Pythagoras’ Theorem: a2+b2=c2
Historical Reference: Attributed to Pythagoras, an ancient Greek mathematician.
Meaning: Describes the relationship between the lengths of the sides of a right triangle, where aa and bb are the lengths of the two shorter sides, and cc is the length of the hypotenuse.
Implication: Fundamental in geometry and trigonometry, providing a method to calculate unknown side lengths in right triangles.
Newton’s Law of Universal Gravitation: F = G(m1m2)/R2
Historical Reference: Proposed by Sir Isaac Newton in his work “Philosophiæ Naturalis Principia Mathematica” in 1687.
Meaning: States that every point mass attracts every other point mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
Implication: Fundamental in classical mechanics, providing a quantitative description of the gravitational force between two objects.
Variation of Gravity with Height: gh = g (r/r+h)2
Historical Reference: Derived from Newton’s law of universal gravitation.
Meaning: The acceleration due to gravity at a height of h above the earth’s surface is gh, and the radius of the earth is R. This equation is used where h is a considerable distance compared to r.
Implication: Provides insight into the decrease in gravitational acceleration as altitude increases, influencing phenomena such as weightlessness in space travel and atmospheric dynamics.
Coulomb’s Law: F= ke*(q1* q2) / r2
Historical Reference: Developed by Charles-Augustin de Coulomb in the 18th century, Coulomb’s law describes the electrostatic force between two charged particles.
Meaning: Organizes energy into inverse square concentrations, illustrating how the strength of the electrostatic force decreases with distance according to the square of the separation distance.
Implication: Fundamental in understanding the behavior of electric charges and the interactions between them, forming the basis for electrostatics and contributing to the development of electromagnetic theory.
Gauss’s Law for Electric Fields: ∇⋅E=ρ/ε0
Historical Reference: Formulated by Carl Friedrich Gauss in the early 19th century, Gauss’s law for electric fields relates the electric flux through a closed surface to the charge enclosed within the surface.
Meaning: In vector calculus, this law describes the flow of electric field lines through a closed surface, providing insights into the distribution of electric charge.
Implication: Essential in analyzing the behavior of electric fields and charges, Gauss’s law helps in solving electrostatic problems and understanding the principles of electric field behavior in various contexts.
Gauss’s Law for Magnetic Fields: ∇⋅B=0
Historical Reference: Formulated by Carl Friedrich Gauss in the early 19th century, Gauss’s law for electric fields relates the electric flux through a closed surface to the charge enclosed within the surface.
Meaning: It states that the magnetic field B has divergence equal to zero, It is equivalent to the statement that magnetic monopoles do not exist.
Implication: The law in this form states that for each volume element in space, there are exactly the same number of “magnetic field lines” entering and exiting the volume. No total “magnetic charge” can build up in any point in space.
Gauss’s Law for Gravitation: ΦD= Qfree
Historical Reference: The name “Gauss’s law for magnetism” is not universally used. The law is also called “Absence of free magnetic poles”.
Meaning: States that the gravitational flux through any closed surface is proportional to the enclosed mass, providing a mathematical representation of gravitational interactions.
Implication: Fundamental in gravitational theory, Gauss’s law for gravitation helps in understanding the distribution of gravitational fields and predicting gravitational effects based on mass distribution.
Faraday’s Law of Electromagnetic Induction: E=dΦB/dt
Historical Reference: Discovered by Michael Faraday in the 19th century, Faraday’s law of electromagnetic induction describes how a changing magnetic field induces an electromotive force (EMF) or electric field.
Meaning: States that a changing magnetic field induces an electric field, demonstrating the connection between magnetic and electric phenomena.
Implication: Crucial in the development of electromagnetism and electrical engineering, Faraday’s law explains the principles behind generators, transformers, and various electrical devices.
Resonance f0=1/2π√(LC)
Resonance: In a circuit, resonance occurs when the frequency of an applied alternating current (AC) matches the natural frequency of the circuit. LC Circuit: In an LC circuit, the inductor stores energy in a magnetic field, while the capacitor stores energy in an electric field. When an AC voltage is applied to the circuit, these components interact to produce oscillations
Meaning: At this frequency, the impedance of the circuit is at a minimum, and the current flowing through it is at a maximum.
Natural Frequency: The natural frequency of an LC circuit is determined by the values of its inductance and capacitance. It is the frequency at which the circuit would oscillate if left to itself after being disturbed.
Resonant Frequency: The resonant frequency of an LC circuit is the frequency at which it exhibits resonance. At this frequency, the inductive reactance (XL) of the inductor and the capacitive reactance (XC) of the capacitor are equal in magnitude but opposite in sign. This means that they cancel each other out, resulting in a purely resistive circuit.
Ohm’s Law: E=IR
Historical Reference: Named after Georg Simon Ohm, Ohm’s law defines the relationship between electric potential (voltage), current, and resistance in an electrical circuit.
