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मुक्त ज्ञानकोश विकिपीडिया से
यहाँ जाएँ: भ्रमण, खोज

एसआई उपसर्ग (प्रीफिक्स)[संपादित करें]

SI उपसर्ग
१०००n १०n उपसर्ग चिन्ह [१] से लागू संख्या दशमलव रूप SI writing style में
१००० १०२४ योट्टा- यो/Y 1991 दस जल्द १ ००० ००० ००० ००० ००० ००० ००० ०००
१००० १०२१ अं- अं/Z 1991 अंक १ ००० ००० ००० ००० ००० ००० ०००
१००० १०१८ एक्जा- ए/E 1975 दस शङ्ख 1 ००० ००० ००० ००० ००० ०००
१००० १०१५ पद्म- प/P 1975 पद्म १ ००० ००० ००० ००० ०००
१००० १०१२ टेरा- टे/T 1960 दस खरब १ ००० ००० ००० ०००
१००० १० अर्ब- अ/G 1960 अरब १ ००० ००० ०००
१००० १० अद- अद/M 1960 अदन्त १ ००० ०००
१००० १० सहस्र- स्र/k 1795 हजार १ ०००
१०००२/३ १० शत- स/h 1795 सौ १००
१०००१/३ १० दश- द/da 1795 दस १०
१०००0 १०0 (none) (none) NA एक
१०००−१/३ १०−१ दशि- दि/d 1795 Tenth ०.१
१०००−२/३ १०−२ शति- शि/c 1795 Hundredth ०.०१
१०००−१ १०−३ सहस्रि- स्रि/m 1795 Thousandth ०.०० १
१०००−२ १०−६ सूक्ष्म- सू/µ 1960[2] Millionth ०.००० ०००१
१०००−३ १०−९ अर्बि- इ/n 1960 Billionth ०.००० ००० ००१
१०००−४ १०−१२ फैम्टो- फ/p 1960 Trillionth ०.००० ००० ००० ००१
१०००−५ १०−१५ पद्मि- पि/f 1964 Quadrillionth ०.००० ००० ००० ००० ००१
१०००−६ १०−१८ एट्टो- a 1964 Quintillionth ०.००० ००० ००० ००० ००० ००१
१०००−७ १०−२१ अंकि- इं/z 1991 Sextillionth ०.००० ००० ००० ००० ००० ००० ००१
१०००−८ १०−२४ योक्टो- y 1991 Septillionth ०.००० ००० ००० ००० ००० ००० ००० ००१
Notes:
१. 1795 की तिथियों से उपरोक्त उपसर्ग प्रयोग में लाए जारहे हैं, जबसे मीट्रिक प्रणाली प्रयोग में आई थी । अन्य तिथियाँ आवश्यक रूप से प्रथम प्रयोग की नहीं रहीं हैं, बलकि CGPM के समझौते द्वारा मान्यता की तिथि हैं जो कि 1889 में हुआ था ।
2. 2. "मैक्रॉन" शब्द को CGPM नें १९४८ में अनुमोदित किया, पर १९६७-६८ में उसे रद्द कर दिया।

आधारभूत यांत्रिकी (Fundamentals of Mechanics)[संपादित करें]

Foundational equations in translation and rotation.

Quantity Translation Rotation
time t t
position x      \theta in radians
mass m m
duration \Delta t \Delta t
displacement \Delta x \Delta \theta
conservation of mass \Delta m = 0 \Delta m = 0
conservation of energy \Delta E = 0 \Delta E = 0
conservation of momentum \Delta P = 0 \Delta L = 0
velocity      v =      dx/dt \omega = d\theta/dt
acceleration      a =      dv/dt \alpha = d\omega/dt
jerk j =     da/dt j = d\alpha/dt
potential energy change \Delta U = -W \Delta U = -W
momentum P = mv      L = I\omega  =|| \mathbf{r} \times \mathbf{P}|| = m|| \mathbf{r} \times \mathbf{v}||
force f    = dP/dt = ma = -dU/dx   \tau = dL/dt = I\alpha        =|| \mathbf{r} \times \mathbf{f}||=m|| \mathbf{r} \times \mathbf{a}||
inertia m = \int     dm = \Sigma    m_i I = \int r^2 dm = \Sigma r^2m_i
impulse J=\int f    dt J=\int \tau dt
work W = \int f    dx       = \mathbf{d} \cdot \mathbf{f} W = \int \tau d\theta
power  P = dW/dt  = fv         P = dW/dt  = \tau\omega
kinetic energy K =      mv^2/2 = P^2/2m K = I \ w^2 / 2 = \Sigma    R^2m
Newton's Third Law    f_{ab} = -   f_{ba} \tau_{ab} = -\tau_{ba}

Every conservative force has a potential energy. By following two principles one can consistently assign a non-relative value to U:

  • Wherever the force is zero, its potential energy is defined to be zero as well.
  • Whenever the force does work, potential energy is lost.

