Speed of light c 3 × 10 8 m/s Planck constant h 6.63 × 10

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.

0.1: Physical Constants

Speed of light c 3 × 10 ⁸ m/s Planck constant h 6.63 × 10

⁻³⁴

J s

hc 1242 eV-nm

Gravitation constant G 6.67×10

⁻¹¹

m ³ kg

⁻¹

s

⁻²

Boltzmann constant k 1.38 × 10

⁻²³

J/K Molar gas constant R 8.314 J/(mol K) Avogadro’s number N _A 6.023 × 10 ²³ mol

⁻¹

Charge of electron e 1.602 × 10

⁻¹⁹

C Permeability of vac-

uum

µ ₀ 4π × 10

⁻⁷

N/A ² Permitivity of vacuum ₀ 8.85 × 10

⁻¹²

F/m Coulomb constant _4π ¹

0

9 × 10 ⁹ N m ² /C ² Faraday constant F 96485 C/mol Mass of electron m e 9.1 × 10

⁻³¹

kg Mass of proton m p 1.6726 × 10

⁻²⁷

kg Mass of neutron m n 1.6749 × 10

⁻²⁷

kg Atomic mass unit u 1.66 × 10

⁻²⁷

kg Atomic mass unit u 931.49 MeV/c ² Stefan-Boltzmann

constant

σ 5.67×10

⁻⁸

W/(m ² K ⁴ ) Rydberg constant R

_∞

1.097 × 10 ⁷ m

⁻¹

Bohr magneton µ B 9.27 × 10

⁻²⁴

J/T Bohr radius a 0 0.529 × 10

⁻¹⁰

m Standard atmosphere atm 1.01325 × 10 ⁵ Pa Wien displacement

constant

b 2.9 × 10

⁻³

m K

1 MECHANICS

1.1: Vectors

Notation: ~a = a x ˆ ı + a y  ˆ + a z ˆ k Magnitude: a = |~a| = q

a ² _x + a ² _y + a ² _z

Dot product: ~a ·~b = a _x b _x + a _y b _y + a _z b _z = ab cos θ Cross product:

~ a

~ b

~ a

×

~ b θ

ˆ ı ˆ ˆ  k

~a ×~b = (a y b z − a z b y )ˆ ı + (a z b x −a x b z )ˆ + (a x b y −a y b x )ˆ k

|~a × ~b| = ab sin θ

1.2: Kinematics

Average and Instantaneous Vel. and Accel.:

~ v av = ∆~ r/∆t, ~ v inst = d~ r/dt

~a _av = ∆~ v/∆t ~a _inst = d~ v/dt

Motion in a straight line with constant a:

v = u + at, s = ut + ¹ ₂ at ² , v ² − u ² = 2as Relative Velocity: ~ v _A/B = ~ v _A − ~ v _B

Projectile Motion: ^x

y

O

u sin θ

u cos θ u

θ R

H

x = ut cos θ, y = ut sin θ − ¹ ₂ gt ² y = x tan θ − g

2u ² cos ² θ x ² T = 2u sin θ

g , R = u ² sin 2θ

g , H = u ² sin ² θ 2g 1.3: Newton’s Laws and Friction

Linear momentum: ~ p = m~ v Newton’s first law: inertial frame.

Newton’s second law: F ~ = ^d~ _dt ^p , F ~ = m~a Newton’s third law: F ~ AB = − F ~ BA

Frictional force: f static, max = µ s N, f kinetic = µ k N Banking angle: ^v _rg

²

= tan θ, ^v _rg

²

= _1−µ ^µ+tan _tan ^θ _θ

Centripetal force: F c = ^mv _r

²

, a c = ^v _r

²

Pseudo force: F ~ _pseudo = −m~a ₀ , F centrifugal = − ^mv _r

²

Minimum speed to complete vertical circle:

v min, bottom = p

5gl, v _{min, top} = p gl

Conical pendulum: T = 2π q

l cos θ g

mg T l θ

θ

1.4: Work, Power and Energy

Work: W = F ~ · S ~ = F S cos θ, W = R F ~ · d S ~ Kinetic energy: K = ¹ ₂ mv ² = _2m ^p

²

Potential energy: F = −∂U/∂x for conservative forces.

