Formulae
S he e t f o r
P h ys i c s
w w w .c o n c e p t s - o
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.
f -p h y s i c s . c om
| pg. 1
Motion in a straight line with constant a:
v = u + at,
s = ut + 21 at2 ,
v 2 − u2 = 2as
Relative Velocity: ~vA/B = ~vA − ~vB
0.1: Physical Constants
c
h
hc
G
k
R
NA
e
µ0
Gravitation constant
Boltzmann constant
Molar gas constant
Avogadro’s number
Charge of electron
Permeability of vacuum
Permitivity of vacuum 0
1
Coulomb constant
4π0
Faraday constant
F
Mass of electron
me
Mass of proton
mp
Mass of neutron
mn
Atomic mass unit
u
Atomic mass unit
u
Stefan-Boltzmann
σ
constant
Rydberg constant
R∞
Bohr magneton
µB
Bohr radius
a0
Standard atmosphere atm
Wien
displacement
b
constant
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
u
u sin θ
Speed of light
Planck constant
Projectile Motion:
θ
u cos θ
R
y = ut sin θ − 21 gt2
g
y = x tan θ − 2
x2
2u cos2 θ
2u sin θ
u2 sin 2θ
u2 sin2 θ
T =
, R=
, H=
g
g
2g
x = ut cos θ,
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 )
1.3: Newton’s Laws and Friction
Linear momentum: p~ = m~v
Newton’s first law: inertial frame.
Newton’s second law: F~ =
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
F~ = m~a
d~
p
dt ,
Newton’s third law: F~AB = −F~BA
Frictional force: fstatic, max = µs N,
Banking angle:
v2
rg
= tan θ,
v2
rg
=
mv 2
r ,
fkinetic = µk N
µ+tan θ
1−µ tan θ
ac =
Pseudo force: F~pseudo = −m~a0 ,
MECHANICS
v2
r
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
~
a × ~b
Cross product:
x
H
O
Centripetal force: Fc =
1
y
l cos θ
g
~
a
k̂
mg
̂
~a ×~b = (ay bz − az by )ı̂ + (az bx − ax bz )̂ + (ax by − ay bx )k̂
|~a × ~b| = ab sin θ
θ
θ T
ı̂
~b
θ
q
1.4: Work, Power and Energy
~ = F S cos θ,
Work: W = F~ · S
W =
Kinetic energy: K = 12 mv 2 =
p2
2m
R
~
F~ · dS
Potential energy: F = −∂U/∂x for conservative forces.
1.2: Kinematics
Ugravitational = mgh,
Average and Instantaneous Vel. and Accel.:
~vav = ∆~r/∆t,
~vinst = d~r/dt
~aav = ∆~v /∆t
~ainst = d~v /dt
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
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Mechanical energy: E = U + K. Conserved if forces are
conservative in nature.
Power Pav =
∆W
∆t
,
Centre of mass: xcm =
P
Pxi mi ,
mi
θ = ωt + 21 αt2 ,
Moment of Inertia: I =
1.5: Centre of Mass and Collision
xcm =
R
R xdm
dm
| pg. 2
Rotation about an axis with constant α:
ω = ω0 + αt,
Pinst = F~ · ~v
f -p h y s i c s . c om
mr 2
2
1
2 mr
2
2
3 mr
P
i
mi ri 2 ,
2
2
5 mr
2
1
12 ml
ω 2 − ω0 2 = 2αθ
I=
R
r2 dm
2
1
2
2
2 mr m(a +b )
12
mr 2
b
a
CM of few useful configurations:
ring
m1
shell
sphere
rod
hollow
solid rectangle
m2
r
1. m1 , m2 separated by r:
disk
C
m2 r
m1 +m2
m1 r
m1 +m2
Ik
Theorem of Parallel Axes: Ik = Icm + md
2
Ic
d
cm
2. Triangle (CM ≡ Centroid) yc =
3. Semicircular ring: yc =
2r
π
4. Semicircular disc: yc =
4r
3π
h
3
h
C
C
Theorem of Perp. Axes: Iz = Ix + Iy
Radius of Gyration: k =
r
r
x
C
p
I/m
4r
3π
~ = ~r × p~,
Angular Momentum: L
C
r
2
Torque: ~τ = ~r × F~ ,
C
3r
8
~τ =
~
dL
dt ,
~ = I~
L
ω
y
P θ ~
F
~
r x
τ = Iα
O
6. Solid Hemisphere: yc =
3r
8
r
y
2r
π
r
r
2
5. Hemispherical shell: yc =
z
h
3
7. Cone: the height of CM from the base is h/4 for
the solid cone and h/3 for the hollow cone.
