भौतिकी के सूत्र

एसआई उपसर्ग (प्रीफिक्स) सम्पादित करें

आधारभूत यांत्रिकी (Fundamentals of Mechanics) सम्पादित करें

Foundational equations in translation and rotation.

Quantity	Translation	Rotation
समय	$t$	$t$
स्थिति	$x$	$θ$ in radians
द्रव्यमान	$m$	$m$
समयान्तर	$Δ t$	$Δ t$
विस्थापन	$Δ x$	$Δ θ$
द्रव्यमान संरक्षण	$Δ m = 0$	$Δ m = 0$
ऊर्जा संरक्षण	$Δ E = 0$	$Δ E = 0$
संवेग संरक्षण	$Δ P = 0$	$Δ L = 0$
वेग	$v = d x / d t$	$ω = d θ / d t$
त्वरण	$a = d v / d t$	$α = d ω / d t$
झटका	$j = d a / d t$	$j = d α / d t$
स्थितिज ऊर्जा परिवर्तन	$Δ U = - W$	$Δ U = - W$
संवेग	$P = m v$	$L = I ω$ $= \| \| 𝐫 \times 𝐏 \| \| = m \| \| 𝐫 \times 𝐯 \| \|$
बल	$f = d P / d t = m a = - d U / d x$	$τ = d L / d t = I α$ $= \| \| 𝐫 \times 𝐟 \| \| = m \| \| 𝐫 \times 𝐚 \| \|$
जड़त्व आघूर्ण	$m = \int d m = Σ m_{i}$	$I = \int r^{2} d m = Σ r^{2} m_{i}$
आवेग	$J = \int f d t$	$J = \int τ d t$
कार्य	$W = \int f d x = 𝐝 \cdot 𝐟$	$W = \int τ d θ$
शक्ति	$P = d W / d t = f v$	$P = d W / d t = τ ω$
गतिज ऊर्जा	$K = m v^{2} / 2 = P^{2} / 2 m$	$K = I w^{2} / 2 = Σ R^{2} m$
न्यूटन का तीसरा नियम	$f_{a b} = - f_{b a}$	$τ_{a b} = - τ_{b a}$

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) सम्पादित करें

Equations in translation and rotation, assuming constant acceleration.

भौतिक राशि	रेखीय गति	घुर्णन गति
विस्थापन	$Δ v = a t$	$Δ ω = α t$
समय	$Δ (v^{2}) = 2 a Δ x$	$Δ (ω^{2}) = 2 α Δ θ$
त्वरण	$Δ x = t Δ v / 2$	$Δ θ = t Δ ω / 2$
प्रा०वेग	$Δ x = - a t^{2} / 2 + v_{2} t$	$Δ θ = - α t^{2} / 2 + ω_{2} t$
अंतिमवेग	$Δ x = + a t^{2} / 2 + v_{1} t$	$Δ θ = + α t^{2} / 2 + ω_{1} t$

एकसमान वृत्तीय गति (Uniform circular motion) सम्पादित करें

uniform circular motion angular to linear displacement	$x = θ r$
uniform circular motion angular to linear speed	$v = θ ω$
uniform circular motion angular to linear acceleration normal component	$a_{r} = ω^{2} r$
uniform circular motion	$𝐝 = 𝐢 c o s ω t + 𝐣 s i n ω t$
uniform circular motion tangential speed	$𝐯 = 𝐝^{'} = - ω r (𝐢 \sin ω t - 𝐣 \cos ω t)$
uniform circular motion tangential component, scalar	$a_{t} = α r$
uniform circular motion centripetal acceleration	$𝐚 = 𝐝^{″} = - ω^{2} 𝐝 = - v^{2} 𝐧 / r$
uniform circular motion centripetal acceleration scalar	$α = v^{2} / r$
uniform circular motion centripetal force	$f = - m v^{2} / r$
uniform circular motion revolution time	$T = 2 π r / v$

Elasticity सम्पादित करें

elastic force, lies parallel to spring	$f = - k d$
elastic potential energy	$U = k x^{2} / 2$
elastic work, positive when relaxes	$W = - k Δ (x^{2}) / 2$

घर्षण (Friction) सम्पादित करें

normal force	$f_{n} = 𝐟 \cdot 𝐧$
static friction maximum, lies tangent to the surface	$f = μ_{s} f_{n}$
kinetic friction, lies tangent to the surface	$f = μ_{k} f_{n}$
drag force, tangent to the path	$f = μ_{d} ρ a v^{2} / 2$
terminal velocity	$v_{t} = \sqrt{2 f g / (μ_{d} ρ A)}$
friction creates heat and sound	$Δ E = f_{k} d$