U gravitational = mgh, U _spring = ¹ ₂ kx ²

Work done by conservative forces is path indepen- dent 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 P av = ^∆W _∆t , P inst = F ~ · ~ v 1.5: Centre of Mass and Collision Centre of mass: x _cm =

^PP

^x m

ⁱ

^m

iⁱ

, x _cm =

R

xdm

R

dm

CM of few useful configurations:

1. m 1 , m 2 separated by r:

m

1

m

2

C r

m₂r m₁+m₂

m₁r m₁+m₂

2. Triangle (CM ≡ Centroid) y c = ^h ₃

C

h 3

h

3. Semicircular ring: y _c = ^2r _π ^C

2r

r

π

4. Semicircular disc: y _c = _3π ^4r _C

4r

r

3π

5. Hemispherical shell: y _c = ^r ₂ C r

^r2

6. Solid Hemisphere: y c = ^3r ₈ C r

^3r8

7. Cone: the height of CM from the base is h/4 for the solid cone and h/3 for the hollow cone.

Motion of the CM: M = P m i

~ v cm =

P m i ~ v i

M , ~ p cm = M~ v cm , ~a cm = F ~ ext

M Impulse: J ~ = R F ~ dt = ∆~ p

Collision: _m

₁

_m

₂

v

1

v

2

Before collision After collision m

1

m

2

v

₁⁰

v

⁰₂

Momentum conservation: m 1 v 1 +m 2 v 2 = m 1 v

⁰

₁ +m 2 v ₂

⁰

Elastic Collision: ¹ ₂ m 1 v 1 2 + ¹ ₂ m 2 v 2 2 = ¹ ₂ m 1 v

⁰

₁ ² + ¹ ₂ m 2 v

⁰

₂ ² Coefficient of restitution:

e = −(v ₁

⁰

− v ₂

⁰

) v ₁ − v ₂ =

1, completely elastic 0, completely in-elastic If v 2 = 0 and m 1 m 2 then v ₁

⁰

= −v 1 .

If v 2 = 0 and m 1 m 2 then v ₂

⁰

= 2v 1 .

Elastic collision with m 1 = m 2 : v ₁

⁰

= v 2 and v ₂

⁰

= v 1 .

1.6: Rigid Body Dynamics

Angular velocity: ω av = ^∆θ _∆t , ω = ^dθ _dt , ~ v = ~ ω × ~ r Angular Accel.: α av = ^∆ω _∆t , α = ^dω _dt , ~a = ~ α × ~ r

Rotation about an axis with constant α:

ω = ω ₀ + αt, θ = ωt + ¹ ₂ αt ² , ω ² − ω ₀ ² = 2αθ

Moment of Inertia: I = P

i m _i r _i ² , I = R r ² dm

ring mr

²

disk

1 2

mr

²

shell

2 3

mr

²

sphere

2 5

mr

²

rod

1 12

ml

²

hollow mr

²

solid

1 2

mr

²

rectangle

m(a2 +b2 ) 12

a b

Theorem of Parallel Axes: I

_k

= I _cm + md ²

cm I

k

d I

c

Theorem of Perp. Axes: I z = I x + I y

x z y

Radius of Gyration: k = p I/m

Angular Momentum: ~ L = ~ r × ~ p, L ~ = I~ ω Torque: ~ τ = ~ r × F , ~ ~ τ = ^d _dt ^{L ~} , τ = Iα

O x

y P

~ r F ~ θ

Conservation of ~ L: ~ τ _ext = 0 = ⇒ ~ L = const.

Equilibrium condition: P F ~ = ~ 0, P

~ τ = ~ 0 Kinetic Energy: K rot = ¹ ₂ Iω ²

Dynamics:

~ τ _cm = I _cm ~ α, F ~ _ext = m~a _cm , ~ p _cm = m~ v _cm K = ¹ ₂ mv cm 2 + ¹ ₂ I cm ω ² , L ~ = I cm ~ ω + ~ r cm × m~ v cm

1.7: Gravitation

Gravitational force: F = G ^m

¹

_r ^m

2 ²

m

1

F F m

2

r

Potential energy: U = − ^{GM m} _r Gravitational acceleration: g = ^GM _R

2

Variation of g with depth: g inside ≈ g 1 − ^2h _R Variation of g with height: g _outside ≈ g 1 − _R ^h Effect of non-spherical earth shape on g:

g _{at pole} > g _{at equator} ( ∵ R _e − R _p ≈ 21 km)

Effect of earth rotation on apparent weight:

mg _θ

⁰

=mg − mω ² R cos ² θ

R

mω

²

R cos θ mg

~ ω

θ

Orbital velocity of satellite: v o = q GM

R

Escape velocity: v _e = q

2GM R

Kepler’s laws:

v

o

a

First: Elliptical orbit with sun at one of the focus.

Second: Areal velocity is constant. ( ∵ d ~ L/dt = 0).

Third: T ² ∝ a ³ . In circular orbit T ² = _GM ^4π

²

a ³ .