Meaning: States that the electric potential (voltage) across a resistor is directly proportional to the current flowing through it and the resistance of the resistor.
Implication: Fundamental in circuit analysis and electrical engineering, Ohm’s law governs the behavior of electrical circuits and is used extensively in the design and analysis of electrical systems.
Stefan-Boltzmann Law: E=σT4
Historical Reference: Developed by Josef Stefan and Ludwig Boltzmann in the late 19th century, the Stefan-Boltzmann law relates the radiant power emitted by a surface to its temperature.
Meaning: States that the total radiant power emitted by a black body per unit surface area is directly proportional to the fourth power of its absolute temperature.
Implication: Essential in thermodynamics and astrophysics, the Stefan-Boltzmann law helps in understanding the energy radiation from various objects, including stars and planets
Maxwell’s Equations
Historical Reference: Formulated by James Clerk Maxwell in the 19th century, Maxwell’s equations describe the behavior of electric and magnetic fields in classical electromagnetism.
First: Gauss’s law for static electric fields – Electric fiends emanate from electric charges with an electrical charge there is an electrical field emanating from it. The strength of this field is proportional to e0. Static charges only affect other charges, not magnets:
∇⋅E=ρ/ε0
Where ∇·E is the divergence of the electric field, ε0 is the vacuum permittivity and ρ is the total volume charge density (charge per unit volume).
Second: Gauss’s law for static magnetic fields – There are no magnetic (monopoles) charges in the universe. There is always as much field pointed in as there is pointed out. Static magnets will affect only other magnets, not charges:
∇⋅B=0
Where ∇·B is the divergence of the magnetic field.
Third: Faraday’s law states a changing (in time) magnetic field produces an electric field. A moving magnet will create an electrical field. A moving magnet it will affect a charge, creating a current of electricity.
∇xE = dB/dt
Where E is the electromotive force (emf) and dB is the magnetic flux.
Fourth: Ampere-Maxwell’s law states a changing electric field produces a magnetic field. The first term describes a moving charge it will generate a magnetic field. Two wires will be attracted to each other if they have a current flowing through them. The second term says a magnetic field is created by moving charges which are created by moving electrical fields (i.e., an electromagnetic wave is propagated). This term recognized the current flow through capacitors (displacement capacitance) can create magnetic fields, The idea that changing magnetic fields are created by changing electrical fields was Maxwell’s addition to Amperes law. This fourth equation is a wave equation.
∇×B = μ0J+ϵ0 ∂E/∂t
Where: E is the electric field, B is the magnetic field, ρ is the charge density, J is the current density, ϵ0 is the permittivity of free space, and μ0 is the permeability of free space.
Equations involving the speed and density of energy
Speed of energy:
c2 = 1/μ0ε0
c = 1/√μ0ε0
Impedance of space:
Z0 = √(μ0/ε0)
Admittance of space: The Rate at Which Energy is Accepted by Time: Y0
Y0 = √(ε0/μ0)
These equations form the bedrock of classical electromagnetism and are crucial for understanding the behavior of electric and magnetic field.
Where: Y0 represents the Admittance of a Field to a Change in Energy Density, ε0 represents Permittivity, and μ0 represents Permeability.
This equation unveils the dynamic interplay between energy and the electromagnetic (EM) field. It offers insights into how time accommodates and interacts with energy concentrations, shaping the gravitational phenomena we observe.
The Energy Momentum Equation: E = √m2c4 + p2c2
Where: E represents energy, m represents mass, and c represents the speed of light in a vacuum, P is the momentum of an object.
Then translated to energy at rest gives us this famous equation:
Einstein’s Mass-Energy Equivalence in Quantum Admittance: E = m/Z02ε02
This equation expresses the relationship between mass (m) and energy (E), proposing that the energy of an object is equal to its mass multiplied by a new constant—where the phase angle of the wave (ϕ) serves as a fundamental constant, rather than the speed of light (c).
Einstein’s Mass-Energy Equivalence: E=mc2
Historical Reference: Proposed by Albert Einstein in his theory of special relativity in 1905, the mass-energy equivalence principle states that mass and energy are equivalent and interchangeable.
Einstein’s Field Equations: Rμν − 1/2(gμνR) + Λgμν = (8πG/c4) Tμν
The Einstein Field Equations are ten equations, contained in the tensor equation shown above, which describe gravity as a result of spacetime being curved by mass and energy. Guv is determined by the curvature of space and time at a particular point in space and time, and is equated with the energy and momentum at that point. The solutions to these equations are the components of the metric tensor guv, which specifies the spacetime geometry. The inertial trajectories of particles can then be found using the geodesic equation.
Despite extensive experimental validation, these equations face fundamental clashes with quantum mechanics. They assume precise localization of energy and momentum for particles, conflicting with the uncertainty principle of quantum theory, which limits our ability to know both properties simultaneously. This contradiction necessitates resolution.