प्रतिबाधा एवं विकृत्ति (Stress and strain) सम्पादित करें

stress
strain
modulus of elasticity	$λ = s t r e s s / s t r a i n$
yield strength
ultimate strength
Young's modulus	$F / A = E Δ L / L$
shear modulus	$F / A = G Δ x / L$
bulk modulus	$F / A = B Δ V / V$

अन्य सम्पादित करें

inertial frames	$x_{P A} = x_{P B} + x_{A B}$
. . .	$v_{P A} = v_{P B} + v_{A B}$
. . .	$a_{P A} = a_{P B} + 0$
trajectory	$y = x \tan θ - g x^{2} / 2 (V_{0} \cos θ)^{2}$
flight distance	$v_{0}^{2} \sin 2 θ / g$
tension, lies within the cord	$f_{t} = f$
mechanical energy	$E_{m e c} = K + U$
mechanical energy is conserved	$Δ E_{m e c} = 0$ when all forces are conservative
thrust	$t = R v_{r e l} = m a$
ideal rocket equation	$Δ v = l n (m_{i} / m_{f}) v_{r e l}$
parallel axis theorem	$I = I_{c o m} + m r^{2}$
list of moments of inertia
indeterminate systems

द्रब्यमान केन्द्र एवं संघट्ट (Center of mass and collisions) सम्पादित करें

center of mass COM	$𝐫_{c o m} = M^{- 1} Σ m_{i} 𝐫_{i}$
. . .	$x_{c o m} = M^{- 1} \int x d m, \dots$
for constant density:	$x_{c o m} = V^{- 1} \int x d V, \dots$
COM is in all planes of symmetry
elastic collision	$Δ E_{k} = 0$
inelastic collision	$Δ E_{k} =$ maximum
conservation of momentum in a two body collision	$𝐏_{1 i} + 𝐏_{2 i} = 𝐏_{1 f} + 𝐏_{2 f}$
system COM remains inert	$𝐯_{c o m} = \frac{(𝐏_{1 i} + 𝐏_{2 i})}{(M_{1} + M_{2})} = c o n s t$
elastic collision, 1D, M2 stationary	$v_{1 f} = \frac{(m_{1} - m_{2})}{(m_{1} + m_{2})} v_{1 i}$
. . .	$v_{2 f} = \frac{(2 m_{1})}{(m_{1} + m_{2})} v_{1 i}$

चिकने तल पर लुढ़कना (Smooth rolling) सम्पादित करें

rolling distance	$x_{a r c} = R θ$
rolling distance ?	$x_{c o m} = R α$
rolling velocity	$v_{c o m} = R ω$
rolling ?	$K = I_{c o m} ω^{2} / 2 + M v_{c o m}^{2} / 2$
rolling down a ramp along axis x	$a_{c o m, x} = - \frac{g \sin θ}{1 + I_{c o m} / M R^{2}}$