1.8: Simple Harmonic Motion

Hooke’s law: F = −kx (for small elongation x.) Acceleration: a = ^d _dt

²

^x

2

= − _m ^k x = −ω ² x

Time period: T = ^2π _ω = 2π p m k

Displacement: x = A sin(ωt + φ) Velocity: v = Aω cos(ωt + φ) = ±ω √

A ² − x ²

Potential energy: U = ¹ ₂ kx ²

−A U 0 A x

Kinetic energy K = ¹ ₂ mv ²

−A K 0 A x

Total energy: E = U + K = ¹ ₂ mω ² A ²

Simple pendulum: T = 2π q

l

g l

Physical Pendulum: T = 2π q

I mgl

Torsional Pendulum T = 2π q I

k

Springs in series: _k ¹

eq

= _k ¹

1

+ _k ¹

2

k

1

k

2

Springs in parallel: k eq = k 1 + k 2

k

1

k

2

Superposition of two SHM’s:

A ~

1

A ~

2

A ~ δ

x ₁ = A ₁ sin ωt, x ₂ = A ₂ sin(ωt + δ) x = x 1 + x 2 = A sin(ωt + )

A = q

A 1 2 + A 2 2 + 2A 1 A 2 cos δ tan = A 2 sin δ

A ₁ + A ₂ cos δ

1.9: Properties of Matter

Modulus of rigidity: Y = ^{F /A} _∆l/l , B = −V _∆V ^∆P , η = _Aθ ^F Compressibility: K = _B ¹ = − _V ^{1 dV} _dP

Poisson’s ratio: σ = lateral strain

longitudinal strain = ^∆D/D _∆l/l Elastic energy: U = ¹ ₂ stress × strain × volume

Surface tension: S = F/l Surface energy: U = SA Excess pressure in bubble:

∆p _air = 2S/R, ∆p _soap = 4S/R Capillary rise: h = ^2S _rρg ^{cos θ}

Hydrostatic pressure: p = ρgh

Buoyant force: F B = ρV g = Weight of displaced liquid Equation of continuity: A 1 v 1 = A 2 v 2 v

1

v

2

Bernoulli’s equation: p + ¹ ₂ ρv ² + ρgh = constant Torricelli’s theorem: v efflux = √

2gh Viscous force: F = −ηA ^dv _dx

Stoke’s law: F = 6πηrv

F

v

Poiseuilli’s equation: Volume flow time = ^πpr _8ηl

⁴

l r

Terminal velocity: v t = ^2r

²

^(ρ−σ)g _9η

2 Waves

2.1: Waves Motion

General equation of wave: ^∂ _∂x

²

^y

₂

= _v ¹

₂

^∂ _∂t

²₂

^y .

Notation: Amplitude A, Frequency ν, Wavelength λ, Pe- riod T , Angular Frequency ω, Wave Number k,

T = 1 ν = 2π

ω , v = νλ, k = 2π λ Progressive wave travelling with speed v:

y = f(t − x/v), +x; y =f (t + x/v), −x

Progressive sine wave:

^λ2

^λ

x y

A

y = A sin(kx − ωt) = A sin(2π (x/λ − t/T ))

2.2: Waves on a String

Speed of waves on a string with mass per unit length µ and tension T : v = p

T /µ

Transmitted power: P av = 2π ² µvA ² ν ² Interference:

y ₁ = A ₁ sin(kx − ωt), y ₂ = A ₂ sin(kx − ωt + δ) y = y 1 + y 2 = A sin(kx − ωt + )

A = q

A 1 2

+ A 2 2

+ 2A 1 A 2 cos δ tan = A ₂ sin δ

A 1 + A 2 cos δ δ =

2nπ, constructive;

(2n + 1)π, destructive.

Standing Waves:

2Acoskx

A N A N A

x λ/4

y 1 = A 1 sin(kx − ωt), y 2 = A 2 sin(kx + ωt) y = y 1 + y 2 = (2A cos kx) sin ωt

x =

n + ¹ _{2 λ}

2 , nodes; n = 0, 1, 2, . . . n ^λ ₂ , antinodes. n = 0, 1, 2, . . .

String fixed at both ends:

L

N A N A N

λ/2

1. Boundary conditions: y = 0 at x = 0 and at x = L 2. Allowed Freq.: L = n ^λ ₂ , ν = _2L ⁿ q

T

µ , n = 1, 2, 3, . . ..

3. Fundamental/1 ^st harmonics: ν 0 = _2L ¹ q

T µ

4. 1 ^st overtone/2 ^nd harmonics: ν ₁ = _2L ² q _T

µ

5. 2 ^nd overtone/3 ^rd harmonics: ν 2 = _2L ³ q

T µ

6. All harmonics are present.

String fixed at one end:

L

N A N A

λ/2

1. Boundary conditions: y = 0 at x = 0

2. Allowed Freq.: L = (2n + 1) ^λ ₄ , ν = ²ⁿ⁺¹ _4L q

T µ , n = 0, 1, 2, . . ..

