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.
Newton’s second law: F~ =
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
=
--------------------------------------------------
Centripetal force: Fc =
MECHANICS
Pseudo force: F~pseudo = −m~a0 ,
mv 2
r ,
fkinetic = µk N
µ+tan θ
1−µ tan θ
ac =
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
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
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 =
Kinetic energy: K = 12 mv 2 =
p2
2m
R
~
F~ · dS
Potential energy: F = −∂U/∂x for conservative forces.
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 =
P
Pxi mi ,
mi
mr 2
xcm =
C
m2 r
m1 +m2
4. Semicircular disc: yc =
4r
3π
disk
shell
6. Solid Hemisphere: yc =
m1 r
m1 +m2
~vcm =
Impulse: J~ =
R
mi~vi
,
M
sphere
rod
hollow
h
C
Ik
Theorem of Parallel Axes: Ik = Icm + md
C
2r
π
cm
Theorem of Perp. Axes: Iz = Ix + Iy
Radius of Gyration: k =
r
r
3r
8
r
p~cm = M~vcm ,
~acm
x
~ = I~
L
ω
4r
3π
C
r
2
C
3r
8
~τ =
y
p
I/m
~ = ~r × p~,
Angular Momentum: L
C
Ic
d
h
3
r
r
2
b
solid rectangle
z
h
3
~
dL
dt ,
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
P
2
1
2
2
2 mr m(a +b )
12
mr 2
2
Torque: ~τ = ~r × F~ ,
5. Hemispherical shell: yc =
2
1
12 ml
m2
r
1. m1 , m2 separated by r:
2r
π
2
2
5 mr
a
ring
m1
3. Semicircular ring: yc =
2
2
3 mr
R
R xdm
dm
CM of few useful configurations:
2. Triangle (CM ≡ Centroid) yc =
2
1
2 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 = 2 mvcm + 2 Icm ω , L
~ + ~rcm × m~vcm
~τcm = Icm α
~,
F~ext
=
M
F~ dt = ∆~
p
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
v10
v20
If v2 = 0 and m1 m2 then
= −v1 .
= 2v1 .
If v2 = 0 and m1 m2 then
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
Variation of g with height: goutside ≈ g 1 −
h
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 =
Angular Accel.: αav =
∆θ
∆t ,
∆ω
∆t ,
ω=
α=
dθ
dt ,
dω
dt ,
~v = ω
~ × ~r
mg
mgθ0 = mg − mω 2 R cos2 θ
θ
R
~a = α
~ × ~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
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
Surface tension: S = F/l
k
= −m
x = −ω 2 x
p
= 2π m
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
Capillary rise: h =
2S cos θ
rρg
U
Potential energy: U = 12 kx2
−A
Kinetic energy K = 12 mv 2
0
x
A
0
x
A
Total energy: E = U + K = 12 mω 2 A2
q
Hydrostatic pressure: p = ρgh
Buoyant force: FB = ρV g = Weight of displaced liquid
K
−A
Simple pendulum: T = 2π
∆psoap = 4S/R
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
Poiseuilli’s equation:
Torsional Pendulum T = 2π
q
Volume flow
time
I
k
Terminal velocity: vt =
Springs in series:
1
keq
=
1
k1
+
1
k2
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
2r 2 (ρ−σ)g
9η
=
πpr 4
8ηl
r
l
Waves
2.1: Waves Motion
General equation of wave:
2
∂ y
∂x2
=
2
1 ∂ y
v 2 ∂t2 .
q
4. 1st overtone/2nd harmonics: ν1 =
2
2L
5. 2nd overtone/3rd harmonics: ν2 =
3
2L
T
µ
q
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 ∝
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
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.
2A cos kx
Intensity: I =
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
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, . . .
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
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 =
3v
4L
5v
4L
5. 2nd overtone/5th harmonics: ν2 = 5ν0 =
6. Only odd harmonics are present.
dy
D
θ
d
S2
A
N
Open organ pipe:
P
y
S1
Path difference: ∆x =
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, constructive;
n + 12 λ, 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), 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.)