उष्मागतिकी (Thermodynamics) सम्पादित करें

Zeroth Law of Thermodynamics	$(A = B) \land (B = C) \Rightarrow A = C$ (where "=" denotes systems in thermal equilibrium
First Law of Thermodynamics	$Δ E_{i n t} = Q + W$
Second Law of Thermodynamics	$Δ S \geq 0$
Third Law of Thermodynamics	$S = S_{s t r u c t u r a l} + C T$
temperature	$T$
molecules	$N$
degrees of freedom	$f$
heat	$Q$ , $Δ E$ due to $Δ T$ (energy)
thermal mass (extensive property)	$C_{t h} = Q / Δ T$
specific heat capacity (bulk property)	$c_{t h} = Q / Δ T m$
enthalpy of vaporization	$L_{v} = Q / m$
enthalpy of fusion	$L_{f} = Q / m$
thermal conductivity	$κ$
thermal resistance	$R = L / κ$
thermal conduction rate	$P = Q / t = A (T_{H} - T_{C}) / R$
thermal conduction rate through a composite slab	$P = Q / t = A (T_{H} - T_{C}) / Σ (R_{i})$
linear coefficient of thermal expansion	$d L / d t = α L$
volume coefficient of thermal expansion	$d V / d t = 3 α V$
Boltzmann constant	$k$ (energy)/(temperature)
Stefan-Boltzmann constant	$σ$ (power)/(area)(temp)^4
thermal radiation	$P = σ ϵ A T_{s y s}^{4}$
thermal absorption	$P = σ ϵ A T_{e n v}^{4}$
adiabatic	$Δ Q = 0$
ideal gas law	$P V = k T N$
work, constant temperature	$W = k T N l n (V_{f} / V_{i})$
work due to gas expansion	$W = \int_{i}^{f} p d V$
. . . adiabatic	$Δ E_{i n t} = W$
. . . constant volume	$Δ E_{i n t} = Q$
. . . free expansion	$Δ E_{i n t} = 0$
. . . closed cycle	$Q + W = 0$
work, constant volume	$W = 0$
work, constant pressure	$W = p Δ V$
translational energy	$E_{k, a v g} = k T f / 2$
internal energy	$E_{i n t} = N k T f / 2$
mean speed	$v_{a v g} = \sqrt{(k T / m) (8 / π)}$
mode speed	$v_{p r b} = \sqrt{(k T / m) 2}$
root mean square speed	$v_{r m s} = \sqrt{(k T / m) 3}$
mean free path	$λ = 1 / (\sqrt{2} π d^{2} N / V)$ ?
Maxwell–Boltzmann distribution	$P (v) = 4 π (m / (2 π k T))^{3 / 2} V^{2} e^{- (m v^{2} / (2 k T))}$
molecular specific heat at a constant volume	$C_{V} = Q / (N Δ T)$
?	$Δ E_{i n t} = N C_{V} Δ T$
molecular specific heat at a constant pressure	$C_{p} = Q / (N Δ T)$
?	$W = p Δ V = N k Δ T$
?	$k = C_{p} - C_{V}$
adiabatic expansion	$p V^{γ} = c o n s t a n t$
adiabatic expansion	$T V^{γ - 1} = c o n s t a n t$
multiplicity of configurations	$W = N! / n_{1}! n_{2}!$
microstate in one half of the box	$n_{1}, n_{2}$
Boltzmann's entropy equation	$S = k l n W$
irreversibility
entropy	$S = - k \sum_{i} P_{i} \ln P_{i}$
entropy change	$Δ S = \int_{i}^{f} (1 / T) d Q \approx Q / T_{a v g}$
entropy change	$Δ S = k N l n (V_{f} / V_{i}) + N C_{V} l n (T_{f} / T_{i})$
entropic force	$f = - T d S / d x$
engine efficiency	$ϵ = \| W \| / \| Q_{H} \|$
Carnot engine efficiency	$ϵ_{c} = (\| Q_{H} \| - \| Q_{L} \|) / \| Q_{H} \| = (T_{H} - T_{L}) / T_{H}$
refrigeration performance	$K = \| Q_{L} \| / \| W \|$
Carnot refrigeration performance	$K_{C} = \| Q_{L} \| / (\| Q_{H} \| - \| Q_{L} \|) = T_{L} / (T_{H} - T_{L})$

तरंग सम्पादित करें

torsion constant	$κ = - τ / θ$
phasor
node
antinode
period	$T$
amplitude	$x_{m}$
decibel	$d B$
frequency	$f = 1 / T = ω / (2 π)$
angular frequency	$ω = 2 π f = 2 π / T$
phase angle	$ϕ$
phase	$(ω t + ϕ)$
damping force	$f_{d} = - b v$
phase	$k y - ω t$
wavenumber	$k$
phase constant	$ϕ$
linear density	$μ$
harmonic number	$n$
harmonic series	$f = v / λ = n v / (2 L)$
wavelength	$λ = k / (2 π)$
bulk modulus	$B = Δ p / (Δ V / V)$
path length difference	$Δ L$
resonance	$ω_{d} = ω$
phase difference	$ϕ = 2 π Δ L / λ$
fully constructive interference	$Δ L / λ = n$
fully destructive interference	$Δ L / λ = n + 0.5$
sound intensity	$I = P / A = ρ v ω^{2} s_{m}^{2} / 2$
sound power source	$P_{s}$
sound intensity over distance	$I = P_{s} / (4 π r^{2})$
sound intensity standard reference	$I_{0}$
sound level	$B = (10 d B) l o g (I / I_{0})$
pipe, two open ends	$f = v / λ = n v / (2 L)$
pipe, one open end	$f = v / λ = n v / (4 L)$ for n odd
beats	$s (t) = [2 s_{m} \cos ω^{'} t] \cos ω t$
beat frequency	$f_{b e a t} = f_{1} - f_{2}$
Doppler effect	$f^{'} = f (v + - v_{D}) / (v + - v_{S})$
sonic boom angle	$\sin θ = v / v_{s}$
average wave power	$P_{a v g} = μ v ω^{2} x_{m}^{2} / 2$
pressure amplitude	$Δ p_{m} = (v ρ ω) x_{m}$
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	$v = ω / k = λ / T = λ f$
speed of sound	$v = \sqrt{B / ρ}$
wave speed on a stretched string	$v = \sqrt{f_{t} / μ}$
angular frequency of an angular simple harmonic oscillator	$ω = \sqrt{I / κ}$
angular frequency of a low amplitude simple pendulum	$ω = \sqrt{L / g}$
angular frequency of a low amplitude physical pendulum	$ω = \sqrt{I / m g h}$
angular frequency of a linear simple harmonic oscillator	$ω = \sqrt{k / m}$
angular frequency of a linear damped harmonic oscillator	$ω^{'} = \sqrt{(k / m) - (b^{2} / 4 m^{2})}$
wave displacement	$x (t) = x_{m} \cos (ω t + ϕ)$
wave displacement when damped	$x (t) = x_{m} \cos (ω^{'} t + ϕ) (e^{- b t / 2 m})$
wave velocity	$v (t) = x_{m} \sin (ω t + ϕ) (- ω)$
wave acceleration	$a (t) = x_{m} \cos (ω t + ϕ) (- ω^{2})$
transverse wave	$x (y, t) = x_{m} \sin (k y - ω t)$
wave traveling backwards	$x (y, t) = x_{m} \sin (k y + ω t)$
resultant wave	$x^{'} (y, t) = x_{m} \sin (k y - ω t + ϕ / 2) (2 \cos ϕ / 2)$
standing wave	$x^{'} (y, t) = \cos (ω t) (2 y \sin k y)$
sound displacement function	$x (y, t) = x_{m} \cos (k y - ω t)$
sound pressure-variation function	$Δ p (y, t) = \sin (k y - ω t) Δ p_{m}$
potential harmonic energy	$E_{U} (t) = k x^{2} / 2 = k x_{m}^{2} \cos^{2} (ω t + ϕ) / 2$
kinetic harmonic energy	$E_{K} (t) = k x^{2} / 2 = k x_{m}^{2} \sin^{2} (ω t + ϕ) / 2$
total harmonic energy	$E (t) = k x_{m}^{2} / 2 = E_{U} + E_{K}$
damped mechanical energy	$E_{m e c} (t) = k e^{- b t / m} x_{m}^{2} / 2$

