q
A = A1 2 + A2 2 + 2A1 A2 cos δ
Displacement wave: s = s0 sin ω(t − x/v)
Pressure wave: p = p0 cos ω(t − x/v), p0 = (Bω/v)s0
Speed of sound waves:
s
s
B
Y
vliquid =
, vsolid =
,
ρ
ρ
A2 sin δ
tan =
A1 + A2 cos δ
2nπ,
constructive;
δ=
(2n + 1)π, destructive.
2
2
s
vgas =
γP
ρ
2
2A cos kx
p0
0 v
Intensity: I = 2πv B s0 2 ν 2 = p2B
= 2ρv
Standing Waves:
x
A
N
A
N
A
Standing longitudinal waves:
λ/4
y1 = A1 sin(kx − ωt),
p1 = p0 sin ω(t − x/v), p2 = p0 sin ω(t + x/v)
p = p1 + p2 = 2p0 cos kx sin ωt
y2 = A2 sin(kx + 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, . . .
Closed organ pipe:
L
L
String fixed at both ends:
N
A
N
A
λ/2
1. Boundary conditions: y = 0 at x = 0 and at x = L
q
n
T
2. Allowed Freq.: L = n λ2 , ν = 2L
µ , n = 1, 2, 3, . . ..
q
1
T
3. Fundamental/1st harmonics: ν0 = 2L
µ
N
1. Boundary condition: y = 0 at x = 0
v
2. Allowed freq.: L = (2n + 1) λ4 , ν = (2n + 1) 4L
, n=
0, 1, 2, . . .
v
3. Fundamental/1st harmonics: ν0 = 4L
3v
4. 1st overtone/3rd harmonics: ν1 = 3ν0 = 4L
5v
5. 2nd overtone/5th harmonics: ν2 = 5ν0 = 4L
6. Only odd harmonics are present.
θ
d
S2
A
N
Open organ pipe:
P
y
S1
Path difference: ∆x = dy
D
L
A
N
D
Phase difference: δ = 2π
λ ∆x
Interference Conditions: for integer n,
2nπ,
constructive;
δ=
(2n + 1)π, destructive,
A
1. Boundary condition: y = 0 at x = 0
v
, n = 1, 2, . . .
Allowed freq.: L = n λ2 , ν = n 4L
∆x =
v
2. Fundamental/1st harmonics: ν0 = 2L
nλ, constructive;
n + 12 λ, destructive
Intensity:
2v
3. 1st overtone/2nd harmonics: ν1 = 2ν0 = 2L
p
I = I1 + I2 + 2 I1 I2 cos δ,
p
p
p 2
p 2
I1 + I2 , Imin =
I1 − I2
Imax =
3v
4. 2nd overtone/3rd harmonics: ν2 = 3ν0 = 2L
5. All harmonics are present.
Resonance column:
l1 + d
l2 + d
I1 = I2 : I = 4I0 cos2 2δ , Imax = 4I0 , Imin = 0
Fringe width: w = λD
d
Optical path: ∆x0 = µ∆x
Interference of waves transmitted through thin film:
l1 + d = λ2 ,
l2 + d = 3λ
4 ,
v = 2(l2 − l1 )ν
Beats: two waves of almost equal frequencies ω1 ≈ ω2
p1 = p0 sin ω1 (t − x/v), p2 = p0 sin ω2 (t − x/v)
p = p1 + p2 = 2p0 cos ∆ω(t − x/v) sin ω(t − x/v)
ω = (ω1 + ω2 )/2, ∆ω = ω1 − ω2 (beats freq.)
Doppler Effect:
v + uo
ν0
ν=
v − us
where, v is the speed of sound in the medium, u0 is
the speed of the observer w.r.t. the medium, considered 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, considered positive when it moves towards the observer and
negative when it moves away from the observer.
2.4: Light Waves
Plane Wave: E = E0 sin ω(t − xv ), I = I0
I0
r
0
Spherical Wave: E = aE
r sin ω(t − v ), I = r 2
Young’s double slit experiment
∆x = 2µd =
nλ, constructive;
n + 12 λ, destructive.
Diffraction from a single slit:
y
θ
b
y
D
For Minima: nλ = b sin θ ≈ b(y/D)
Resolution: sin θ = 1.22λ
b
Law of Malus: I = I0 cos2 θ
θ
I0
I
Optics
h
Lens maker’s formula: f1 = (µ − 1)
1
1
R1 − R2
i
3.1: Reflection of Light
f
normal
Laws of reflection:
i r
incident
(i)
reflected
Lens formula: v1 − u1 = f1 ,
m = uv
u
Incident ray, reflected ray, and normal lie in the same
plane (ii) ∠i = ∠r
v
Power of the lens: P = f1 , P in diopter if f in metre.