~ ~τext = 0 =⇒ L
~ = const.
Conservation of L:
P~
P
Equilibrium condition:
F = ~0,
~τ = ~0
Kinetic Energy: Krot = 12 Iω 2
Motion of the CM: M =
P
~vcm =
Impulse: J~ =
R
mi~vi
,
M
P
Dynamics:
mi
p~cm = M~vcm ,
~acm
F~ext = m~acm ,
p~cm = m~vcm
2
2
1
1
~
~ + ~rcm × m~vcm
K = 2 mvcm + 2 Icm ω , L = Icm ω
~τcm = Icm α
~,
F~ext
=
M
F~ dt = ∆~
p
1.7: Gravitation
Before collision After collision
Collision:
m1
m2
v1
m1
v2
v10
m2
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 .
Angular Accel.: αav =
Get Formulas
∆θ
∆t ,
∆ω
∆t ,
F F
r
Potential energy: U = − GMr m
Gravitational acceleration: g =
GM
R2
Variation of g with depth: ginside ≈ g 1 −
h
R
Variation of g with height: goutside ≈ g 1 −
2h
R
Effect of non-spherical earth shape on g:
gat pole > gat equator (∵ Re − Rp ≈ 21 km)
Effect of earth rotation on apparent weight:
1.6: Rigid Body Dynamics
Angular velocity: ωav =
m1
2
Gravitational force: F = G mr1 m
2
ω=
α=
dθ
dt ,
dω
dt ,
~v = ω
~ × ~r
~a = α
~ × ~r
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m2
Formulae
S he e t f o r
P h ys i c s
w w w .c o n c e p t s - o
f -p h y s i c s . c om
ω
~
~
A
Superposition of two SHM’s:
2
x1 = A1 sin ωt,
Orbital velocity of satellite: vo =
Escape velocity: ve =
q
δ
~1
A
θ
R
q
~2
A
mω R cos θ
mg
mgθ0 = mg − mω 2 R cos2 θ
| pg. 3
x2 = A2 sin(ωt + δ)
x = x1 + x2 = A sin(ωt + )
q
A = A1 2 + A2 2 + 2A1 A2 cos δ
GM
R
tan =
2GM
R
A2 sin δ
A1 + A2 cos δ
vo
Kepler’s laws:
1.9: Properties of Matter
a
Modulus of rigidity: Y =
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 =
1.8: Simple Harmonic Motion
1
B
F/A
∆l/l ,
= − V1
B = −V
∆P
∆V
F
Aθ
, η=
dV
dP
Poisson’s ratio: σ =
lateral strain
longitudinal strain
Elastic energy: U =
1
2
=
∆D/D
∆l/l
stress × strain × volume
Hooke’s law: F = −kx (for small elongation x.)
Acceleration: a =
d2 x
dt2
Time period: T =
2π
ω
k
= −m
x = −ω 2 x
p
= 2π m
k
Surface tension: S = F/l
Surface energy: U = SA
Excess pressure in bubble:
Displacement: x = A sin(ωt + φ)
√
Velocity: v = Aω cos(ωt + φ) = ±ω A2 − x2
∆pair = 2S/R,
Capillary rise: h =
∆psoap = 4S/R
2S cos θ
rρg
U
Potential energy: U = 12 kx2
−A
0
x
A
Hydrostatic pressure: p = ρgh
Kinetic energy K = 12 mv 2
K
−A
0
x
Buoyant force: FB = ρV g = Weight of displaced liquid
A
Equation of continuity: A1 v1 = A2 v2
Total energy: E = U + K = 12 mω 2 A2
Simple pendulum: T = 2π
q
Physical Pendulum: T = 2π
Bernoulli’s equation: p + 21 ρv 2 + ρgh = constant
√
Torricelli’s theorem: vefflux = 2gh
l
g
q
v2
v1
l
dv
Viscous force: F = −ηA dx
F
I
mgl
Stoke’s law: F = 6πηrv
v
Torsional Pendulum T = 2π
q
Poiseuilli’s equation:
I
k
Volume flow
time
Terminal velocity: vt =
Springs in series:
1
keq
=
1
k1
+
1
k2
Springs in parallel: keq = k1 + k2
Get Formulas
k1
=
πpr 4
8ηl
2r 2 (ρ−σ)g
9η
k2
k2
k1
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r
l
Formulae
P h ys i c s
w w w .c o n c e p t s - o
Waves
2.1: Waves Motion
General equation of wave:
2
∂ y
∂x2
=
2
1 ∂ y
v 2 ∂t2 .