गुरुत्वाकर्षण (Gravitation) सम्पादित करें

gravitational constant	$G$ (force)(distance/mass)^2
gravitational force	$f_{G} = G m_{1} m_{2} / r^{2}$
superposition applies	$𝐅 = Σ 𝐅_{i} = \int d 𝐅$
gravitational acceleration	$a_{g} = G m / r^{2}$
free fall acceleration	$a_{f} = a_{g} - ω^{2} R$
shell theorem for gravitation
potential energy from gravity	$U = - G m_{1} m_{2} / r \approx m a_{g} y$
escape speed	$v = \sqrt{2 G m / r}$
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	$T^{2} = (4 π^{2} / G m) r^{3}$
orbital energy	$E = - G m_{1} m_{2} / a 2$
standard gravity	$a_{g} = G m_{E a r t h} / r_{E a r t h}^{2} \approx 9.81 m / s^{2}$
weight, points toward the center of gravity	$f_{g} = - f_{n} = m g$
path independence	$W_{a b, 1} = W_{a b, 2} = \dots$
Einstein field equations	$R_{μ ν} - \frac{1}{2} g_{μ ν} R + g_{μ ν} Λ = \frac{8 π G}{c^{4}} T_{μ ν}$

तरलगतिकी (Fluid dynamics) सम्पादित करें

density	$ρ = Δ m / Δ V$
pressure	$p = Δ F / Δ A$
pressure difference	$Δ p = ρ g Δ y$
pressure at depth	$p = p_{0} + ρ g h$
barometer versus manometer
Pascal's principle
Archimedes' Principle
buoyant force	$F_{b} = m_{f} g$
gravitational force when floating	$F_{g} = F_{b}$
apparent weight	$w e i g h t_{a p p} = w e i g h t - F_{b}$
ideal fluid
equation of continuity	$R_{V} = A v =$ constant
Bernoulli's equation	$p + ρ v^{2} / 2 + ρ g y =$ constant