3. Fundamental/1 ^st harmonics: ν 0 = _4L ¹ q

T µ

4. 1 ^st overtone/3 ^rd harmonics: ν ₁ = _4L ³ q

T µ

5. 2 ^nd overtone/5 ^th harmonics: ν 2 = _4L ⁵ q

T µ

6. Only odd harmonics are present.

Sonometer: ν ∝ _L ¹ , ν ∝ √

T , ν ∝

^√

¹ _µ . ν = _2L ⁿ q

T µ

2.3: Sound Waves

Displacement wave: s = s ₀ sin ω(t − x/v)

Pressure wave: p = p 0 cos ω(t − x/v), p 0 = (Bω/v)s 0

Speed of sound waves:

v liquid = s B

ρ , v solid = s Y

ρ , v gas = s γP

ρ

Intensity: I = ^2π _v

²

^B s ₀ ² ν ² = ^p _2B

⁰²

^v = ^p _2ρv

⁰²

Standing longitudinal waves:

p 1 = p 0 sin ω(t − x/v), p 2 = p 0 sin ω(t + x/v) p = p ₁ + p ₂ = 2p ₀ cos kx sin ωt

Closed organ pipe: ^L

1. Boundary condition: y = 0 at x = 0

2. Allowed freq.: L = (2n + 1) ^λ ₄ , ν = (2n + 1) _4L ^v , n = 0, 1, 2, . . .

3. Fundamental/1 ^st harmonics: ν 0 = _4L ^v

4. 1 ^st overtone/3 ^rd harmonics: ν 1 = 3ν 0 = _4L ^3v

5. 2 ^nd overtone/5 ^th harmonics: ν 2 = 5ν 0 = _4L ^5v 6. Only odd harmonics are present.

Open organ pipe: _L

A N A N A

1. Boundary condition: y = 0 at x = 0

Allowed freq.: L = n ^λ ₂ , ν = n _4L ^v , n = 1, 2, . . . 2. Fundamental/1 ^st harmonics: ν 0 = _2L ^v

3. 1 ^st overtone/2 ^nd harmonics: ν 1 = 2ν 0 = _2L ^2v 4. 2 ^nd overtone/3 ^rd harmonics: ν 2 = 3ν 0 = _2L ^3v 5. All harmonics are present.

Resonance column:

l

1

+ d l

2

+ d

l ₁ + d = ^λ ₂ , l ₂ + d = ^3λ ₄ , v = 2(l ₂ − l ₁ )ν Beats: two waves of almost equal frequencies ω ₁ ≈ ω ₂

p 1 = p 0 sin ω 1 (t − x/v), p 2 = p 0 sin ω 2 (t − x/v) p = p ₁ + p ₂ = 2p ₀ cos ∆ω(t − x/v) sin ω(t − x/v) ω = (ω 1 + ω 2 )/2, ∆ω = ω 1 − ω 2 (beats freq.) Doppler Effect:

ν = v + u o

v − u _s ν 0

where, v is the speed of sound in the medium, u ₀ is the speed of the observer w.r.t. the medium, consid- ered positive when it moves towards the source and negative when it moves away from the source, and u s

is the speed of the source w.r.t. the medium, consid- ered positive when it moves towards the observer and negative when it moves away from the observer.

2.4: Light Waves

Plane Wave: E = E 0 sin ω(t − ^x _v ), I = I 0

Spherical Wave: E = ^aE _r

⁰

sin ω(t − ^r _v ), I = ^I _r

⁰2

Young’s double slit experiment

Path difference: ∆x = ^dy _D

S

1

P

S

₂

d

y

D

θ

Phase difference: δ = ^2π _λ ∆x

Interference Conditions: for integer n, δ =

2nπ, constructive;

(2n + 1)π, destructive,

∆x =

nλ, constructive;

n + ¹ ₂

λ, destructive Intensity:

I = I 1 + I 2 + 2 p

I 1 I 2 cos δ, I _max = p

I ₁ + p I ₂ 2

, I _min = p I ₁ − p

I ₂ 2

I 1 = I 2 : I = 4I 0 cos ^{2 δ} ₂ , I max = 4I 0 , I min = 0 Fringe width: w = ^λD _d

Optical path: ∆x

⁰

= µ∆x

Interference of waves transmitted through thin film:

∆x = 2µd =

nλ, constructive;

n + ¹ ₂

λ, destructive.

Diffraction from a single slit: b θ

y y D

For Minima: nλ = b sin θ ≈ b(y/D) Resolution: sin θ = ^1.22λ _b

Law of Malus: I = I 0 cos ² θ _I

₀

^θ _I

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3 Optics

3.1: Reflection of Light

Laws of reflection: ^normal

incident i r reflected (i) Incident ray, reflected ray, and normal lie in the same plane (ii) ∠ i = ∠ r

Plane mirror:

d d

(i) the image and the object are equidistant from mir- ror (ii) virtual image of real object

Spherical Mirror: ^O

I

u v f

1. Focal length f = R/2 2. Mirror equation: ¹ _v + _u ¹ = _f ¹ 3. Magnification: m = − _u ^v

3.2: Refraction of Light

Refractive index: µ = speed of light in vacuum speed of light in medium = ^c _v

Snell’s Law: _sin ^sin _r ⁱ = ^µ _µ

²

1

µ

1

µ

2

incident

refracted reflected i

r

Apparent depth: µ = apparent depth ^{real depth} = _d ^d

0

O I d d

⁰

Critical angle: θ _c = sin

^{−1 1}

_µ

θ

c

µ

Deviation by a prism:

µ δ

i i

⁰

A

r r

⁰

δ = i + i

⁰

− A, general result µ = sin ^A+δ ₂

^m

sin ^A ₂ , i = i

⁰

for minimum deviation δ m = (µ − 1)A, for small A

i δ

δ

_m

i

⁰

Refraction at spherical surface:

P O Q

µ

₁

µ

₂

u v

µ ₂ v − µ ₁

u = µ ₂ − µ ₁

R , m = µ ₁ v µ 2 u

Lens maker’s formula: _f ¹ = (µ − 1) h

1 R

₁

− _R ¹

2

i

Lens formula: ¹ _v − _u ¹ = ¹ _f , m = ^v _u

f

u v

Power of the lens: P = _f ¹ , P in diopter if f in metre.

Two thin lenses separated by distance d:

1 F = 1

f ₁ + 1 f ₂ − d

f ₁ f ₂

f

₁

f

₂

d

3.3: Optical Instruments

Simple microscope: m = D/f in normal adjustment.

Compound microscope: ^O

^∞

Objective Eyepiece

u v f

e

D

1. Magnification in normal adjustment: m = ^v _u ^D _f

e

2. Resolving power: R = _∆d ¹ = ^{2µ sin} _λ ^θ

Astronomical telescope:

f

o

f

e

1. In normal adjustment: m = − ^f _f

^o

e

, L = f o + f e

2. Resolving power: R = _∆θ ¹ = _1.22λ ¹

3.4: Dispersion

Cauchy’s equation: µ = µ 0 + _λ ^A

2

, A > 0 Dispersion by prism with small A and i:

1. Mean deviation: δ _y = (µ _y − 1)A 2. Angular dispersion: θ = (µ _v − µ _r )A Dispersive power: ω = ^µ _µ

^v−µr

y−1

≈ _δ ^θ

y

(if A and i small) Dispersion without deviation:

µ µ

⁰

A

A

⁰

(µ _y − 1)A + (µ

⁰

_y − 1)A

⁰

= 0 Deviation without dispersion:

(µ v − µ r )A = (µ

⁰

_v − µ

⁰

_r )A

⁰

4 Heat and Thermodynamics

4.1: Heat and Temperature

Temp. scales: F = 32 + ⁹ ₅ C, K = C + 273.16

Ideal gas equation: pV = nRT , n : number of moles van der Waals equation: p + _V ^a

₂

(V − b) = nRT Thermal expansion: L = L 0 (1 + α∆T ),

A = A 0 (1 + β ∆T ), V = V 0 (1 + γ∆T ), γ = 2β = 3α Thermal stress of a material: ^F _A = Y ^∆l _l

4.2: Kinetic Theory of Gases General: M = mN _A , k = R/N _A

Maxwell distribution of speed:

v n

v

p

v v ¯

rms

RMS speed: v _rms = q

3kT m = q

3RT M

Average speed: v ¯ = q 8kT

πm = q 8RT

πM

Most probable speed: v _p = q

2kT m

Pressure: p = ¹ ₃ ρv ² _rms

Equipartition of energy: K = ¹ ₂ kT for each degree of freedom. Thus, K = ^f ₂ kT for molecule having f de- grees of freedoms.

Internal energy of n moles of an ideal gas is U = ^f ₂ nRT .

4.3: Specific Heat Specific heat: s = _m∆T ^Q Latent heat: L = Q/m

Specific heat at constant volume: C v = _n∆T ^∆Q _V Specific heat at constant pressure: C p = _n∆T ^∆Q

_p Relation between C p and C v : C p − C v = R Ratio of specific heats: γ = C p /C v

Relation between U and C _v : ∆U = nC _v ∆T Specific heat of gas mixture:

C _v = n ₁ C _v1 + n ₂ C _v2 n 1 + n 2

, γ = n ₁ C _p1 + n ₂ C _p2 n 1 C v1 + n 2 C v2

Molar internal energy of an ideal gas: U = ^f ₂ RT , f = 3 for monatomic and f = 5 for diatomic gas.

4.4: Theromodynamic Processes

First law of thermodynamics: ∆Q = ∆U + ∆W Work done by the gas:

∆W = p∆V, W = Z V

2

V

₁

pdV W isothermal = nRT ln

V 2

V ₁

W isobaric = p(V 2 − V 1 ) W adiabatic = p ₁ V ₁ − p ₂ V ₂

γ − 1 W _isochoric = 0

Efficiency of the heat engine:

T

₁

T

₂

Q

₁

Q

₂

W

η = work done by the engine

heat supplied to it = Q 1 − Q 2

Q 1

η carnot = 1 − Q 2

Q ₁ = 1 − T 2

T ₁

Coeff. of performance of refrigerator:

T

1

T

₂

Q

1

Q

₂

W

COP = ^Q _W

²

= _Q ^Q

²

1−Q₂

Entropy: ∆S = ^∆Q _T , S _f − S _i = R f i

∆Q T

Const. T : ∆S = ^Q _T , Varying T : ∆S = ms ln ^T _T

^f

i

Adiabatic process: ∆Q = 0, pV ^γ = constant 4.5: Heat Transfer

Conduction: ^∆Q _∆t = −KA ^∆T _x Thermal resistance: R = _KA ^x

R series = R 1 + R 2 = _A ¹

x

1

K

₁

+ _K ^x

²

2

x

1

A x

2

K

1

K

2

1

R

_parallel

= _R ¹

1

+ _R ¹

2

= _x ¹ (K 1 A 1 + K 2 A 2 ) _K

₁

K

2

x A

1

A

2

Kirchhoff ’s Law: emissive power

absorptive power = ^E _a

^body

body

= E _blackbody

Wien’s displacement law: λ m T = b

λ E

λ

λ

_m

Stefan-Boltzmann law: ^∆Q _∆t = σeAT ⁴

Newton’s law of cooling: ^dT _dt = −bA(T − T 0 )

5 Electricity and Magnetism

5.1: Electrostatics Coulomb’s law: F ~ = _4π ¹

0

q

₁

q

₂

r

²

r ˆ _q

₁

_{r q}

₂

Electric field: E(~ ~ r) = _4π ¹

0

q r

²

r ˆ

~ r E ~ q

Electrostatic energy: U = − _4π ¹

0

q

₁

q

₂

r

Electrostatic potential: V = _4π ¹

0

q r

dV = − E ~ · ~ r, V (~ r) = − Z ~ r

∞

E ~ · d~ r

Electric dipole moment: ~ p = q ~ d _−q ^{~ p} _+q

d

Potential of a dipole: V = _4π ¹

0

p cos θ r

²

~ p

r V (r) θ

Field of a dipole:

~ p

r E

r

E

_θ

θ

E r = _4π ¹

0

2p cos θ

r

³

, E θ = _4π ¹

0

p sin θ r

³

Torque on a dipole placed in E: ~ ~ τ = p ~ × E ~ Pot. energy of a dipole placed in E: ~ U = −~ p · E ~

5.2: Gauss’s Law and its Applications Electric flux: φ = H E ~ · d S ~

Gauss’s law: H E ~ · d S ~ = q _in / ₀

Field of a uniformly charged ring on its axis:

E P = _4π ¹

0

qx (a

²

+x

²

)

^3/2

a

q x P E ~

E and V of a uniformly charged sphere:

E = ( ₁

4π

0

Qr

R

³

, for r < R

1 4π

₀

Q

r

²

, for r ≥ R _O

R r

E

V = ( _Q

8π

₀

R 3 − _R ^r

²2

, for r < R

1 4π

0

Q

r , for r ≥ R

O R r

V

E and V of a uniformly charged spherical shell:

E =

0, for r < R

1 4π

0

Q

r

²

, for r ≥ R

O R r

E

V = ( ₁

4π

0

Q

R , for r < R

1 4π

₀

Q

r , for r ≥ R

O R r

V

Field of a line charge: E = _2π ^λ

0

r

Field of an infinite sheet: E = ₂ ^σ

0

Field in the vicinity of conducting surface: E = ^σ

0

5.3: Capacitors Capacitance: C = q/V

Parallel plate capacitor: C = 0 A/d

A

−q A +q d

Spherical capacitor: C = ^4π _r

⁰

^r

¹

^r

²

2−r1

r

1

r

2

−q +q

Cylindrical capacitor: C = _ln(r ^2π

⁰

^l

2

/r

₁

) l

r

1

r

2

Capacitors in parallel: C _eq = C ₁ + C ₂ ^A _C

₂

B C

₁

Capacitors in series: _C ¹

eq

= _C ¹

1

+ _C ¹

2

C

₁

C

₂

A B

Force between plates of a parallel plate capacitor:

F = _2A ^Q

²

0

Energy stored in capacitor: U = ¹ ₂ CV ² = ^Q _2C

²

= ¹ ₂ QV Energy density in electric field E: U/V = ¹ ₂ 0 E ² Capacitor with dielectric: C =

⁰

^KA _d

5.4: Current electricity Current density: j = i/A = σE Drift speed: v d = ¹ ₂ ^eE _m τ = _neA ⁱ

Resistance of a wire: R = ρl/A, where ρ = 1/σ Temp. dependence of resistance: R = R ₀ (1 + α∆T ) Ohm’s law: V = iR

Kirchhoff ’s Laws: (i) The Junction Law: The algebraic sum of all the currents directed towards a node is zero i.e., Σ node I i = 0. (ii)The Loop Law: The algebraic sum of all the potential differences along a closed loop in a circuit is zero i.e., Σ loop ∆ V i = 0.