Two thin lenses separated by distance d:
Plane mirror:
d
d
(i) the image and the object are equidistant from mirror (ii) virtual image of real object
1
1
1
d
=
+
−
F
f1
f2
f1 f2
d
f1
f2
I
Spherical Mirror:
O
f
u
3.3: Optical Instruments
v
Simple microscope: m = D/f in normal adjustment.
1. Focal length f = R/2
Eyepiece
Objective
2. Mirror equation: v1 + u1 = f1
3. Magnification: m = − uv
Compound microscope:
∞
O
u
v
3.2: Refraction of Light
fe
D
1. Magnification in normal adjustment: m = uv fDe
speed of light in vacuum
c
Refractive index: µ = speed
of light in medium = v
incident
µ1 i
µ2
sin i
Snell’s Law: sin
r = µ1
µ2
reflected
θ
1
2. Resolving power: R = ∆d
= 2µ sin
λ
fo
r
fe
refracted
d0
real depth
d
Apparent depth: µ = apparent
depth = d0
Astronomical telescope:
d I
O
Critical angle: θc = sin−1 µ1
1. In normal adjustment: m = − ffoe , L = fo + fe
µ
θc
1
1
2. Resolving power: R = ∆θ
= 1.22λ
3.4: Dispersion
A
δ
Deviation by a prism:
i
i0
r0
r
µ=
m
sin A+δ
2
,
sin A2
A>0
Dispersion by prism with small A and i:
µ
δ = i + i0 − A,
Cauchy’s equation: µ = µ0 + λA2 ,
1. Mean deviation: δy = (µy − 1)A
general result
i = i0 for minimum deviation
2. Angular dispersion: θ = (µv − µr )A
θ
r
Dispersive power: ω = µµvy−µ
−1 ≈ δy (if A and i small)
δ
δm = (µ − 1)A,
for small A
δm
Dispersion without deviation:
i0
µ2
P
µ2
µ1
µ2 − µ1
−
=
,
v
u
R
m=
µ1 v
µ2 u
Q
O
u
µ0
µ
A0
i
µ1
Refraction at spherical surface:
A
v
(µy − 1)A + (µ0y − 1)A0 = 0
Deviation without dispersion:
(µv − µr )A = (µ0v − µ0r )A0
Heat and Thermodynamics
4.4: Theromodynamic Processes
First law of thermodynamics: ∆Q = ∆U + ∆W
4.1: Heat and Temperature
Temp. scales: F = 32 + 95 C,
Work done by the gas:
K = C + 273.16
Z V2
Ideal gas equation: pV = nRT ,
n : number of moles
van der Waals equation: p + Va2 (V − b) = nRT
pdV
V
1 V2
Wisothermal = nRT ln
V1
∆W = p∆V,
Thermal expansion: L = L0 (1 + α∆T ),
A = A0 (1 + β∆T ), V = V0 (1 + γ∆T ), γ = 2β = 3α
W =
Wisobaric = p(V2 − V1 )
p1 V1 − p2 V2
Wadiabatic =
γ−1
Wisochoric = 0
∆l
Thermal stress of a material: F
A =Y l
4.2: Kinetic Theory of Gases
General: M = mNA , k = R/NA
T1
Q1
Efficiency of the heat engine:
n
W
Q2
Maxwell distribution of speed:
T2
vp v̄ vrms
RMS speed: vrms =
Average speed: v̄ =
q
3kT
m
=
q
8kT
πm
=
Most probable speed: vp =
q
3RT
M
q
8RT
πM
q
η=
T1
Q1
Coeff. of performance of refrigerator:
2kT
m
W
Q2
T2
2
Pressure: p = 13 ρvrms
Equipartition of energy: K = 12 kT for each degree of
freedom. Thus, K = f2 kT for molecule having f degrees of freedoms.
Internal energy
work done by the engine
Q1 − Q2
=
heat supplied to it
Q1
Q2
T2
ηcarnot = 1 −
=1−
Q1
T1
v
of n moles of an ideal gas is U = f2 nRT .