विद्युतचुम्बकत्व (Electromagnetism) सम्पादित करें

Lorentz force	$𝐅 = q (𝐄 + 𝐯 \times 𝐁)$
Gauss' law	$\oint 𝐄 \cdot d 𝐀 = Φ_{E} = q_{e n c} / ϵ_{0}$
Gauss' law for magnetic fields	$\oint 𝐁 \cdot d 𝐀 = Φ_{B} = 0$
Faraday's law of induction	$\oint 𝐄 \cdot d 𝐬 = - d Φ_{B} / d t = - ℰ$
Ampere-maxwell law	$\oint 𝐁 \cdot d 𝐬 = μ_{0} (i_{e n c} + i_{d, e n c})$
elementary charge	$e$
electric charge	$q = n e$
conservation of charge	$Δ q = 0$
linear charge density	$λ = q / l^{1}$
surface charge density	$σ = q / l^{2}$
volume charge density	$ρ = q / l^{3}$
electric constant	$ϵ_{0}$ (time)^2(charge)^2/(mass)(volume)
magnetic constant	$μ_{0}$ (force)(time)^2/(charge)^2
Coulomb's law	$F = q_{1} q_{2} / (4 π ϵ_{0}) r^{2}$
electric field	$𝐄 = 𝐅 / q$
electric field lines	end at a negative charge
Gaussian surface	$𝐀$
flux notation implies a normal unit vector	$\cdot d 𝐀 \to \cdot 𝐧 d 𝐀$
electric flux	$Φ_{E} = \oint 𝐄 \cdot d 𝐀$
magnetic flux	$Φ_{B} = \int 𝐁 \cdot d 𝐀$
magnetic flux given assumptions	$Φ_{B} = B A$
dielectric constant	$κ \geq 1$
dielectric	$ϵ_{0} \to ϵ_{0} κ$
Gauss' law with dialectric	$q_{e n c} = ϵ_{0} \oint κ 𝐄 \cdot d 𝐀$
Biot-Savart law	$𝐁 = \int \frac{μ_{0}}{4 π} \frac{(i d 𝐬) \times 𝐫}{r^{3}},$
Lenz's law	induced current always opposes its cause
inductance (with respect to time)	$L = - ℰ / q^{″}$
inductance from coils	$L = N Φ_{B} / i$
inductance of a solenoid	$L / l = μ_{0} n^{2} A$
displacement current	$i_{d} = ϵ_{0} d Φ_{E} / d t$
displacement vector	$𝐝$
electric dipole moment	$𝐩 = q 𝐝$
electric dipole torque	$τ = 𝐩 \times 𝐄$
electric dipole potential energy	$U = - 𝐩 \cdot 𝐄$
magnetic dipole moment of a coil, magnitude only	$μ = i N A$
magnetic dipole moment torque	$τ = μ \times 𝐁$
magnetic dipole moment potential energy	$U = - μ \cdot 𝐁$
electric field accelerating a charged mass	$a = q E / m$
electric field of a charged point	$E = q / ϵ_{0} 4 π r^{2} \hat{r}$
electric field of a dipole moment	$E = p / ϵ_{0} 2 π z^{3}$
electric field of a charged line	$E = λ / ϵ_{0} 2 π r$
electric field of a charged ring	$E = q z / ϵ_{0} 4 π (z^{2} + R^{2})^{3 / 2}$
electric field of a charged conducting surface	$E = σ / ϵ_{0}$
electric field of a charged non-conducting surface	$E = σ / ϵ_{0} 2$
electric field of a charged disk	$E = σ (1 - z) / ϵ_{0} 2 \sqrt{z^{2} + R^{2}}$
electric field outside spherical shell r>=R	$E = q / ϵ_{0} 4 π r^{2}$
electric field inside spherical shell r<R	$E = 0$
electric field of uniform charge r<=R	$E = q r / ϵ_{0} 4 π R^{3}$
electric field energy density	$u = ϵ_{0} E^{2} / 2$
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	$Δ V = - W / q = Δ U / q$
electric potential from electric field	$Δ V = - \int_{i}^{f} 𝐄 \cdot d 𝐬$
electric field from electric potential	$\nabla V = - 𝐄$
electric potential of a charged point	$V = q / ϵ_{0} 4 π r$
electric potential of a set of charged points	$V = Σ V_{i} = (1 / ϵ_{0} 4 π) Σ q_{i} / r_{i}$
electric potential of a dipole	$V = p \cos θ / ϵ_{0} 4 π r^{2}$
electric potential of continuous charge	$V = \int d V = (1 / ϵ_{0} 4 π) \int d q / r$
electric potential energy of a pair of charged points	$V q_{2} = U = W = q_{1} q_{2} / ϵ_{0} 4 π r$
capacitance	$C = q / V$ (charge)^2/(energy)
capacitance of parallel plates	$C = ϵ_{0} A / d$
capacitance of a cylinder	$C = ϵ_{0} 2 π L / \ln (b / a)$
capacitance of a sphere	$C = ϵ_{0} 4 π b a / (b - a)$
capacitance of an isolated sphere	$C = ϵ_{0} 4 π R$