स्थिर त्वरण (Constant acceleration)[संपादित करें]

Equations in translation and rotation, assuming constant acceleration.

Quantity Translation Rotation
displacement \Delta v = at \Delta \omega = \alpha t
time \Delta(v^2) = 2a\Delta x \Delta(\omega^2) = 2\alpha\Delta \theta
acceleration \Delta x = t\Delta v/2 \Delta \theta = t\Delta \omega/2
initial velocity \Delta x = -at^2/2 + v_2t \Delta \theta = -\alpha t^2/2 + \omega_2t
final velocity \Delta x = +at^2/2 + v_1t \Delta \theta = +\alpha t^2/2 + \omega_1t

एकसमान वृत्तीय गति (Uniform circular motion)[संपादित करें]

uniform circular motion angular to linear displacement x = \theta r
uniform circular motion angular to linear speed v = \theta \omega
uniform circular motion angular to linear acceleration normal component a_r = \omega^2r
uniform circular motion \mathbf{d} = \mathbf{i}cos\omega t + \mathbf{j}sin\omega t
uniform circular motion tangential speed \mathbf{v} = \mathbf{d}' = -\omega r (\mathbf{i}\sin\omega t - \mathbf{j}\cos\omega t)
uniform circular motion tangential component, scalar a_t = \alpha r
uniform circular motion centripetal acceleration \mathbf{a} = \mathbf{d}'' = -\omega^2\mathbf{d} = -v^2\mathbf{n}/r
uniform circular motion centripetal acceleration scalar \alpha=v^2/r
uniform circular motion centripetal force f = -mv^2/r
uniform circular motion revolution time T=2\pi r/v

Elasticity[संपादित करें]

elastic force, lies parallel to spring f = -kd
elastic potential energy U=kx^2/2
elastic work, positive when relaxes W = -k\Delta(x^2)/2

घर्षण (Friction)[संपादित करें]

normal force f_n = \mathbf{f}\cdot\mathbf{n}
static friction maximum, lies tangent to the surface f=\mu_sf_n
kinetic friction, lies tangent to the surface f=\mu_kf_n
drag force, tangent to the path f =\mu_d\rho a v^2/2
terminal velocity v_t=\sqrt{2fg/(\mu_d\rho A)}
friction creates heat and sound \Delta E = f_kd

प्रतिबाधा एवं विकृत्ति (Stress and strain)[संपादित करें]

stress
strain
modulus of elasticity \lambda = {stress}/{strain}
yield strength
ultimate strength
Young's modulus F/A = E\Delta L/L
shear modulus F/A = G\Delta x/L
bulk modulus F/A = B\Delta V/V

अन्य[संपादित करें]

inertial frames x_{PA} = x_{PB} + x_{AB}
. . . v_{PA} = v_{PB} + v_{AB}
. . . a_{PA} = a_{PB} + 0
trajectory y=x\tan\theta-gx^2/2(V_0\cos\theta)^2
flight distance v_0^2\sin{2\theta}/g
tension, lies within the cord f_t = f
mechanical energy  E_{mec}=K + U
mechanical energy is conserved  \Delta E_{mec} = 0 when all forces are conservative
thrust t = Rv_{rel}=ma
ideal rocket equation \Delta v = ln(m_i/m_f)v_{rel}
parallel axis theorem I = I_{com} + mr^2
list of moments of inertia
indeterminate systems

द्रब्यमान केन्द्र एवं संघट्ट (Center of mass and collisions)[संपादित करें]

center of mass COM \mathbf{r}_{com}=M^{-1}\Sigma m_i \mathbf{r}_i
. . . x_{com}=M^{-1}\int x dm, \cdots
for constant density: x_{com}=V^{-1}\int x dV, \cdots
COM is in all planes of symmetry
elastic collision \Delta E_k = 0
inelastic collision \Delta E_k = maximum
conservation of momentum in a two body collision \mathbf{P}_{1i}+\mathbf{P}_{2i}=\mathbf{P}_{1f}+\mathbf{P}_{2f}
system COM remains inert \mathbf{v}_{com}={(\mathbf{P}_{1i}+\mathbf{P}_{2i})\over(M_1+M_2)} = const
elastic collision, 1D, M2 stationary v_{1f}={(m_1 - m_2)\over(m_1 + m_2)}v_{1i}
. . . v_{2f}={(2m_1)\over(m_1 + m_2)}v_{1i}

चिकने तल पर लुढ़कना (Smooth rolling)[संपादित करें]

rolling distance x_{arc}=R\theta
rolling distance ? x_{com}=R\alpha
rolling velocity v_{com}=R\omega
rolling ? K = I_{com}\omega^2/2 + Mv^2_{com}/2
rolling down a ramp along axis x a_{com,x}=-\frac{g\sin\theta}{1+I_{com}/MR^2}

