Physics Equations and Diagrams — Documenting Scientific Concepts

Newton's Second Law states that the force acting on an object is equal to the mass of the object multiplied by its acceleration:

$\vec{F} = m\vec{a}$F=ma

Where:

• $\vec{F}$F is the net force vector (in Newtons, N)
• $m$m is the mass (in kilograms, kg)
• $\vec{a}$a is the acceleration vector (in m/s²)

In component form:

$F_x = ma_x, \quad F_y = ma_y, \quad F_z = ma_z$Fx​=max​,Fy​=may​,Fz​=maz​

The kinetic energy of an object with mass $m$m moving at velocity $v$v is:

$KE = \frac{1}{2}mv^2$KE=21​mv2

For a system of particles, the total kinetic energy is:

$KE_{total} = \sum_{i=1}^{n} \frac{1}{2}m_i v_i^2$KEtotal​=∑i=1n​21​mi​vi2​

Derivation from Work-Energy Theorem:

Starting with $W = \int F \, dx$W=∫Fdx and $F = ma = m\frac{dv}{dt}$F=ma=mdtdv​:

$W = \int m\frac{dv}{dt} \, dx = \int m\frac{dx}{dt} \, dv = \int mv \, dv = \frac{1}{2}mv^2$W=∫mdtdv​dx=∫mdtdx​dv=∫mvdv=21​mv2

Rendering diagram...

Equations of Motion:

Parallel to incline: $ma = mg\sin\theta - f$ma=mgsinθ−f

Perpendicular to incline: $N = mg\cos\theta$N=mgcosθ

Maxwell's equations describe how electric and magnetic fields are generated and altered by charges and currents:

1. Gauss's Law (Electric):

$\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}$∇⋅E=ϵ0​ρ​

Electric field divergence equals charge density divided by permittivity.

2. Gauss's Law (Magnetic):

$\nabla \cdot \vec{B} = 0$∇⋅B=0

No magnetic monopoles exist; magnetic field lines form closed loops.

3. Faraday's Law:

$\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$∇×E=−∂t∂B​

A changing magnetic field induces an electric field.

4. Ampère-Maxwell Law:

$\nabla \times \vec{B} = \mu_0\vec{J} + \mu_0\epsilon_0\frac{\partial \vec{E}}{\partial t}$∇×B=μ0​J+μ0​ϵ0​∂t∂E​

Magnetic fields are generated by currents and changing electric fields.

Where:

• $\vec{E}$E = electric field (V/m)
• $\vec{B}$B = magnetic field (T)
• $\rho$ρ = charge density (C/m³)
• $\vec{J}$J = current density (A/m²)
• $\epsilon_0$ϵ0​ = permittivity of free space
• $\mu_0$μ0​ = permeability of free space

Taking the curl of Faraday's law and substituting Ampère-Maxwell law (in vacuum, $\rho = 0$ρ=0, $\vec{J} = 0$J=0):

$\nabla \times (\nabla \times \vec{E}) = -\frac{\partial}{\partial t}(\nabla \times \vec{B})$∇×(∇×E)=−∂t∂​(∇×B)

Using the vector identity $\nabla \times (\nabla \times \vec{E}) = \nabla(\nabla \cdot \vec{E}) - \nabla^2\vec{E}$∇×(∇×E)=∇(∇⋅E)−∇2E and $\nabla \cdot \vec{E} = 0$∇⋅E=0:

$-\nabla^2\vec{E} = -\mu_0\epsilon_0\frac{\partial^2 \vec{E}}{\partial t^2}$−∇2E=−μ0​ϵ0​∂t2∂2E​

This gives the wave equation:

$\nabla^2\vec{E} = \mu_0\epsilon_0\frac{\partial^2 \vec{E}}{\partial t^2}$∇2E=μ0​ϵ0​∂t2∂2E​

The wave speed is:

$c = \frac{1}{\sqrt{\mu_0\epsilon_0}} \approx 3 \times 10^8 \text{ m/s}$c=μ0​ϵ0​​1​≈3×108 m/s

Rendering diagram...