capacitors in parallel	$C_{e q}^{+ 1} = Σ C_{i}^{+ 1}$
capacitors in series	$C_{e q}^{- 1} = Σ C_{i}^{- 1}$
capacitor potential energy	$U = q^{2} / C 2 = C V^{2} / 2$
current	$i = d q / d t$
drift speed	$𝐯_{d}$
current density	$𝐉 = n e 𝐯_{d} / m^{3}$
current density magnitude	$J = i / A$
current density to get current	$i = \int J d A$
resistance	$R = V / i$
resistivity	$ρ = 𝐄 / 𝐉$
resistivity temperature coefficient	$α$
resistivity across temperature	$ρ - ρ_{0} = ρ_{0} α (T - T_{0})$
resistivity and resistance	$R A = ρ L$
electrical conductivity	$σ = 1 / ρ = 𝐉 / 𝐄$
resistor power dissipation	$P = i^{2} R = V^{2} / R$
internal resistance	$i = ℰ / (R + r)$
resistors in series	$R_{e q}^{+ 1} = Σ R_{i}^{+ 1}$
resistors in parallel	$R_{e q}^{- 1} = Σ R_{i}^{- 1}$
Kirchoff's current law	$i_{i n} = i_{o u t}$
Ohm's law	$V = i R$
emf	$ℰ = d W / d q = i R$
emf rules	loop, resistance, emf
electrical power	$P = i V$
emf power	$P_{e m f} = i ℰ$
electric potential difference across a real battery	$p = ℰ - i R$
magnetic field force on a moving charge	$𝐅_{B} = q 𝐯 \times 𝐁$
magnetic field force on a current	$𝐅_{B} = i 𝐋 \times 𝐁$
Hall effect	$n = B i / V l e$
circulating charged particle	$\| q \| v B = m v^{2} / r$
cyclotron resonance condition	$f = f_{o s c}$
magnetic field of a line	$B = μ_{0} i / 2 π R$
magnetic field of a ray	$B = μ_{0} i / 4 π R$
magnetic field at the center of a circular arc	$B = μ_{0} i ϕ / 4 π R$
magnetic field of a solenoid	$B = μ_{0} i n$
magnetic field of a toroid	$B = μ_{0} i N / 2 π r$
magnetic field of a current carrying coil	$𝐁 = μ_{0} μ / 2 π z^{3}$
self induction of emf	$ℰ_{L} = - L d i / d t$
magnetic energy	$U_{B} = L i^{2} / 2$
magnetic energy density	$u_{B} = B^{2} / 2 μ_{0}$
mutual induction	$ℰ_{1} = - M d i_{2} / d t, ℰ_{2} = - M d i_{1} / d t$
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	$R_{e q} = (N p / N s)^{2} R$
induced magnetic field inside a circular capacitor	$B = (μ_{0} i_{d} / 2 π R^{2}) r$
induced magnetic field outside a circular capacitor	$B = μ_{0} i_{d} / 2 π r r$
RC circuit ODE with respect to time	$R q^{'} + C^{- 1} q = ℰ$
RC circuit capacitive time constant	$τ = R C$
RC circuit charging a capacitor	$q = C ℰ (1 - e^{- t / R C})$
RL circuit ODE with respect to time	$L i^{″} + R i^{'} = ℰ$
RL circuit time constant	$τ_{L} = L / R$
RL circuit rise of current	$i = ℰ / R (1 - e^{- t / τ_{L}})$
RL circuit decay of current	$i = ℰ e^{- t / τ_{L}} / R = i_{0} e^{- t / τ_{L}}$
LC circuit ODE with respect to time	$L q^{″} + C^{- 1} q = ℰ$
LC circuit	$ω = 1 / \sqrt{L C}$
LC circuit charge	$q = Q c o s (ω t + ϕ)$
LC circuit current	$i = - ω Q s i n (ω t + ϕ)$
LC circuit electrical potential energy	$U_{E} = q^{2} / 2 C = Q^{2} c o s^{2} (ω t + ϕ) / 2 C$
LC circuit magnetic potential energy	$U_{B} = Q^{2} s i n^{2} (ω t + ϕ) / 2 C$
RLC circuit ODE with respect to time	$L q^{″} + R q^{'} + C^{- 1} q = ℰ$
RLC circuit charge	$q = Q e T^{- R t / 2 L} c o s (ω^{'} t + ϕ)$
resistive load	$V_{R} = I_{R} R$
capacitive load	$V_{C} = I_{C} X_{C}$
inductive load	$V_{L} = I_{L} X_{L}$
resistive reactance	$X_{R} = ?$
capacitive reactance	$X_{C} = 1 / ω_{d} C$
inductive reactance	$X_{L} = ω_{d} L$
phase constant	$t a n ϕ = X_{L} - X_{C} / R$
electromagnetic resonance	$ω_{d} = ω = 1 / \sqrt{L C}$
AC current	$I_{r m s} = I / \sqrt{2}$
AC voltage	$V_{r m s} = V / \sqrt{2}$
AC emf	$ℰ_{r m s} = ℰ_{m} / \sqrt{2}$
AC power	$P_{a v g} = ℰ I_{r m s} c o s ϕ$