उष्मागतिकी (Thermodynamics)[संपादित करें]

Zeroth Law of Thermodynamics (A = B) \and (B=C) \Rightarrow A=C
(where "=" denotes systems in thermal equilibrium
First Law of Thermodynamics \Delta E_{int} = Q + W
Second Law of Thermodynamics \Delta S \ge 0
Third Law of Thermodynamics S = S_{structural} + CT
temperature T
molecules N
degrees of freedom f
heat Q, \Delta E due to \Delta T (energy)
thermal mass (extensive property) C_{th} = Q/\Delta T
specific heat capacity (bulk property) c_{th} = Q/\Delta Tm
enthalpy of vaporization L_v = Q/m
enthalpy of fusion L_f = Q/m
thermal conductivity \kappa
thermal resistance R=L/ \kappa
thermal conduction rate P = Q/t = A(T_H - T_C)/R
thermal conduction rate through a composite slab P = Q/t = A(T_H - T_C)/\Sigma(R_i)
linear coefficient of thermal expansion  dL/dt = \alpha L
volume coefficient of thermal expansion dV/dt = 3 \alpha V
Boltzmann constant k (energy)/(temperature)
Stefan-Boltzmann constant \sigma (power)/(area)(temp)^4
thermal radiation P = \sigma \epsilon A T ^4_{sys}
thermal absorption P = \sigma \epsilon A T ^4_{env}
adiabatic \Delta Q = 0
ideal gas law PV = kTN
work, constant temperature W=kTNln(V_f/V_i)
work due to gas expansion W = \int_{i}^{f}pdV
. . . adiabatic \Delta E_{int} = W
. . . constant volume \Delta E_{int} = Q
. . . free expansion \Delta E_{int} = 0
. . . closed cycle Q + W = 0
work, constant volume W=0
work, constant pressure W=p\Delta V
translational energy E_{k,avg} = kTf/2
internal energy E_{int} = NkTf/2
mean speed v_{avg}= \sqrt{(kT/m)(8/\pi)}
mode speed v_{prb} = \sqrt{(kT/m)2}
root mean square speed v_{rms} = \sqrt{(kT/m)3}
mean free path \lambda = 1/(\sqrt{2} \pi d^2 N / V)?
Maxwell–Boltzmann distribution P(v)=4\pi(m/(2\pi kT))^{3/2}V^2e^{-(mv^2/(2kT))}
molecular specific heat at a constant volume C_V = Q/(N\Delta T)
? \Delta E_{int} = NC_V \Delta T
molecular specific heat at a constant pressure C_p = Q/(N\Delta T)
? W = p \Delta V = Nk \Delta T
? k = C_p - C_V
adiabatic expansion pV^{\gamma} = constant
adiabatic expansion TV^{\gamma - 1} = constant
multiplicity of configurations W = N!/n_1!n_2!
microstate in one half of the box n_1, n_2
Boltzmann's entropy equation S = klnW
irreversibility
entropy S = - k\sum_i P_i \ln P_i \!
entropy change \Delta S = \int_i^f(1/T)dQ \approx Q/T_{avg}
entropy change \Delta S = kNln(V_f/V_i) + NC_Vln(T_f/T_i)
entropic force f = -TdS/dx
engine efficiency \epsilon = |W|/|Q_H|
Carnot engine efficiency \epsilon_c = (|Q_H|-|Q_L|)/|Q_H| = (T_H-T_L)/T_H
refrigeration performance K = |Q_L|/|W|
Carnot refrigeration performance K_C = |Q_L|/(|Q_H|-|Q_L|) = T_L/(T_H-T_L)

तरंग[संपादित करें]