Q2
2
COP = Q
W = Q1 −Q2
Entropy: ∆S = ∆Q
T , Sf − Si =
R f ∆Q
i
T
T
Const. T : ∆S = Q
T,
Varying T : ∆S = ms ln Tfi
Adiabatic process: ∆Q = 0, pV γ = constant
4.3: Specific Heat
Specific heat:
4.5: Heat Transfer
Q
s = m∆T
∆T
Conduction: ∆Q
∆t = −KA x
Latent heat: L = Q/m
x
Thermal resistance: R = KA
∆Q
Specific heat at constant volume: Cv = n∆T
V
∆Q
Specific heat at constant pressure: Cp = n∆T
Rseries = R1 + R2 = A1
x1
x2
K1 + K2
K2
x1
x2
1
1
1
Rparallel = R1 + R2 = x (K1 A1 + K2 A2 )
1
A2
K1
A1
x
E
emissive power
body
Kirchhoff ’s Law: absorptive
power = abody = Eblackbody
Specific heat of gas mixture:
n1 Cv1 + n2 Cv2
,
n1 + n2
A
K2
γ = Cp /Cv
Relation between U and Cv : ∆U = nCv ∆T
Cv =
K1
p
Relation between Cp and Cv : Cp − Cv = R
Ratio of specific heats:
Eλ
γ=
n1 Cp1 + n2 Cp2
n1 Cv1 + n2 Cv2
Molar internal energy of an ideal gas: U = f2 RT ,
f = 3 for monatomic and f = 5 for diatomic gas.
Wien’s displacement law: λm T = b
λm
4
Stefan-Boltzmann law: ∆Q
∆t = σeAT
Newton’s law of cooling: dT
dt = −bA(T − T0 )
λ
Electricity and Magnetism
5.3: Capacitors
Capacitance: C = q/V
5.1: Electrostatics
1 q1 q2
Coulomb’s law: F~ = 4π
2 r̂
0 r
Electric field:
q1
r
q2
q
~ · ~r,
dV = −E
A
~
E
~
r
r2
1 q1 q2
U = − 4π
r
0
Electrostatic potential: V
+q
A
d
~ r) = 1 q2 r̂
E(~
4π0 r
Electrostatic energy:
−q
Parallel plate capacitor: C = 0 A/d
Spherical capacitor:
0 r1 r2
C = 4π
r2 −r1
−q +q
r1
1 q
= 4π
0 r
Z ~r
V (~r) = −
~ · d~r
E
2π0 l
Cylindrical capacitor: C = ln(r
2 /r1 )
∞
r2
l
r1
p
~
Electric dipole moment: p~ = q d~
−q
+q
d
Capacitors in parallel: Ceq = C1 + C2
A
C1
C2
B
1 p cos θ
Potential of a dipole: V = 4π
r2
0
V (r)
θ r
Capacitors in series: C1eq = C11 + C12
p
~
Er
Field of a dipole:
θ r
Eθ
p
~
1 2p cos θ
Er = 4π
,
r3
0
C1
C2
A
B
Force between plates of a parallel plate capacitor:
Q2
F = 2A
0
2
Q
= 12 QV
Energy stored in capacitor: U = 12 CV 2 = 2C
1 p sin θ
Eθ = 4π
r3
0
~ ~τ = p~ × E
~
Torque on a dipole placed in E:
Energy density in electric field E: U/V = 12 0 E 2
~ U = −~
~
Pot. energy of a dipole placed in E:
p·E
Capacitor with dielectric: C = 0 KA
d
5.2: Gauss’s Law and its Applications
H
~ · dS
~
Electric flux: φ = E
5.4: Current electricity
Gauss’s law:
H
Current density: j = i/A = σE
~ · dS
~ = qin /0
E
i
Drift speed: vd = 12 eE
m τ = neA
Field of a uniformly charged ring on its axis:
Resistance of a wire: R = ρl/A, where ρ = 1/σ
a
qx
1
EP = 4π
2
2 3/2
0 (a +x )
q
x
P
~
E
Temp. dependence of resistance: R = R0 (1 + α∆T )
Ohm’s law: V = iR
E and V (of a uniformly charged sphere:
1 Qr
4π0 R3 , for r < R
E
E=
1 Q
for r ≥ R
4π0 r 2 ,
O
(
2
1 Qr
V
4π0 R3 , for r < R
V =
1 Q
for r ≥ R
O
4π0 r ,
r
R
r
R
E and V of a uniformly charged spherical shell:
0,
for r < R
E
E=
1 Q
,
for r ≥ R
4π0 r 2
O
R
(
1 Q
4π0 R , for r < R
V
V =
1 Q
,
for
r
≥
R
4π0 r
O
Field of a line charge:
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.