Key Properties:

• Electric and magnetic fields oscillate perpendicular to each other
• Both fields are perpendicular to the direction of propagation
• Energy flows in the direction of $\vec{E} \times \vec{B}$E×B (Poynting vector)

The time-dependent Schrödinger equation describes how quantum states evolve:

$i\hbar\frac{\partial}{\partial t}\Psi(\vec{r}, t) = \hat{H}\Psi(\vec{r}, t)$iℏ∂t∂​Ψ(r,t)=H^Ψ(r,t)

Where:

• $\Psi(\vec{r}, t)$Ψ(r,t) is the wave function
• $\hbar = h/2\pi$ℏ=h/2π is the reduced Planck constant
• $\hat{H}$H^ is the Hamiltonian operator

Time-Independent Form:

For stationary states with $\Psi(\vec{r}, t) = \psi(\vec{r})e^{-iEt/\hbar}$Ψ(r,t)=ψ(r)e−iEt/ℏ:

$\hat{H}\psi(\vec{r}) = E\psi(\vec{r})$H^ψ(r)=Eψ(r)

For a Particle in a Potential:

$-\frac{\hbar^2}{2m}\nabla^2\psi + V(\vec{r})\psi = E\psi$−2mℏ2​∇2ψ+V(r)ψ=Eψ

The uncertainty principle states that certain pairs of physical properties cannot be simultaneously known to arbitrary precision:

Position-Momentum Uncertainty:

$\Delta x \cdot \Delta p \geq \frac{\hbar}{2}$Δx⋅Δp≥2ℏ​

Energy-Time Uncertainty:

$\Delta E \cdot \Delta t \geq \frac{\hbar}{2}$ΔE⋅Δt≥2ℏ​

General Form for Operators:

For any two observables $\hat{A}$A^ and $\hat{B}$B^:

$\Delta A \cdot \Delta B \geq \frac{1}{2}|\langle[\hat{A}, \hat{B}]\rangle|$ΔA⋅ΔB≥21​∣⟨[A^,B^]⟩∣

Where $[\hat{A}, \hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A}$[A^,B^]=A^B^−B^A^ is the commutator.

Rendering diagram...

Key Concepts:

• Before measurement: system exists in superposition of states
• Measurement causes wave function collapse
• Outcome is probabilistic, determined by $|\langle\psi_i|\psi\rangle|^2$∣⟨ψi​∣ψ⟩∣2
• After measurement: system is in definite eigenstate

Zeroth Law (Thermal Equilibrium):

If systems A and B are each in thermal equilibrium with system C, then A and B are in thermal equilibrium with each other.

First Law (Energy Conservation):

$\Delta U = Q - W$ΔU=Q−W

Where:

• $\Delta U$ΔU = change in internal energy
• $Q$Q = heat added to system
• $W$W = work done by system

Second Law (Entropy):

For any thermodynamic process:

$\Delta S_{universe} \geq 0$ΔSuniverse​≥0

The entropy of an isolated system never decreases. For a reversible process:

$dS = \frac{dQ_{rev}}{T}$dS=TdQrev​​

Third Law (Absolute Zero):

As temperature approaches absolute zero, the entropy of a perfect crystal approaches zero:

$\lim_{T \to 0} S = 0$limT→0​S=0

The Carnot cycle represents the most efficient heat engine possible between two temperature reservoirs.

Efficiency:

$\eta = 1 - \frac{T_C}{T_H} = \frac{W}{Q_H}$η=1−TH​TC​​=QH​W​

Where:

• $T_H$TH​ = hot reservoir temperature (K)
• $T_C$TC​ = cold reservoir temperature (K)
• $W$W = work output
• $Q_H$QH​ = heat input from hot reservoir

Carnot Cycle Stages:

Rendering diagram...

Key Insight: No real engine can exceed Carnot efficiency.

Ideal Gas Law:

$PV = nRT$PV=nRT

Where:

• $P$P = pressure (Pa)
• $V$V = volume (m³)
• $n$n = number of moles
• $R$R = universal gas constant (8.314 J/(mol·K))
• $T$T = temperature (K)

Alternative Forms:

Using $n = N/N_A$n=N/NA​ where $N$N is number of molecules:

$PV = NkT$PV=NkT

Where $k = R/N_A$k=R/NA​ is Boltzmann's constant.

Internal Energy of Ideal Gas:

For a monatomic ideal gas:

$U = \frac{3}{2}nRT$U=23​nRT

For a diatomic ideal gas:

$U = \frac{5}{2}nRT$U=25​nRT