torsion constant \kappa = -\tau / \theta
phasor
node
antinode
period T
amplitude x_m
decibel dB
frequency f = 1/T = \omega /(2\pi)
angular frequency \omega = 2\pi f = 2\pi / T
phase angle \phi
phase (\omega t + \phi)
damping force f_d = -bv
phase ky -\omega t
wavenumber k
phase constant \phi
linear density \mu
harmonic number n
harmonic series f = v/\lambda = nv/(2L)
wavelength \lambda = k/(2\pi)
bulk modulus B = \Delta p /(\Delta V / V)
path length difference \Delta L
resonance \omega_d = \omega
phase difference \phi = 2 \pi \Delta L / \lambda
fully constructive interference \Delta L/\lambda = n
fully destructive interference \Delta L/\lambda = n+0.5
sound intensity I = P/A = \rho v \omega^2 s^2_m/2
sound power source P_s
sound intensity over distance I = P_s/(4\pi r^2)
sound intensity standard reference I_0
sound level \Beta = (10 dB)log(I/I_0)
pipe, two open ends f=v/\lambda = nv/(2L)
pipe, one open end f = v/\lambda = nv/(4L) for n odd
beats s(t) = [2s_m\cos\omega ' t ] \cos \omega t
beat frequency f_{beat} = f_1 - f_2
Doppler effect f' = f(v+-v_D)/(v+-v_S)
sonic boom angle \sin \theta = v/v_s
average wave power P_{avg}=\mu v \omega^2 x_m^2/2
pressure amplitude \Delta p_m = (v\rho \omega)x_m
wave equation \frac{\partial y}{\partial x^2} = \frac{1}{v^2} \frac{\partial ^2 y}{\partial t^2}
wave superposition x'(y,t) = x_1(y,t) + x_2(y,t)
wave speed v = \omega/k = \lambda/T = \lambda f
speed of sound v = \sqrt{B/ \rho }
wave speed on a stretched string v=\sqrt{f_t/\mu}
angular frequency of an angular simple harmonic oscillator \omega = \sqrt{I/\kappa}
angular frequency of a low amplitude simple pendulum \omega = \sqrt{L/g}
angular frequency of a low amplitude physical pendulum \omega = \sqrt{I/mgh}
angular frequency of a linear simple harmonic oscillator \omega = \sqrt{k/m}
angular frequency of a linear damped harmonic oscillator \omega ' = \sqrt{(k/m)-(b^2/4m^2)}
wave displacement x(t)=x_m\cos(\omega t + \phi)
wave displacement when damped x(t)=x_m\cos(\omega 't+\phi)(e^{-bt/2m})
wave velocity v(t)=x_m\sin(\omega t + \phi)(- \omega)
wave acceleration a(t)=x_m\cos(\omega t + \phi)(- \omega^2 )
transverse wave x(y,t) = x_m\sin(ky-\omega t)
wave traveling backwards x(y,t) = x_m\sin(ky+\omega t)
resultant wave x'(y,t) = x_m\sin(ky-\omega t + \phi/2)(2\cos\phi/2)
standing wave x'(y,t) = \cos(\omega t)(2y\sin ky)
sound displacement function x(y,t) = x_m\cos(ky-\omega t)
sound pressure-variation function \Delta p(y,t) = \sin(ky-\omega t)\Delta p_m
potential harmonic energy E_U(t) = kx^2/2 = kx_m^2\cos^2(\omega t + \phi)/2
kinetic harmonic energy E_K(t) = kx^2/2 = kx_m^2\sin^2(\omega t + \phi)/2
total harmonic energy E(t) = kx_m^2/2 = E_U + E_K
damped mechanical energy E_{mec}(t) = ke^{-bt/m}x^2_m/2

गुरुत्वाकर्षण (Gravitation)[संपादित करें]

gravitational constant G (force)(distance/mass)^2
gravitational force f_G = Gm_1m_2/r^2
superposition applies \mathbf{F} = \Sigma \mathbf{F}_i = \int d\mathbf{F}
gravitational acceleration a_g = Gm/r^2
free fall acceleration a_f = a_g - \omega^2R
shell theorem for gravitation
potential energy from gravity U = -Gm_1m_2/r \approx ma_gy
escape speed v = \sqrt{2Gm/r}
Kepler's law 1 planets move in an ellipse, with the star at a focus
Kepler's law 2 A'' = 0
Kepler's law 3 T^2 = (4\pi^2/Gm)r^3
orbital energy E = - Gm_1m_2/a2
standard gravity  a_g = Gm_{Earth}/r_{Earth}^2  \approx 9.81m/s^2
weight, points toward the center of gravity f_g = -f_n = mg
path independence W_{ab,1}=W_{ab,2}=\cdots
Einstein field equations R_{\mu \nu} - {1 \over 2}g_{\mu \nu}\,R + g_{\mu \nu} \Lambda = {8 \pi G \over c^4} T_{\mu \nu}

तरलगतिकी (Fluid dynamics)[संपादित करें]

density \rho = \Delta m / \Delta V
pressure p = \Delta F / \Delta A
pressure difference \Delta p = \rho g\Delta y
pressure at depth p = p_0 + \rho gh
barometer versus manometer
Pascal's principle
Archimedes' Principle
buoyant force F_b = m_fg
gravitational force when floating F_g = F_b
apparent weight weight_{app} = weight - F_b
ideal fluid
equation of continuity R_V = Av = constant
Bernoulli's equation p + \rho v^2/2 + \rho  gy = constant

विद्युतचुम्बकत्व (Electromagnetism)[संपादित करें]