R
Resistors in parallel: R1eq = R11 + R12
Field in the vicinity of conducting surface: E = σ0
R1
R2
B
r
Resistors in series: Req = R1 + R2
A
R1
R2
R1
r
Wheatstone bridge:
E = 2πλ0 r
Field of an infinite sheet: E = 2σ0
A
R2
↑ G
R3
R4
V
Balanced if R1 /R2 = R3 /R4 .
Electric Power: P = V 2 /R = I 2 R = IV
B
i
Galvanometer as an Ammeter:
ig G
i
i − ig
~
Energy of a magnetic dipole placed in B:
~
U = −~
µ·B
S
ig G = (i − ig )S
R
Galvanometer as a Voltmeter:
G
i
↑
A ig
~
B
l
Bi
Hall effect: Vw = ned
y
w
d
x
z
B
VAB = ig (R + G)
5.6: Magnetic Field due to Current
C
R
Charging of capacitors:
~
⊗B
i
r
~ = µ0 i d~l×~
Biot-Savart law: dB
4π r 3
θ
d~l
V
~
r
i
h
t
q(t) = CV 1 − e− RC
θ2
C
Field due to a straight conductor:
t
Discharging of capacitors: q(t) = q0 e− RC
d
i
q(t)
~
⊗B
θ1
R
µ0 i
(cos θ1 − cos θ2 )
B = 4πd
Time constant in RC circuit: τ = RC
µ0 i
Field due to an infinite straight wire: B = 2πd
Peltier heat
Peltier effect: emf e = ∆H
∆Q = charge transferred .
i1
µ0 i1 i2
Force between parallel wires: dF
dl = 2πd
d
e
Seeback effect:
T0
Tn
T
Ti
Field on the axis of a ring:
1. Thermo-emf: e = aT + 12 bT 2
a
P
i
~
B
d
2. Thermoelectric power: de/dt = a + bT .
2
0 ia
BP = 2(a2µ+d
2 )3/2
3. Neutral temp.: Tn = −a/b.
4. Inversion temp.: Ti = −2a/b.
Thomson effect:
i2
Thomson heat
emf e = ∆H
∆Q = charge transferred = σ∆T .
Faraday’s law of electrolysis: The mass deposited is
a
0 iθ
Field at the centre of an arc: B = µ4πa
~
B
i
θ
a
0i
Field at the centre of a ring: B = µ2a
m = Zit = F1 Eit
Ampere’s law:
where i is current, t is time, Z is electrochemical equivalent, E is chemical equivalent, and F = 96485 C/g is
Faraday constant.
H
~ · d~l = µ0 Iin
B
Field inside a solenoid: B = µ0 ni, n = Nl
l
0N i
Field inside a toroid: B = µ2πr
5.5: Magnetism
r
~ + qE
~
Lorentz force on a moving charge: F~ = q~v × B
~2
B
Charged particle in a uniform magnetic field:
v
q
Field of a bar magnet:
S
2πm
r = mv
qB , T = qB
d
N
~1
B
d
µ0 2M
B1 = 4π
d3 ,
~⊗ r
B
~
B
Force on a current carrying wire:
Angle of dip: Bh = B cos δ
~l
~
F
µ0 M
B2 = 4π
d3
Horizontal
Bv
i
~
F~ = i ~l × B
Magnetic moment of a current loop (dipole):
~
µ
~ A
~
µ
~ = iA
i
~ ~τ = µ
~
Torque on a magnetic dipole placed in B:
~ ×B
δ
Bh
B
0 ni
Tangent galvanometer: Bh tan θ = µ2r
,
i = K tan θ
Moving coil galvanometer: niAB = kθ,
k
i = nAB
θ
Time period of magnetometer: T = 2π
~ = µH
~
Permeability: B
q
I
M Bh
C
5.7: Electromagnetic Induction
H
~ · dS
~
Magnetic flux: φ = B
RC circuit:
Z=
R
l
~
v
1
tan φ = ωCR
R2 + (1/ωC)2 ,
L
Z=
Motional emf: e = Blv
p
LR circuit:
+
√
R
R
i
φ
ωL
˜
Z
e0 sin ωt
R2 + ω 2 L2 ,
~
⊗B
tan φ = ωL
R
L
−
LCR Circuit:
di
e = −L dt
C
R
Z
φ
˜
e0 sin ωt
q
1
ωC
i
Self inductance of a solenoid: L = µ0 n2 (πr2 l)
h
i
t
Growth of current in LR circuit: i = Re 1 − e− L/R
2
1
− ωL ,
Z = R2 + ωC
q
1
1
νresonance = 2π
LC
i
Power factor: P = erms irms cos φ
R
φ
˜
Lenz’s Law: Induced current create a B-field that opposes the change in magnetic flux.