f -p h y s i c s . c om
q
4. 1st overtone/2nd harmonics: ν1 =
2
2L
5. 2nd overtone/3rd harmonics: ν2 =
3
2L
1
2π
T = =
,
ν
ω
q
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
2
Progressive sine wave:
λ
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 ∝
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
6. All harmonics are present.
2π
k=
λ
v = νλ,
| pg. 4
T
µ
Notation: Amplitude A, Frequency ν, Wavelength λ, Period T , Angular Frequency ω, Wave Number k,
1
L,
ν∝
√
T, ν ∝
√1 .
µ
ν=
n
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.
Intensity: I =
2A cos kx
2
S he e t f o r
Standing Waves:
2π 2 B
2 2
v s0 ν
=
p0 2 v
2B
=
s
vgas =
γP
ρ
p0 2
2ρv
x
A
N
A
N
A
Standing longitudinal waves:
λ/4
p1 = p0 sin ω(t − x/v),
y1 = A1 sin(kx − ωt),
y2 = A2 sin(kx + ωt)
p2 = p0 sin ω(t + x/v)
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
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
Get Formulas
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, . . .
3. Fundamental/1st harmonics: ν0 =
v
4L
4. 1st overtone/3rd harmonics: ν1 = 3ν0 =
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3v
4L
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5v
4L
5. 2nd overtone/5th harmonics: ν2 = 5ν0 =
f -p h y s i c s . c om
P
y
S1
Path difference: ∆x =
6. Only odd harmonics are present.
dy
D
θ
d
S2
A
N
Open organ pipe:
| pg. 5
L
A
N
Phase difference: δ =
D
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
2. Fundamental/1st harmonics: ν0 =
∆x =
v
2L
3. 1st overtone/2nd harmonics: ν1 = 2ν0 =
2v
2L
4. 2nd overtone/3rd harmonics: ν2 = 3ν0 =
3v
2L
nλ, n + 21 λ,
constructive;
destructive
Intensity:
p
I = I1 + I2 + 2 I1 I2 cos δ,
p
p
p 2
p 2
I1 + I2 , Imin =
I1 − I2
Imax =
5. All harmonics are present.
l2 + d
l1 + d
I1 = I2 : I = 4I0 cos2 2δ , Imax = 4I0 , Imin = 0
Resonance column:
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),
∆ω = ω1 − ω2
constructive;
destructive.
Diffraction from a single slit:
y
θ
b
y
(beats freq.)
D
For Minima: nλ = b sin θ ≈ b(y/D)
Doppler Effect:
Resolution: sin θ =
v + uo
ν0
ν=
v − us
1.22λ
b
Law of Malus: I = I0 cos2 θ
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
Spherical Wave: E =
nλ, n + 21 λ,
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,
∆x = 2µd =
aE0
r
sin ω(t − vr ), I =
I0
r2
θ
I0
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Young’s double slit experiment
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I
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3
S he e t f o r
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Optics
f -p h y s i c s . c om
1
f
Lens maker’s formula:
h
= (µ − 1)
1
R1
−
1
R2
| pg. 6
i
3.1: Reflection of Light
f
Lens formula:
normal
Laws of reflection:
i r
incident
(i)
reflected
1
v
−
1
u
= f1 ,
v
u
m=
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
2. Mirror equation:
1
v
3. Magnification: m =
Eyepiece
Objective
1
u
− uv
+
=
1
f
∞
O