Lorentz force \mathbf{F} = q (\mathbf{E} + \mathbf{v} \times \mathbf{B})
Gauss' law \oint\mathbf{E}\cdot d \mathbf{A} = \Phi_E = q_{enc}/\epsilon_0
Gauss' law for magnetic fields \oint \mathbf{B} \cdot d \mathbf{A} = \Phi_B = 0
Faraday's law of induction \oint\mathbf{E}\cdot d\mathbf{s} = -d\Phi_B/dt = -\mathcal{E}
Ampere-maxwell law \oint \mathbf{B} \cdot d\mathbf{s} = \mu_0(i_{enc} + i_{d,enc})
elementary charge e
electric charge q = ne
conservation of charge \Delta q = 0
linear charge density \lambda = q/l^1
surface charge density \sigma = q/l^2
volume charge density \rho = q/l^3
electric constant \epsilon_0 (time)^2(charge)^2/(mass)(volume)
magnetic constant \mu_0 (force)(time)^2/(charge)^2
Coulomb's law F = q_1q_2/(4\pi\epsilon_0)r^2
electric field \mathbf{E} =\mathbf{F}/q
electric field lines end at a negative charge
Gaussian surface \mathbf{A}
flux notation implies a normal unit vector \cdot d \mathbf{A} \to \cdot \mathbf{n} d \mathbf{A}
electric flux \Phi_E = \oint\mathbf{E}\cdot d \mathbf{A}
magnetic flux \Phi_B = \int \mathbf{B}\cdot d\mathbf{A}
magnetic flux given assumptions \Phi_B = BA
dielectric constant \kappa \ge 1
dielectric \epsilon_0 \to \epsilon_0\kappa
Gauss' law with dialectric q_{enc} = \epsilon_0 \oint \kappa\mathbf{E}\cdot d \mathbf{A}
Biot-Savart law  \mathbf{B} = \int\frac{\mu_0}{4\pi}\ \frac{(id\mathbf{s}) \times \mathbf{r}}{r^3},
Lenz's law induced current always opposes its cause
inductance (with respect to time) L=-\mathcal{E}/q''
inductance from coils L=N\Phi_B/i
inductance of a solenoid L/l=\mu_0n^2A
displacement current i_d = \epsilon_0 d\Phi_E/dt
displacement vector \mathbf{d}
electric dipole moment \mathbf{p} = q\mathbf{d}
electric dipole torque \mathbf{\tau}=\mathbf{p}\times\mathbf{E}
electric dipole potential energy U = -\mathbf{p}\cdot\mathbf{E}
magnetic dipole moment of a coil, magnitude only \mu=iNA
magnetic dipole moment torque \mathbf{\tau}=\mathbf{\mu}\times\mathbf{B}
magnetic dipole moment potential energy U=-\mathbf{\mu}\cdot\mathbf{B}
electric field accelerating a charged mass a = qE/m
electric field of a charged point E = q / \epsilon_0 4 \pi r^2 \hat{r}
electric field of a dipole moment E = p / \epsilon_0 2 \pi z^3
electric field of a charged line E = \lambda / \epsilon_0 2\pi r
electric field of a charged ring E = qz/\epsilon_04\pi(z^2 + R^2)^{3/2}
electric field of a charged conducting surface E = \sigma / \epsilon_0
electric field of a charged non-conducting surface E = \sigma /\epsilon_0 2
electric field of a charged disk E = \sigma (1 - z)/ \epsilon_0 2 \sqrt{z^2 + R^2}
electric field outside spherical shell r>=R E = q/\epsilon_0 4 \pi r^2
electric field inside spherical shell r<R E = 0
electric field of uniform charge r<=R E = qr/\epsilon_0 4 \pi R^3
electric field energy density u = \epsilon_0 E^2/2
electric potential versus electric potential energy (energy)/(charge) versus (energy)
electric potential energy U = - W_{\infty}
electric potential V = -W_{\infty}/q = U/q
electric potential difference \Delta V = -W/q = \Delta U/q
electric potential from electric field \Delta V = -\int_i^f \mathbf{E}\cdot d\mathbf{s}
electric field from electric potential \nabla V = -\mathbf{E}
electric potential of a charged point V = q/\epsilon_0 4 \pi r
electric potential of a set of charged points V = \Sigma V_i = (1/\epsilon_0 4 \pi) \Sigma q_i/r_i
electric potential of a dipole V = p\cos\theta/\epsilon_0 4 \pi r^2
electric potential of continuous charge V = \int dV = (1/\epsilon_0 4 \pi)\int dq/r
electric potential energy of a pair of charged points Vq_2 = U = W = q_1q_2/\epsilon_04\pi r