L
Z
1
ωC
e0 sin ωt
Faraday’s law: e = − dφ
dt
Self inductance: φ = Li,
R
i
ωL
tan φ =
1
ωC − ωL
R
1
ωC −ωL
R
e
0.63 R
e
i
S
N1
= ee12 , e1 i1 = e2 i2
Transformer: N
2
t
L
R
Decay of current in LR circuit: i = i0 e
L
t
− L/R
i0
0.37i0
S
i
L
R
t
Time constant of LR circuit: τ = L/R
Energy stored in an inductor: U = 21 Li2
2
B
Energy density of B field: u = VU = 2µ
0
Mutual inductance: φ = M i,
di
e = −M dt
EMF induced in a rotating coil: e = N ABω sin ωt
i
Alternating current:
t
T
i = i0 sin(ωt + φ),
T = 2π/ω
RT
Average current in AC: ī = T1 0 i dt = 0
RMS current: irms =
h R
i1/2
1 T 2
i
dt
= √i02
T 0
i2
t
T
Energy: E = irms 2 RT
1
Capacitive reactance: Xc = ωC
Inductive reactance: XL = ωL
Imepedance: Z = e0 /i0
˜
N1
N2
i1
i2
˜
√
Speed of the EM waves in vacuum: c = 1/ µ0 0
i
R
e1
e2
Modern Physics
N
N0
Population at time t: N = N0 e−λt
6.1: Photo-electric effect
N0
2
O
Photon’s energy: E = hν = hc/λ
t1/2
Photon’s momentum: p = h/λ = E/c
Half life: t1/2 = 0.693/λ
Max. KE of ejected photo-electron: Kmax = hν − φ
Average life: tav = 1/λ
Threshold freq. in photo-electric effect: ν0 = φ/h
Population after n half lives: N = N0 /2n .
Mass defect: ∆m = [Zmp + (A − Z)mn ] − M
V0
Stopping potential:
Vo = hc
e
1
λ
hc
e
− φe
−φ
e
t
φ
hc
1
λ
Binding energy: B = [Zmp + (A − Z)mn − M ] c2
Q-value: Q = Ui − Uf
de Broglie wavelength: λ = h/p
= ∆mc2
Energy released in nuclear reaction: ∆E
where ∆m = mreactants − mproducts .
6.2: The Atom
6.4: Vacuum tubes and Semiconductors
Energy in nth Bohr’s orbit:
En = −
mZ 2 e4
,
80 2 h2 n2
13.6Z 2
eV
n2
Half Wave Rectifier:
a0 = 0.529 Å
Full Wave Rectifier:
En = −
D
Radius of the nth Bohr’s orbit:
rn =
0 h2 n2
,
πmZe2
rn =
n2 a0
,
Z
R Output
˜
˜
Output
Quantization of the angular momentum: l = nh
2π
Grid
Triode Valve:
Photon energy in state transition: E2 − E1 = hν
E2
Plate
E2
hν
E1
Cathode
Filament
hν
E1
Absorption
Emission
∆V
Plate resistance of a triode: rp = ∆ipp
Wavelength of emitted radiation: for
from nth to mth state:
1
1
1
2
= RZ
− 2
λ
n2
m
a
transition
∆Vg =0
∆i
Transconductance of a triode: gm = ∆Vpg
∆Vp =0
∆V
Amplification by a triode: µ = − ∆Vpg
∆ip =0
Relation between rp , µ, and gm : µ = rp × gm
Kα
Kβ
I
X-ray spectrum:
hc
λmin = eV
Ie
Ic
Current in a transistor: Ie = Ib + Ic
λmin
λα
λ
Ib
√
Moseley’s law: ν = a(Z − b)
X-ray diffraction: 2d sin θ = nλ
α and β parameters of a transistor: α =
Ic
α
Ib , β = 1−α
Heisenberg uncertainity principle:
∆p∆x ≥ h/(2π),
∆E∆t ≥ h/(2π)
∆Ic
Transconductance: gm = ∆V
be
Ic
Ie ,
Logic Gates:
6.3: The Nucleus
Nuclear radius: R = R0 A1/3 ,
Decay rate: dN
dt = −λN
R0 ≈ 1.1 × 10−15 m
A
0
0
1
1
B
0
1
0
1
AND
AB
0
0
0
1
OR
A+B
0
1
1
1
NAND
AB
1
1
1
0
NOR
A+B
1
0
0
0
XOR
AB̄ + ĀB
0
1
1
0
β =