Compound microscope:
u
v
3.2: Refraction of Light
speed of light in vacuum
speed of light in medium
Refractive index: µ =
Snell’s Law:
sin i
sin r
=
c
v
=
incident
µ1 i
µ2
µ1
Apparent depth: µ =
µ2
real depth
apparent depth
Critical angle: θc = sin−1
=
1. Magnification in normal adjustment: m =
reflected
d0
d I
O
µ
A
µ=
,
Astronomical telescope:
i0
r0
r
=
1
1.22λ
A
λ2 ,
A>0
Dispersion by prism with small A and i:
general result
1. Mean deviation: δy = (µy − 1)A
i = i0 for minimum deviation
2. Angular dispersion: θ = (µv − µr )A
Dispersive power: ω =
for small A
µ2
P
m=
Q
O
u
µ2
µ1
µ2 − µ1
−
=
,
v
u
R
≈
Dispersion without deviation:
i
µ1
Refraction at spherical surface:
µv −µr
µy −1
δm
i0
Get Formulas
1
∆θ
Cauchy’s equation: µ = µ0 +
δ
δm = (µ − 1)A,
fe
3.4: Dispersion
δ
µ
m
sin A+δ
2
sin A2
2µ sin θ
λ
1. In normal adjustment: m = − ffoe , L = fo + fe
θc
i
=
refracted
d
d0
Deviation by a prism:
1
∆d
v D
u fe
fo
r
1
µ
2. Resolving power: R =
2. Resolving power: R =
δ = i + i0 − A,
fe
D
v
θ
δy
(if A and i small)
A
µ0
µ
A0
(µy − 1)A + (µ0y − 1)A0 = 0
Deviation without dispersion:
(µv − µr )A = (µ0v − µ0r )A0
µ1 v
µ2 u
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S he e t f o r
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Heat and Thermodynamics
| pg. 7
4.4: Theromodynamic Processes
First law of thermodynamics: ∆Q = ∆U + ∆W
4.1: Heat and Temperature
Temp. scales: F = 32 + 95 C,
f -p h y s i c s . c om
Work done by the gas:
K = C + 273.16
n : number of moles
van der Waals equation: p + Va2 (V − b) = nRT
F
A
=Y
pdV
V
1 V2
= nRT ln
V1
∆W = p∆V,
Wisothermal
Thermal expansion: L = L0 (1 + α∆T ),
A = A0 (1 + β∆T ), V = V0 (1 + γ∆T ), γ = 2β = 3α
Thermal stress of a material:
V2
Z
Ideal gas equation: pV = nRT ,
W =
Wisobaric = p(V2 − V1 )
p1 V1 − p2 V2
Wadiabatic =
γ−1
Wisochoric = 0
∆l
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
work done by the engine
Q1 − Q2
=
heat supplied to it
Q1
Q2
T2
ηcarnot = 1 −
=1−
Q1
T1
v
η=
T1
Q1
Coeff. of performance of refrigerator:
2kT
m
W
Q2
T2
2
Pressure: p = 13 ρvrms
COP =
Equipartition of energy: K = 12 kT for each degree of
freedom. Thus, K = f2 kT for molecule having f degrees of freedoms.
Internal energy of n moles of an ideal gas is U =
f
2 nRT .
Q2
W
Q2
Q1 −Q2
=
Entropy: ∆S =
∆Q
T ,
Sf − Si =
Const. T : ∆S =
Q
T,
Rf
i
∆Q
T
Varying T : ∆S = ms ln
Tf
Ti
Adiabatic process: ∆Q = 0, pV γ = constant
4.3: Specific Heat
Specific heat: s =
4.5: Heat Transfer
Q
m∆T
Conduction:
Latent heat: L = Q/m
= −KA ∆T
x
x
KA
Thermal resistance: R =
Specific heat at constant volume: Cv =
∆Q
n∆T
Specific heat at constant pressure: Cp =
V
∆Q
n∆T
Rseries = R1 + R2 =
1
A
x1
K1
+
x2
K2
1
Rparallel
=
1
R1
+
1
R2
=
1
x
x1
x2
(K1 A1 + K2 A2 )
n1 Cv1 + n2 Cv2
,
n1 + n2
A
K2
A2
K1
A1
x
Kirchhoff ’s Law:
emissive power
absorptive power
=
Ebody
abody
Specific heat of gas mixture:
= Eblackbody
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.