capacitance C = q/V (charge)^2/(energy)
capacitance of parallel plates C = \epsilon_0A/d
capacitance of a cylinder C = \epsilon_0 2 \pi L/\ln(b/a)
capacitance of a sphere C = \epsilon_0 4 \pi ba/(b-a)
capacitance of an isolated sphere C = \epsilon_0 4 \pi R
capacitors in parallel C_{eq}^{+1} = \Sigma C_i^{+1}
capacitors in series C_{eq}^{-1} = \Sigma C_i^{-1}
capacitor potential energy U=q^2/C2 = CV^2/2
current i = dq/dt
drift speed \mathbf{v}_d
current density \mathbf{J} = ne\mathbf{v}_d/m^3
current density magnitude J = i/A
current density to get current i = \int JdA
resistance R = V/i
resistivity \rho = \mathbf{E}/\mathbf{J}
resistivity temperature coefficient \alpha
resistivity across temperature \rho - \rho_0 = \rho_0\alpha(T-T_0)
resistivity and resistance R A = \rho L
electrical conductivity \sigma = 1/\rho = \mathbf{J}/\mathbf{E}
resistor power dissipation P = i^2R = V^2/R
internal resistance i = \mathcal{E}/(R+r)
resistors in series R_{eq}^{+1}=\Sigma R_i^{+1}
resistors in parallel R_{eq}^{-1} =\Sigma R_i^{-1}
Kirchoff's current law i_{in} = i_{out}
Ohm's law V=iR
emf \mathcal{E} = dW/dq = iR
emf rules loop, resistance, emf
electrical power P=iV
emf power P_{emf} = i\mathcal{E}
electric potential difference across a real battery p = \mathcal{E} - iR
magnetic field force on a moving charge \mathbf{F}_B = q\mathbf{v}\times\mathbf{B}
magnetic field force on a current \mathbf{F}_B=i\mathbf{L}\times\mathbf{B}
Hall effect n = Bi/Vle
circulating charged particle |q|vB=mv^2/r
cyclotron resonance condition f = f_{osc}
magnetic field of a line B = \mu_0i/2\pi R
magnetic field of a ray B=\mu_0i/4\pi R
magnetic field at the center of a circular arc B=\mu_0i\phi/4\pi R
magnetic field of a solenoid B=\mu_0in
magnetic field of a toroid B=\mu_0iN/2\pi r
magnetic field of a current carrying coil \mathbf{B}=\mu_0\mathbf{\mu}/2\pi z^3
self induction of emf \mathcal{E}_L = -Ldi/dt
magnetic energy U_B=Li^2/2
magnetic energy density u_B=B^2/2\mu_0
mutual induction \mathcal{E}_1=-Mdi_2/dt,\mathcal{E}_2=-Mdi_1/dt
transformation of voltage V_s N_p = V_p N_s
transformation of current I_s N_s = I_p N_p
transformation of reistance R_{eq} = (Np/Ns)^2R
induced magnetic field inside a circular capacitor B = (\mu_0i_d/2\pi R^2)r
induced magnetic field outside a circular capacitor B = \mu_0i_d/2\pi rr
RC circuit ODE with respect to time Rq' + C^{-1}q=\mathcal{E}
RC circuit capacitive time constant \tau = RC
RC circuit charging a capacitor q = C\mathcal{E}(1-e^{-t/RC})
RL circuit ODE with respect to time Li''+Ri'=\mathcal{E}
RL circuit time constant \tau_L=L/R
RL circuit rise of current i = \mathcal{E}/R(1-e^{-t/\tau_L})
RL circuit decay of current i=\mathcal{E}e^{-t/\tau_L}/R=i_0e^{-t/\tau_L}
LC circuit ODE with respect to time Lq''+C^{-1}q = \mathcal{E}
LC circuit \omega = 1/\sqrt{LC}
LC circuit charge q = Qcos(\omega t + \phi)
LC circuit current i=-\omega Q sin(\omega t + \phi)
LC circuit electrical potential energy U_E=q^2/2C=Q^2cos^2(\omega t + \phi)/2C
LC circuit magnetic potential energy U_B=Q^2sin^2(\omega t + \phi)/2C
RLC circuit ODE with respect to time Lq'' + Rq' +C^{-1}q = \mathcal{E}
RLC circuit charge q = QeT^{-Rt/2L}cos(\omega't+\phi)
resistive load V_R=I_RR
capacitive load V_C = I_C X_C
inductive load V_L = I_L X_L
resistive reactance X_R = ?
capacitive reactance X_C = 1/\omega_d C
inductive reactance X_L = \omega_d L
phase constant tan\phi=X_L - X_C /R
electromagnetic resonance \omega_d = \omega = 1/\sqrt{LC}
AC current I_{rms}=I/\sqrt{2}
AC voltage V_{rms}=V/\sqrt{2}
AC emf \mathcal{E}_{rms}=\mathcal{E}_m/\sqrt{2}
AC power P_{avg}=\mathcal{E}I_{rms}cos\phi