प्रकाश (Light) सम्पादित करें

electric light component	$E = E_{m} s i n (k x - ω t)$
magnetic light component	$B = B_{m} s i n (k x - ω t)$
speed of light	$c = 1 / \sqrt{μ_{0} ϵ_{0}} = E / B$
Poynting vector	$𝐒 = μ_{0}^{- 1} 𝐄 \times 𝐁$
Poynting vector magnitude	$S = E B / μ_{0} = E^{2} / c μ_{0}$
rms electric field of light	$E_{r m s} = E / \sqrt{2}$
light intensity	$I = E_{r m s}^{2} / c μ_{0}$
light intensity at the sphere	$I = P_{s} / 4 π r^{2}$
radiation momentum with total absorption (inelastic)	$Δ p = Δ U / c$
radiation momentum with total reflection (elastic)	$Δ p = 2 Δ U / c$
radiation pressure with total absorption (inelastic)	$p_{r} = I / c$
radiation pressure with total reflection (elastic)	$p_{r} = 2 I / c$
intensity from polarizing unpolarized light	$I = I_{0} / 2$
intensity from polarizing polarized light	$I = I_{0} c o s^{2} θ$
index of refraction of substance f	$n_{f} = c / v_{f}$
angle of reflection	$θ_{1} = θ_{2}$
angle of refraction	$n_{1} s i n θ_{1} = n_{2} s i n θ_{2}$
angle of total reflection	$θ_{c} = s i n^{- 1} n_{2} / n_{1}$
angle of total polarisation	$θ_{B} = t a n^{- 1} n_{2} / n_{1}$
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	$1 / f = (n_{l e n s} / n_{m e d} - 1) (1 / r_{1} - 1 / r_{2})$
path length difference	$Δ L = d s i n θ$
double slit minima	$d s i n θ = (N + 1 / 2) λ$
double slit maxima	$d s i n θ = N λ$
double-slit interference intensity	$I = 4 I_{0} c o s^{2} (π d s i n θ / λ)$
thin film in air minima	$(N + 0 / 2) λ / n_{2}$
thin film in air maxima	$2 L = (N + 1 / 2) λ / n_{2}$
single-slit minima	$a s i n θ = N λ$
single-slit intensity	$I (θ) = I_{0} (s i n α / α)^{2}$
double slit intensity	$I (θ) = I_{0} (c o s^{2} B) (s i n α / α)^{2}$
. . .	$α = π a s i n θ / λ$
circular aperture first minimum	$s i n θ = 1.22 λ / d$
Rayleigh's criterion	$θ_{R} = 1.22 λ / d$
diffraction grating maxima lines	$d s i n θ = N λ$
diffraction grating half-width	$Δ θ_{h w} = λ / N d c o s θ$
diffraction grating dispersion	$D = N / d c o s θ$
diffraction grating resolving power	$R = N n$
diffraction grating lattice distance	$d = N λ / 2 s i n θ$

विशिष्ट आपेक्षिकता (Special Relativity) सम्पादित करें

Lorentz factor	$γ = 1 / \sqrt{1 - (v / c)^{2}}$
Lorentz transformation	$t^{'} = γ (t - x v / c^{2})$
. . .	$x^{'} = γ (x - v t)$
. . .	$y^{'} = y$
. . .	$z^{'} = z$
time dilation	$Δ t = γ Δ t_{0}$
length contraction	$L = L_{0} / γ$
relativistic Doppler effect	$f = f_{0} \sqrt{1 - (v / c) / 1 + (v / c)}$
Doppler shift	$v = \| Δ λ \| c / λ_{0}$
momentum	$𝐩 = γ m 𝐯$
rest energy	$E_{0} = m c^{2}$
total energy	$E = E_{0} + K = m c^{2} + K = γ m c^{2} = \sqrt{(p c)^{2} + (m c^{2})^{2}}$
Energy Removed	$Q = - Δ m c^{2}$
kinetic energy	$K = E - m c^{2} = γ m c^{2} - m c^{2} = m c^{2} (γ - 1)$