Resistors in parallel: _R ¹

eq

= _R ¹

1

+ _R ¹

2

R

₂

A B

R

₁

Resistors in series: R eq = R 1 + R 2 R

₁

R

₂

A B

Wheatstone bridge:

R

₁

R

₂

R

₄

R

₃

V

↑

G

Balanced if R ₁ /R ₂ = R ₃ /R ₄ .

Electric Power: P = V ² /R = I ² R = IV

Galvanometer as an Ammeter:

i i

g

G

S i

−

i

g

i

i _g G = (i − i _g )S

Galvanometer as a Voltmeter:

A i

_g

R G

↑

B

V AB = i g (R + G)

Charging of capacitors:

R C

V

q(t) = CV h

1 − e

^−RCt

i

Discharging of capacitors: q(t) = q ₀ e

^−RCt

C q(t)

R

Time constant in RC circuit: τ = RC

Peltier effect: emf e = ^∆H _∆Q = Peltier heat charge transferred . Seeback effect:

T

0

T

n

T

i

T e

1. Thermo-emf: e = aT + ¹ ₂ bT ²

2. Thermoelectric power: de/dt = a + bT . 3. Neutral temp.: T _n = −a/b.

4. Inversion temp.: T i = −2a/b.

Thomson effect: emf e = ^∆H _∆Q = Thomson heat

charge transferred = σ∆T . Faraday’s law of electrolysis: The mass deposited is

m = Zit = _F ¹ Eit

where i is current, t is time, Z is electrochemical equiv- alent, E is chemical equivalent, and F = 96485 C/g is Faraday constant.

5.5: Magnetism

Lorentz force on a moving charge: F ~ = q~ v × B ~ + q ~ E Charged particle in a uniform magnetic field:

r v q

⊗

B ~

r = ^mv _qB , T = ^2πm _qB

Force on a current carrying wire:

i

~ l B ~

F ~

F ~ = i ~l × B ~

Magnetic moment of a current loop (dipole):

i

~

µ A ~ ~ µ = i ~ A

Torque on a magnetic dipole placed in B: ~ ~ τ = ~ µ× B ~

Energy of a magnetic dipole placed in B: ~ U = −~ µ · B ~

Hall effect: V _w = _ned ^Bi

i

l d

w x

y z B ~

5.6: Magnetic Field due to Current Biot-Savart law: d B ~ = ^µ _4π

⁰

^{i d} _r ^{~ l×~}

3

^r

i d ~ l

θ

~ r

⊗

B ~

Field due to a straight conductor: _i d

⊗

B ~ θ

₁

θ

₂

B = _4πd ^µ

⁰

ⁱ (cos θ ₁ − cos θ ₂ )

Field due to an infinite straight wire: B = _2πd ^µ

⁰

ⁱ Force between parallel wires: ^dF _dl = ^µ

⁰

_2πd ⁱ

¹

ⁱ

²

ⁱ

¹

ⁱ

²

d

Field on the axis of a ring: ^a B ~ P d i

B P = _2(a

₂

^µ _+d

⁰

^ia

₂²

₎

_3/2

Field at the centre of an arc: B = ^µ _4πa

⁰

^iθ B ~

i a a θ

Field at the centre of a ring: B = ^µ _2a

⁰

ⁱ Ampere’s law: H B ~ · d~l = µ ₀ I _in

Field inside a solenoid: B = µ ₀ ni, n = ^N _l

l

Field inside a toroid: B = ^µ _2πr

⁰

^{N i}

r

Field of a bar magnet: _~

B

₁

B ~

2

d d N S

B ₁ = ^µ _4π

⁰

^2M _d

₃

, B ₂ = ^µ _4π

⁰

^M _d

₃

Angle of dip: B h = B cos δ

B B

h

B

_v

Horizontal δ

Tangent galvanometer: B _h tan θ = ^µ _2r

⁰

ⁿⁱ , i = K tan θ Moving coil galvanometer: niAB = kθ, i = _nAB ^k θ Time period of magnetometer: T = 2π q

I M B

h

Permeability: B ~ = µ ~ H

5.7: Electromagnetic Induction Magnetic flux: φ = H B ~ · d S ~ Faraday’s law: e = − ^dφ _dt

Lenz’s Law: Induced current create a B-field that op- poses the change in magnetic flux.