प्रकाश (Light)[संपादित करें]

electric light component E = E_m sin(kx-\omega t)
magnetic light component B = B_m sin(kx-\omega t)
speed of light c = 1/\sqrt{\mu_0\epsilon_0} = E/B
Poynting vector \mathbf{S} = \mu_0^{-1}\mathbf{E}\times\mathbf{B}
Poynting vector magnitude S = EB/\mu_0 = E^2/c\mu_0
rms electric field of light E_{rms} = E/\sqrt{2}
light intensity I = E^2_{rms}/c\mu_0
light intensity at the sphere I = P_s/4\pi r^2
radiation momentum with total absorption (inelastic) \Delta p = \Delta U/c
radiation momentum with total reflection (elastic) \Delta p = 2 \Delta U/c
radiation pressure with total absorption (inelastic) p_r = I/c
radiation pressure with total reflection (elastic) p_r = 2I/c
intensity from polarizing unpolarized light I = I_0/2
intensity from polarizing polarized light I = I_0cos^2\theta
index of refraction of substance f n_f = c/v_f
angle of reflection \theta_1=\theta_2
angle of refraction n_1sin\theta_1 = n_2sin\theta_2
angle of total reflection \theta_c = sin^{-1}n_2/n_1
angle of total polarisation \theta_B = tan^{-1}n_2/n_1
image distance in a plane mirror d_i = -d_o
image distance in a spherical mirror n_1/d_o + n_2/d_i = (n_2 - n_1)/r
spherical mirror focal length f =r/2
spherical mirror 1/d_o + 1/d_i = 1/f
lateral magnification m and h negative when upside down m=h_i/h_o = -d_i/d_o
lens focal length 1/f = 1/d_o +1/d_i
lens focal length from refraction indexes 1/f = (n_{lens}/n_{med}-1)(1/r_1 - 1/r_2)
path length difference \Delta L = d sin\theta
double slit minima d sin\theta = (N + 1/2)\lambda
double slit maxima d sin\theta = N\lambda
double-slit interference intensity I = 4I_0cos^2(\pi d sin\theta / \lambda)
thin film in air minima (N + 0/2)\lambda/n_2
thin film in air maxima 2L = (N + 1/2)\lambda/n_2
single-slit minima a sin \theta = N\lambda
single-slit intensity I(\theta)=I_0(sin\alpha/\alpha)^2
double slit intensity I(\theta) = I_0(cos^2\Beta)(sin\alpha/\alpha)^2
. . . \alpha = \pi a sin\theta/\lambda
circular aperture first minimum sin\theta = 1.22\lambda/d
Rayleigh's criterion \theta_R = 1.22\lambda/d
diffraction grating maxima lines dsin\theta = N\lambda
diffraction grating half-width \Delta\theta_{hw} = \lambda/Ndcos\theta
diffraction grating dispersion D=N/d cos\theta
diffraction grating resolving power R=Nn
diffraction grating lattice distance d = N\lambda/2sin\theta

विशिष्ट आपेक्षिकता (Special Relativity)[संपादित करें]

Lorentz factor \gamma = 1/\sqrt{1-(v/c)^2}
Lorentz transformation t' = \gamma(t-xv/c^2)
. . . x'=\gamma(x-vt)
. . . y' = y
. . . z' = z
time dilation \Delta t = \gamma \Delta t_0
length contraction L = L_0/\gamma
relativistic Doppler effect f=f_0\sqrt{1-(v/c)/1+(v/c)}
Doppler shift v=|\Delta\lambda|c/\lambda_0
momentum \mathbf{p}=\gamma m\mathbf{v}
rest energy E_0 = mc^2
total energy E = E_0 + K = mc^2 + K = \gamma mc^2 = \sqrt{(pc)^2 + (mc^2)^2}
Energy Removed Q = -\Delta mc^2
kinetic energy K = E - mc^2 = \gamma mc^2 - mc^2 = mc^2(\gamma -1)

कण भौतिकी (Particle Physics)[संपादित करें]