Get Formulas
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:
∆Q
∆t
Wien’s displacement law: λm T = b
λm
Stefan-Boltzmann law:
∆Q
∆t
Newton’s law of cooling:
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= σeAT 4
dT
dt
= −bA(T − T0 )
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λ
Formulae
5
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Electricity and Magnetism
| pg. 8
5.3: Capacitors
Capacitance: C = q/V
5.1: Electrostatics
Coulomb’s law: F~ =
f -p h y s i c s . c om
1 q1 q2
4π0 r 2 r̂
~ r) =
Electric field: E(~
q1
q
−q
Parallel plate capacitor: C = 0 A/d
+q
A
A
~
E
~
r
r2
1 q1 q2
− 4π
r
0
Electrostatic potential: V =
~ · ~r,
dV = −E
q2
d
1 q
4π0 r 2 r̂
Electrostatic energy: U =
r
Spherical capacitor: C =
4π0 r1 r2
r2 −r1
−q +q
r1
1 q
4π0 r
~
r
Z
~ · d~r
E
V (~r) = −
Cylindrical capacitor: C =
∞
2π0 l
ln(r2 /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
4π0 r 2
Potential of a dipole: V =
V (r)
θ r
Capacitors in series:
p
~
Er
Field of a dipole:
θ r
Eθ
p
~
Er =
1 2p cos θ
,
4π0
r3
Eθ =
1
Ceq
=
1
C1
+
1
C2
C1
B
Force between plates of a parallel plate capacitor:
Q2
F = 2A
0
Energy stored in capacitor: U = 12 CV 2 =
1 p sin θ
4π0 r 3
C2
A
Q2
2C
= 12 QV
~ ~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
H
~ · dS
~ = qin /0
Gauss’s law: E
Drift speed: vd =
Field of a uniformly charged ring on its axis:
Resistance of a wire: R = ρl/A, where ρ = 1/σ
EP =
5.4: Current electricity
Current density: j = i/A = σE
a
qx
1
4π0 (a2 +x2 )3/2
q
x
P
~
E
1 eE
2 mτ
=
i
neA
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
2
4π0 r
O
(
2 Q
3 − Rr 2 , for r < R V
0R
V = 8π
1 Q
for r ≥ R
4π0 r ,
O
r
R
r
R
Resistors in parallel:
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
(
Q
1
4π0 R , for r < R
V
V =
1 Q
,
for
r
≥
R
4π0 r
O
Field of a line charge: E =
r
=
1
R1
+
1
R2
A
R1
Resistors in series: Req = R1 + R2
A
R1
R2
R1
r
R
R2
R2
Wheatstone bridge:
R3
R4
V
σ
20
σ
0
Electric Power: P = V 2 /R = I 2 R = IV
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B
↑ G
Balanced if R1 /R2 = R3 /R4 .
Field in the vicinity of conducting surface: E =
Get Formulas
1
Req
B
λ
2π0 r
Field of an infinite sheet: E =
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.
Formulae
S he e t f o r
P h ys i c s
w w w .c o n c e p t s - o
i
Galvanometer as an Ammeter:
ig G
i
i − ig
f -p h y s i c s . c om
| pg. 9
~
Energy of a magnetic dipole placed in B:
~
U = −~
µ·B
S
ig G = (i − ig )S
Hall effect: Vw =
R
Galvanometer as a Voltmeter:
G
i
↑
A ig
~
B
l
Bi
ned
y
w
d
x
z
B
VAB = ig (R + G)
5.6: Magnetic Field due to Current
C
R
Charging of capacitors:
~ =
Biot-Savart law: dB
~
⊗B
i
µ0 i d~l×~
r
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
B=
Time constant in RC circuit: τ = RC
µ0 i
4πd (cos θ1
− cos θ2 )
Field due to an infinite straight wire: B =
Peltier effect: emf e =
∆H
∆Q
=
Peltier heat
charge transferred .
Force between parallel wires:
dF
dl
=
µ0 i
2πd
i1
µ0 i1 i2
2πd
d
e
Seeback effect:
T0
Tn
T
Ti
a
Field on the axis of a ring:
1. Thermo-emf: e = aT + 12 bT 2
P
i
BP =
3. Neutral temp.: Tn = −a/b.
µ0 ia2
2(a2 +d2 )3/2
4. Inversion temp.: Ti = −2a/b.
Thomson effect: emf e =
∆H
∆Q
=
Thomson heat
charge transferred
= σ∆T .
