In the field of physics, numerous formulas and equations are fundamental to solving problems, describing physical phenomena, and understanding the behavior of various systems. This in-depth compilation aims to provide a quick reference to some of the essential formulas and equations used in classical mechanics, electromagnetism, thermodynamics, optics, quantum mechanics, and other branches of physics:

1. Classical Mechanics:

Newton’s Second Law of Motion:

F = ma

The force (F) acting on an object is equal to its mass (m) multiplied by its acceleration (a).

Gravitational Force:

F = G * (m1 * m2) / r^2

The gravitational force (F) between two objects with masses (m1 and m2) is proportional to the product of their masses and inversely proportional to the square of the distance (r) between their centers. G is the universal gravitational constant.

Kinetic Energy:

KE = 0.5 * m * v^2

The kinetic energy (KE) of an object with mass (m) moving at a velocity (v) is given by half the product of mass and the square of velocity.

Work-Energy Theorem:

W = ΔKE = KE_final – KE_initial

The work (W) done on an object is equal to the change in its kinetic energy (ΔKE).

Conservation of Momentum:

Σ(m * v) = Σ(m * v)_initial = Σ(m * v)_final

In a closed system, the total momentum (Σ(m * v)) remains constant before and after a collision or interaction.

2. Electromagnetism:

Coulomb’s Law:

F = k * (|q1 * q2|) / r^2

The electrostatic force (F) between two point charges (q1 and q2) is proportional to the product of their magnitudes and inversely proportional to the square of the distance (r) between them. k is the Coulomb’s constant.

Ohm’s Law:

V = I * R

The voltage (V) across a resistor is equal to the product of the current (I) passing through it and its resistance (R).

Gauss’s Law:

Φ = ε₀ * Σ(q) / εᵣ

The electric flux (Φ) through a closed surface is equal to the total charge (Σ(q)) enclosed by the surface divided by the permittivity of free space (ε₀) multiplied by the relative permittivity (εᵣ) of the material.

Faraday’s Law of Electromagnetic Induction:

ε = -dΦ / dt

The induced electromotive force (ε) in a coil is equal to the negative rate of change of magnetic flux (Φ) through the coil.

3. Thermodynamics:

First Law of Thermodynamics:

ΔU = Q – W

The change in internal energy (ΔU) of a system is equal to the heat (Q) added to the system minus the work (W) done by the system on its surroundings.

Ideal Gas Law:

PV = nRT

The product of pressure (P) and volume (V) of an ideal gas is proportional to the number of moles (n) of the gas, the ideal gas constant (R), and the absolute temperature (T) in Kelvin.

Entropy Change:

ΔS = Q_rev / T

The change in entropy (ΔS) of a system is equal to the heat (Q_rev) absorbed or released in a reversible process divided by the temperature (T) at which the process occurs.

4. Optics:

Snell’s Law of Refraction:

n₁ * sin(θ₁) = n₂ * sin(θ₂)

The ratio of the refractive indices (n) of two media is equal to the ratio of the sines of the angles of incidence (θ₁) and refraction (θ₂) of a light ray passing from one medium to another.

Lens Maker’s Formula:

1 / f = (n – 1) * ((1 / R₁) – (1 / R₂))

The focal length (f) of a thin lens is inversely proportional to the difference between the refractive index of the lens (n) and the refractive index of the medium (usually air) and the difference of the radii of curvature (R₁ and R₂) of its two surfaces.

Diffraction Grating Equation:

m * λ = d * sin(θ)

The wavelength (λ) of light diffracted by a grating with slit spacing (d) at an angle (θ) is given by an integer (m) multiple of the wavelength.

5. Quantum Mechanics:

de Broglie Wavelength:

λ = h / p

The de Broglie wavelength (λ) of a particle with momentum (p) is inversely proportional to its momentum and directly proportional to Planck’s constant (h).

Schrödinger Equation:

iħ * ∂Ψ / ∂t = -ħ² / 2m * ∂²Ψ / ∂x² + V * Ψ

The time-dependent Schrödinger equation describes how the wavefunction (Ψ) of a quantum system evolves over time in terms of its energy (E), mass (m), and potential energy (V).

Heisenberg Uncertainty Principle:

Δx * Δp ≥ ħ / 2

The product of the uncertainties in the position (Δx) and momentum (Δp) of a particle is greater than or equal to Planck’s constant divided by 2.

6. General Relativity:

Einstein Field Equations:

Gᵤᵥ + Λgᵤᵥ = 8πG / c^⁴ * Tᵤᵥ

The Einstein field equations relate the curvature of spacetime (Gᵤᵥ) to the energy-momentum tensor (Tᵤᵥ), cosmological constant (Λ), Newton’s gravitational constant (G), and the speed of light (c).

Schwarzschild Metric:

ds² = -(1 – 2GM / c²) * dt² + (1 – 2GM / c²)^⁻¹ * dr² + r² * (dθ² + sin²θ * dφ²)

The Schwarzschild metric describes the spacetime around a spherically symmetric mass (M), such as a non-rotating black hole.

This comprehensive compilation of key formulas and equations covers a wide range of physics topics and principles. By providing a quick reference to these fundamental relationships, students, researchers, and practitioners can efficiently apply them to various problem-solving scenarios and gain a deeper understanding of the underlying physical phenomena. Always remember that understanding the concepts and assumptions behind the equations is equally important for effective application in the field of physics.