standard model see 4x4 chart of particles
Planck's constant h, in energy/frequency
Reduced Planck's constant \hbar = h/2\pi, in energy/frequency
Planck–Einstein equation E = hf
threshold frequency f_0
work function \Phi = hf_0
photoelectric kinetic energy K_{max} = hf - \Phi
photon momentum p = hf/c = h/\lambda
de Broglie wavelength \lambda = h/p
Schrodinger's equation i\hbar\frac{\partial}{\partial t} \Psi(\mathbf{r},\,t) = \hat H \Psi(\mathbf{r},t)
Schrodinger's equation one dimensional motion d^2\psi/dx^2 + 8\pi^2m[E-U(x)]\psi/h^2 = 0
Schrodinger's equation free particle d^2\psi/dx^2 + k^2\psi = 0
Heisenberg's uncertainty principle \Delta x \cdot \Delta p_x \ge \hbar
infinite potential well E_n = (hn/2L)^2/2m
wavefunction of a trapped electron \psi_n(x) = A sin(n\pi x/L), for positive int n
wavefunction probability density p(x) = \psi^2_n(x)dx
normalization \int \psi^2_n(x)dx = 1
hydrogen atom orbital energy E_n = -me^4/8\epsilon_0^2h^2n^2 = 13.61eV/n^2, for positive int n
hydrogen atom spectrum 1/\lambda = R(1/n^2_{low} - 1/n^2_{high})
hydrogen atom radial probability density P(r) = 4r^2/a^3e^{2r/a}
spin projection quantum number m_s \in \{-1/2,+1/2\}
orbital magnetic dipole moment \mathbf{\mu}_{orb} = -e\mathbf{L}/2m
orbital magnetic dipole moment components \mathbf{\mu}_{orb,z} = -m_\mathcal{L}\mu_B
spin magnetic dipole moment \mathbf{\mu_s} = -e\mathbf{S}/m = gq\mathbf{S}/2m
orbital magnetic dipole moment \mathbf{\mu}_{orb}=-e\mathbf{L}_{orb}/2m
spin magnetic dipole moment potential U = -\mathbf{\mu}_s\cdot\mathbf{B}_{ext} = -\mu_{s,z}B_{ext}
orbital magnetic dipole moment potential U = -\mathbf{\mu}_{orb}\cdot\mathbf{B}_{ext} = -\mu_{orb,z}B_{ext}
Bohr magneton \mu_B = e\hbar/2m
angular momentum components L_z = m\mathcal{L}\hbar
spin angular momentum magnitude S = \hbar\sqrt{s(s+1)}
cutoff wavelength \lambda_{min} = hc/K_0
density of states N(E) = 8\sqrt{2}\pi m^{3/2}E^{1/2}/h^3
occupancy probability P(E) = 1/(e^{(E-E_F)/kT}+1)
Fermi energy E_F = (3/16\sqrt{2}\pi)^{2/3}h^2n^{2/3}m
mass number A = Z+N
nuclear radius r=r_0A^{1/3}, r_0 \approx 1.2fm
mass excess \Delta = M - A
radioactive decay N = N_0e^{-\lambda t}
Hubble constant H = 71.0km/s
Hubble's law v=Hr
conservation of lepton number
conservation of baryon number
conservation of strangeness
eightfold way
weak force
strong force 
\begin{align}
\mathcal{L}_\mathrm{QCD} 
& = \bar{\psi}_i\left(i \gamma^\mu (D_\mu)_{ij} - m\, \delta_{ij}\right) \psi_j - \frac{1}{4}G^a_{\mu \nu} G^{\mu \nu}_a \\
& = \bar{\psi}_i (i \gamma^\mu \partial_\mu  -  m)\psi_i - g G^a_\mu \bar{\psi}_i \gamma^\mu T^a_{ij} \psi_j - \frac{1}{4}G^a_{\mu \nu} G^{\mu \nu}_a \,,\\
\end{align}
Noether's theorem
Electroweak interaction :\mathcal{L}_{EW} = \mathcal{L}_g + \mathcal{L}_f + \mathcal{L}_h + \mathcal{L}_y.
\mathcal{L}_g = -\frac{1}{4}W_a^{\mu\nu}W_{\mu\nu}^a - \frac{1}{4}B^{\mu\nu}B_{\mu\nu}
\mathcal{L}_f =   \overline{Q}_i iD\!\!\!\!/\; Q_i+ \overline{u}_i^c iD\!\!\!\!/\; u^c_i+ \overline{d}_i^c iD\!\!\!\!/\; d^c_i+ \overline{L}_i iD\!\!\!\!/\; L_i+ \overline{e}^c_i iD\!\!\!\!/\; e^c_i
\mathcal{L}_h = |D_\mu h|^2 - \lambda \left(|h|^2 - \frac{v^2}{2}\right)^2
\mathcal{L}_y = - y_{u\, ij} \epsilon^{ab} \,h_b^\dagger\, \overline{Q}_{ia} u_j^c - y_{d\, ij}\, h\, \overline{Q}_i d^c_j - y_{e\,ij} \,h\, \overline{L}_i e^c_j + h.c.
Quantum electrodynamics :\mathcal{L}=\bar\psi(i\gamma^\mu D_\mu-m)\psi -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\;,

क्वांटम यांत्रिकी (Quantum Mechanics)[संपादित करें]

Postulate 1: State of a system A system is completely specified at any one time by a Hilbert space vector.
Postulate 2: Observables of a system A measurable quantity corresponds to an operator with eigenvectors spanning the space.
Postulate 3: Observation of a system Measuring a system applies the observable's operator to the system and the system collapses into the observed eigenvector.
Postulate 4: Probabilistic result of measurement The probability of observing an eigenvector is derived from the square of its wavefunction.
Postulate 5: Time evolution of a system The way the wavefunction evolves over time is determined by Shrodinger's equation.

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