Motional emf: e = Blv

−

+

~ v

l

⊗

B ~

Self inductance: φ = Li, e = −L ^di _dt

Self inductance of a solenoid: L = µ 0 n ² (πr ² l) Growth of current in LR circuit: i = _R ^e h

1 − e

^−L/Rt

i

e

L R

S i

t i

L R

0.63

_R^e

Decay of current in LR circuit: i = i ₀ e

^−L/Rt

L R

S i

t i

i

₀

L R

0.37i

0

Time constant of LR circuit: τ = L/R Energy stored in an inductor: U = ¹ ₂ Li ² Energy density of B field: u = ^U _V = _2µ ^B

²

0

Mutual inductance: φ = M i, e = −M _dt ^di

EMF induced in a rotating coil: e = N ABω sin ωt

Alternating current: ⁱ _t

T

i = i ₀ sin(ωt + φ), T = 2π/ω Average current in AC: ¯ i = _T ¹ R T

0 i dt = 0 RMS current: i rms = h

1 T

R T

0 i ² dt i ^1/2

=

^√

ⁱ

⁰

2 t

i

²

T

Energy: E = i rms 2

RT

Capacitive reactance: X c = _ωC ¹ Inductive reactance: X _L = ωL Imepedance: Z = e 0 /i 0

RC circuit: i

C R

e

₀

sin ˜ ωt ^R

1 ωC

Z φ

Z = p

R ² + (1/ωC) ² , tan φ = _ωCR ¹

LR circuit: ⁱ

L R

e

₀

sin ˜ ωt

R ωL

Z φ

Z = √

R ² + ω ² L ² , tan φ = ^ωL _R

LCR Circuit: _i

L C R

e

₀

sin ˜ ωt ^R

1 ωC

ωL

Z

₁

ωC−

ωL φ

Z = q

R ² + _ωC ¹ − ωL ²

, tan φ =

^ωC1

_R

^−ωL

ν resonance = _2π ¹

q 1 LC

Power factor: P = e rms i rms cos φ Transformer: ^N _N

¹

2

= ^e _e

¹

2

, e 1 i 1 = e 2 i 2

i

₁

N

₁

i

₂

N

₂

e

₁

˜ ˜ ^e

²

Speed of the EM waves in vacuum: c = 1/ √ µ 0 0

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6 Modern Physics

6.1: Photo-electric effect Photon’s energy: E = hν = hc/λ Photon’s momentum: p = h/λ = E/c

Max. KE of ejected photo-electron: K _max = hν − φ Threshold freq. in photo-electric effect: ν 0 = φ/h

Stopping potential: V o = ^hc _e ¹ _λ

− ^φ _e

1 λ

V

0 φ hc

hc e

−^φ_e

de Broglie wavelength: λ = h/p

6.2: The Atom

Energy in nth Bohr’s orbit:

E _n = − mZ ² e ⁴

8 ₀ ² h ² n ² , E _n = − 13.6Z ² n ² eV Radius of the nth Bohr’s orbit:

r _n = 0 h ² n ²

πmZe ² , r _n = n ² a 0

Z , a ₀ = 0.529 ˚ A Quantization of the angular momentum: l = ^nh _2π Photon energy in state transition: E ₂ − E ₁ = hν

E

1

E

₂

hν

Emission E

1

E

₂

hν Absorption

Wavelength of emitted radiation: for a transition from nth to mth state:

1

λ = RZ ² 1

n ² − 1 m ²

X-ray spectrum: λ min = _eV ^hc

λ I

λ

_min

λ

_α

K

α

K

_β

Moseley’s law: √

ν = a(Z − b) X-ray diffraction: 2d sin θ = nλ Heisenberg uncertainity principle:

∆p∆x ≥ h/(2π), ∆E∆t ≥ h/(2π)

6.3: The Nucleus

Nuclear radius: R = R ₀ A ^1/3 , R ₀ ≈ 1.1 × 10

⁻¹⁵

m Decay rate: ^dN _dt = −λN

Population at time t: N = N 0 e

^−λt

O t

N

₀

N

N0 2

t

_1/2

Half life: t _1/2 = 0.693/λ Average life: t _av = 1/λ

Population after n half lives: N = N 0 /2 ⁿ . Mass defect: ∆m = [Zm p + (A − Z)m n ] − M Binding energy: B = [Zm _p + (A − Z)m _n − M ] c ² Q-value: Q = U i − U f

Energy released in nuclear reaction: ∆E = ∆mc ² where ∆m = m reactants − m products .

6.4: Vacuum tubes and Semiconductors Half Wave Rectifier:

˜

D

R Output

Full Wave Rectifier:

˜ ^Output

Triode Valve:

Filament Plate

Grid Cathode

Plate resistance of a triode: r p = ^∆V _∆i

^p

p

_∆V

g

=0

Transconductance of a triode: g m = _∆V ^∆i

^p

g

_∆V

p

=0

Amplification by a triode: µ = − ^∆V _∆V

^p

g

_∆i

p

=0

Relation between r p , µ, and g m : µ = r p × g m

Current in a transistor: I _e = I _b + I _c

I

c

I

_b

I

e

α and β parameters of a transistor: α = ^I _I

^c

e

, β =

I

_c

I

_b

, β = _1−α ^α

Transconductance: g m = _∆V ^∆I

^c

be

Logic Gates:

AND OR NAND NOR XOR

A B AB A+B AB A + B A ¯ B + ¯ AB

0 0 0 0 1 1 0

0 1 0 1 1 0 1

1 0 0 1 1 0 1

1 1 1 1 0 0 0