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)
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 =
∆x = 2µd =
aE0
r
sin ω(t − vr ), I =
Young’s double slit experiment
I0
r2
θ
I0
I
Optics
1
f
Lens maker’s formula:
h
= (µ − 1)
1
R1
−
1
R2
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
=
µ2
real depth
apparent depth
Critical angle: θc = sin−1
=
1. Magnification in normal adjustment: m =
c
v
=
incident
µ1 i
µ2
µ1
Apparent depth: µ =
reflected
2. Resolving power: R =
d0
d
d0
2µ sin θ
λ
fe
Astronomical telescope:
d I
O
1
µ
1. In normal adjustment: m = − ffoe , L = fo + fe
µ
θc
1
∆θ
=
i
i0
r0
r
1
1.22λ
3.4: Dispersion
δ
Deviation by a prism:
Cauchy’s equation: µ = µ0 +
A
λ2 ,
A>0
Dispersion by prism with small A and i:
µ
m
sin A+δ
2
,
sin A2
=
refracted
A
µ=
1
∆d
v D
u fe
fo
r
2. Resolving power: R =
δ = i + i0 − A,
fe
D
1. Mean deviation: δy = (µy − 1)A
general result
i = i0 for minimum deviation
2. Angular dispersion: θ = (µv − µr )A
Dispersive power: ω =
µv −µr
µy −1
≈
θ
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
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
K2
x1
x2
1
Rparallel
=
1
R1
+
1
R2
=
1
x
(K1 A1 + K2 A2 )
A2
K1
A1
x
Kirchhoff ’s Law:
emissive power
absorptive power
=
Ebody
abody
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:
∆Q
∆t
= 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.
Wien’s displacement law: λm T = b
λm
Stefan-Boltzmann law:
∆Q
∆t
Newton’s law of cooling:
= σeAT 4
dT
dt
= −bA(T − T0 )
λ
Electricity and Magnetism
5.3: Capacitors
Capacitance: C = q/V
5.1: Electrostatics
Coulomb’s law: F~ =
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
Z
~
r
~ · 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
Current density: j = i/A = σE
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
Drift speed: vd =
1 eE
2 mτ
=
i
neA
a
qx
1
4π0 (a2 +x2 )3/2
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
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
(
1 Q
4π0 R , for r < R
V
V =
1 Q
,
for
r
≥
R
4π0 r
O
Field of a line charge: 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.
1
Req
=
1
R1
+
1
R2
r
Resistors in series: Req = R1 + R2
A
R1
R2
R1
r
R
Wheatstone bridge:
R2
σ
0
Electric Power: P = V 2 /R = I 2 R = IV
B
R2
↑ G
R3
R4
V
Balanced if R1 /R2 = R3 /R4 .
σ
20
Field in the vicinity of conducting surface: E =
R1
B
λ
2π0 r
Field of an infinite sheet: E =
A
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
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:
d
N
S
2πm
qB
~1
B
d
~⊗ r
B
B1 =
~
B
Force on a current carrying wire:
B2 =
µ0 M
4π d3
Angle of dip: Bh = B cos δ
~l
~
F
µ0 2M
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
Tangent galvanometer: Bh tan θ =
µ0 ni
2r ,
~ = µH
~
Permeability: B
B
i = K tan θ
Moving coil galvanometer: niAB = kθ,
Time period of magnetometer: T = 2π
Bh
i=
q
k
nAB θ
I
M Bh
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 ,
R
√
R
R2 + ω 2 L2 ,
tan φ =
L
Z
C
ωL
R
R
˜
R2 +
νresonance =
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
tan φ =
L
Z=
Motional emf: e = Blv
p
LR circuit:
+
L
φ
˜
Z=
e=
Z
1
ωC
e0 sin ωt
Faraday’s law: e = − dφ
dt
Self inductance: φ = Li,
R
i
1
ωC
q
1
2π
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
i0
0.37i0
S
i
t
L
R
Time constant of LR circuit: τ = L/R
Energy stored in an inductor: U = 21 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
Imepedance: Z = e0 /i0
=
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
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
φ
e
−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
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
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)
Transconductance: gm =
Ic
Ie ,
∆Ic
∆Vbe
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 =