कण भौतिकी (Particle Physics) सम्पादित करें

standard model	see 4x4 chart of particles
Planck's constant	$h$ , in energy/frequency
Reduced Planck's constant	$ℏ = h / 2 π$ , in energy/frequency
Planck–Einstein equation	$E = h f$
threshold frequency	$f_{0}$
work function	$Φ = h f_{0}$
photoelectric kinetic energy	$K_{m a x} = h f - Φ$
photon momentum	$p = h f / c = h / λ$
de Broglie wavelength	$λ = h / p$
Schrodinger's equation	$i ℏ \frac{\partial}{\partial t} Ψ (𝐫, t) = \hat{H} Ψ (𝐫, t)$
Schrodinger's equation one dimensional motion	$d^{2} ψ / d x^{2} + 8 π^{2} m [E - U (x)] ψ / h^{2} = 0$
Schrodinger's equation free particle	$d^{2} ψ / d x^{2} + k^{2} ψ = 0$
Heisenberg's uncertainty principle	$Δ x \cdot Δ p_{x} \geq ℏ$
infinite potential well	$E_{n} = (h n / 2 L)^{2} / 2 m$
wavefunction of a trapped electron	$ψ_{n} (x) = A s i n (n π x / L)$ , for positive int n
wavefunction probability density	$p (x) = ψ_{n}^{2} (x) d x$
normalization	$\int ψ_{n}^{2} (x) d x = 1$
hydrogen atom orbital energy	$E_{n} = - m e^{4} / 8 ϵ_{0}^{2} h^{2} n^{2} = 13.61 e V / n^{2}$ , for positive int n
hydrogen atom spectrum	$1 / λ = R (1 / n_{l o w}^{2} - 1 / n_{h i g h}^{2})$
hydrogen atom radial probability density	$P (r) = 4 r^{2} / a^{3} e^{2 r / a}$
spin projection quantum number	$m_{s} \in {- 1 / 2, + 1 / 2}$
orbital magnetic dipole moment	$μ_{o r b} = - e 𝐋 / 2 m$
orbital magnetic dipole moment components	$μ_{o r b, z} = - m_{ℒ} μ_{B}$
spin magnetic dipole moment	$μ_{𝐬} = - e 𝐒 / m = g q 𝐒 / 2 m$
orbital magnetic dipole moment	$μ_{o r b} = - e 𝐋_{o r b} / 2 m$
spin magnetic dipole moment potential	$U = - μ_{s} \cdot 𝐁_{e x t} = - μ_{s, z} B_{e x t}$
orbital magnetic dipole moment potential	$U = - μ_{o r b} \cdot 𝐁_{e x t} = - μ_{o r b, z} B_{e x t}$
Bohr magneton	$μ_{B} = e ℏ / 2 m$
angular momentum components	$L_{z} = m ℒ ℏ$
spin angular momentum magnitude	$S = ℏ \sqrt{s (s + 1)}$
cutoff wavelength	$λ_{m i n} = h c / K_{0}$
density of states	$N (E) = 8 \sqrt{2} π m^{3 / 2} E^{1 / 2} / h^{3}$
occupancy probability	$P (E) = 1 / (e^{(E - E_{F}) / k T} + 1)$
Fermi energy	$E_{F} = (3 / 16 \sqrt{2} π)^{2 / 3} h^{2} n^{2 / 3} m$
mass number	$A = Z + N$
nuclear radius	$r = r_{0} A^{1 / 3}, r_{0} \approx 1.2 f m$
mass excess	$Δ = M - A$
radioactive decay	$N = N_{0} e^{- λ t}$
Hubble constant	$H = 71.0 k m / s$
Hubble's law	$v = H r$
conservation of lepton number
conservation of baryon number
conservation of strangeness
eightfold way
weak force
strong force	$\begin{aligned} ℒ_{Q C D} & = {\bar{ψ}}_{i} (i γ^{μ} (D_{μ})_{i j} - m δ_{i j}) ψ_{j} - \frac{1}{4} G_{μ ν}^{a} G_{a}^{μ ν} \\ = {\bar{ψ}}_{i} (i γ^{μ} \partial_{μ} - m) ψ_{i} - g G_{μ}^{a} {\bar{ψ}}_{i} γ^{μ} T_{i j}^{a} ψ_{j} - \frac{1}{4} G_{μ ν}^{a} G_{a}^{μ ν} \end{aligned}$
Noether's theorem
Electroweak interaction	: $ℒ_{E W} = ℒ_{g} + ℒ_{f} + ℒ_{h} + ℒ_{y} .$ $ℒ_{g} = - \frac{1}{4} W_{a}^{μ ν} W_{μ ν}^{a} - \frac{1}{4} B^{μ ν} B_{μ ν}$ $ℒ_{f} = {\overline{Q}}_{i} i D / Q_{i} + {\overline{u}}_{i}^{c} i D / u_{i}^{c} + {\overline{d}}_{i}^{c} i D / d_{i}^{c} + {\overline{L}}_{i} i D / L_{i} + {\overline{e}}_{i}^{c} i D / e_{i}^{c}$ $ℒ_{h} = \| D_{μ} h \|^{2} - λ {(\| h \|^{2} - \frac{v^{2}}{2})}^{2}$ $ℒ_{y} = - y_{u i j} ϵ^{a b} h_{b}^{†} {\overline{Q}}_{i a} u_{j}^{c} - y_{d i j} h {\overline{Q}}_{i} d_{j}^{c} - y_{e i j} h {\overline{L}}_{i} e_{j}^{c} + h . c .$
Quantum electrodynamics	: $ℒ = \bar{ψ} (i γ^{μ} D_{μ} - m) ψ - \frac{1}{4} F_{μ ν} F^{μ ν},$

क्वांटम यांत्रिकी (Quantum Mechanics) सम्पादित 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.

इन्हें भी देखें सम्पादित करें

भौतिकी में प्रायः प्रयुक्त चर

सन्दर्भ सम्पादित करें

Halliday, David Fundamentals of Physics,. Chichester:John Wiley & Sons.ISBN 9780470044742.
Zettili, Nouredine Quantum Mechanics. New York:Wiley.ISBN 0470026782.

बाहरी कड़ियाँ सम्पादित करें

Physics formulae at xs4all.nl