Field at the centre of an arc: B =
a
µ0 iθ
4πa
~
B
i
θ
a
Faraday’s law of electrolysis: The mass deposited is
Field at the centre of a ring: B =
1
F
~
B
d
2. Thermoelectric power: de/dt = a + bT .
m = Zit =
i2
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 =
N
l
l
Field inside a toroid: B =
5.5: Magnetism
µ0 i
2a
µ0 N i
2πr
r
~ + qE
~
Lorentz force on a moving charge: F~ = q~v × B
~2
B
Charged particle in a uniform magnetic field:
v
q
r=
mv
qB ,
T =
Field of a bar magnet:
2πm
qB
~1
B
d
~⊗ r
B
B1 =
~
B
Force on a current carrying wire:
µ0 2M
4π d3 ,
B2 =
µ0 M
4π d3
Angle of dip: Bh = B cos δ
~l
~
F
Horizontal
δ
Bv
i
~
F~ = i ~l × B
Tangent galvanometer: Bh tan θ =
Magnetic moment of a current loop (dipole):
~
µ
~ A
~
µ
~ = iA
i
µ0 ni
2r ,
i=
q
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1
B
k
nAB θ
I
M Bh
~ = µH
~
Permeability: B
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Bh
i = K tan θ
Moving coil galvanometer: niAB = kθ,
Time period of magnetometer: T = 2π
~ ~τ = µ
~
Torque on a magnetic dipole placed in B:
~ ×B
Get Formulas
d
N
S
Formulae
S he e t f o r
P h ys i c s
w w w .c o n c e p t s - o
C
5.7: Electromagnetic Induction
H
~ · dS
~
Magnetic flux: φ = B
RC circuit:
Lenz’s Law: Induced current create a B-field that opposes the change in magnetic flux.
l
~
v
R2 + (1/ωC)2 ,
φ
R
tan φ =
√
R
R2 + ω 2 L2 ,
Z
ωL
R
tan φ =
L
C
R
Z
φ
˜
e0 sin ωt
q
1
ωC
i
di
−L dt
i
ωL
˜
LCR Circuit:
Z=
R
φ
e0 sin ωt
~
⊗B
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
1
ωCR
i
−
R
Z
1
ωC
L
Z=
Motional emf: e = Blv
p
LR circuit:
+
L
R
i
˜
Z=
e=
| pg. 10
e0 sin ωt
Faraday’s law: e = − dφ
dt
Self inductance: φ = Li,
f-p h y s i c s . c om
R2 +
1
ωC
q
1
2π
νresonance =
2
− ωL ,
ωL
tan φ =
1
ωC
− ωL
N2
˜
R
1
ωC
−ωL
R
1
LC
Power factor: P = erms irms cos φ
e
0.63 R
e
i
S
t
L
R
Transformer:
Decay of current in LR circuit: i = i0 e
L
t
− L/R
=
e1
e2 ,
e1 i1 = e2 i2
e1
˜
N1
i1
i2
√
Speed of the EM waves in vacuum: c = 1/ µ0 0
i
R
N1
N2
i0
0.37i0
S
i
t
L
R
Time constant of LR circuit: τ = L/R
Energy stored in an inductor: U = 12 Li2
Energy density of B field: u =
Mutual inductance: φ = M i,
U
V
=
B2
2µ0
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
1 T
T
0
i1/2
i2 dt
=
i0
√
2
i2
t
T
Energy: E = irms 2 RT
Capacitive reactance: Xc =
1
ωC
Inductive reactance: XL = ωL
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Imepedance: Z = e0 /i0
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e2
Formulae
6
P h ys i c s
S he e t f o r
w w w .c o n c e p t s - o
f-p h y s i c s . c om
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 .
1
λ
hc
e
−
hc
e
φ
e
−φ
e
t
Mass defect: ∆m = [Zmp + (A − Z)mn ] − M
V0
Stopping potential: Vo =
| pg. 11
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
Quantization of the angular momentum: l =
Output
Grid
Triode Valve:
Cathode
Filament
Plate
E2
hν
E1
˜
nh
2π
Photon energy in state transition: E2 − E1 = hν
E2
R Output
˜
hν
E1
Absorption
Emission
Plate resistance of a triode: rp =
Wavelength of emitted radiation: for
from nth to mth state:
1
1
1
2
= RZ
− 2
λ
n2
m
a
transition
∆Vp
∆ip
Transconductance of a triode: gm =
Amplification by a triode: µ = −
∆Vp
∆Vg
∆Vg =0
∆ip
∆Vg
∆Vp =0
∆ip =0
Relation between rp , µ, and gm : µ = rp × gm
Kα
Kβ
I
X-ray spectrum: λmin =
hc
eV
Ie
Ic
Current in a transistor: Ie = Ib + Ic
λmin
Moseley’s law:
√
λα
λ
Ib
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)
Transconductance: gm =
Ic
Ie ,
∆Ic
∆Vbe
Logic Gates:
6.3: The Nucleus
Nuclear radius: R = R0 A1/3 ,
Decay rate:
dN
dt
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
= −λN
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β =