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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.

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 ˆı + ay ˆ + az kˆ 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

ˆ k

mg ˆ

~a ×~b = (ay bz − az by )ˆı + (az bx − ax bz )ˆ  + (ax by − ay bx )kˆ |~a × ~b| = ab sin θ

θ

θ T

ˆ ı

~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 v10

v2

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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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

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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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w w w .c o n c e p t s - o 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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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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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 v ¯ vrms

RMS speed: vrms = Average speed: v¯ =

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 =

∆Q Specific heat at constant volume: Cv = n∆T

V

∆Q Specific heat at constant pressure: Cp = 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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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 ˆ 4π0 r 2 r

~ 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 ˆ 4π0 r 2 r

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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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

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d N

S

Formulae

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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: ¯i = 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 ˚ A

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

Plate resistance of a triode: rp =

E1 Absorption

Emission

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 =

∆Vg =0



∆ip ∆Vg

∆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: 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 ¯ + AB ¯ AB 0 1 1 0

= −λN

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β =