The equations seem to have a fundamental flaw. Even though both the theory and its equations have been repeatedly confirmed through experiments, they clash with the well-established principles of quantum mechanics. This clash isn’t just theoretical – it applies to any experiment in a lab.
The equations assume we can pinpoint the energy and momentum of a particle at any exact point in space and time. However, the uncertainty principle, a cornerstone of quantum theory, tells us this is impossible. There’s a fundamental limit to how precisely we can know both a particle’s energy and momentum.
Planck’s Equation: E=hf
Historical Reference: Introduced by Max Planck in 1900.
Meaning: Relates the energy of a photon to its frequency, where h is Planck’s constant.
Implication: Foundational in quantum mechanics, providing a key relationship between the energy and frequency of electromagnetic radiation
Schwarzschild radius Rs=2GM/c2)
Historical Reference: The Schwarzschild radius was named after the German astronomer Karl Schwarzschild, who calculated this exact solution for the theory of general relativity in 1916.
Meaning: The Schwarzschild radius or the gravitational radius is a physical parameter in the Schwarzschild solution to Einstein’s field equations that corresponds to the radius defining the event horizon of a Schwarzschild black hole.
Implication: It is a characteristic radius associated with any quantity of mass.
Lorentz Factor γ=1/√(1- v2/c2)
Historical Reference: Introduced in the context of special relativity by Hendrik Lorentz and confirmed by Albert Einstein.
Meaning: The Lorentz factor (γ) quantifies the effect of time dilation and length contraction on moving objects relative to a stationary observer, increasing with velocity according to the formula , where v is the velocity and c is the speed of light.
Implication: The Lorentz factor plays a crucial role in relativistic mechanics, accounting for the observed phenomena of time dilation and length contraction at speeds approaching the speed of light. It’s essential for understanding the behavior of particles in particle accelerators, the stability of high-speed spacecraft, and the fundamentals of cosmology.
Lorentz Force Law: F=qE+qv*B
Historical Reference: Historians suggest that the law is implicit in a paper by James Clerk Maxwell, published in 1865. Hendrik Lorentz arrived at a complete derivation in 1895, identifying the contribution of the electric force a few years after Oliver Heaviside correctly identified the contribution of the magnetic force.
Meaning: Lorentz’s force explains the mathematical equations along with the physical importance of forces acting on the charged particles that are traveling through the space containing electric as well as the magnetic field.
Implication: It says that the electromagnetic force on a charge q is a combination of (1) a force in the direction of the electric field E (proportional to the magnitude of the field and the quantity of charge), and (2) a force at right angles to both the magnetic field B and the velocity v of the charge (proportional to the magnitude of the field, the charge, and the velocity)
Lorentz Magnetic Force (Torque) Equation τ=q (r×B)
Historical Reference: This concept is derived from the Lorentz force law and is fundamental in understanding the rotational motion of charged particles in magnetic fields.
Meaning: This equation describes the additional rotational force experienced by a charged particle moving through a magnetic field. It accounts for the interaction between the magnetic field and the motion of the particle, resulting in a twisting or rotational motion.
Implication: The magnetic Lorentz force (torque) is essential for understanding phenomena such as the behavior of charged particles in cyclotrons, where particles are accelerated in circular paths by magnetic fields, as well as the dynamics of magnetic materials and electromagnetic devices.
Lorentz Time Dilation Equation Δt′=Δt/√ (v2/c2)
Historical Reference: Developed by Hendrik Lorentz as part of his transformations to account for the effects of relative motion between inertial frames of reference, contributing to the foundation of special relativity.
Meaning: The Lorentz time dilation equation describes how time intervals appear to be dilated or stretched when observed from a frame of reference moving at a significant fraction of the speed of light relative to a stationary frame.
Implication: This equation has profound implications for the nature of time and motion, leading to phenomena such as time dilation in high-speed travel and relativistic effects in particle accelerators and astrophysical phenomena.
Schwarzschild metric r=2Gm/c2
Historical Reference: Developed by Karl Schwarzschild in 1916 as a solution to Einstein’s field equations of general relativity.
Meaning: Represents the Schwarzschild radius (r), which defines the size of the event horizon of a non-rotating black hole. It relates the mass (m) of an object to its Schwarzschild radius, gravitational constant (G), and the speed of light (c).
Implication: Fundamental in the study of black holes and gravitational phenomena, providing a theoretical framework for understanding the curvature of spacetime around massive objects.
Schrödinger Equation: HΨ=iℏ(∂t/∂)Ψ
Historical Reference: Proposed by Erwin Schrödinger in 1926 as part of the development of quantum mechanics.
Meaning: Represents the fundamental wave equation of quantum mechanics, where H is the Hamiltonian operator, Ψ is the wave function, i is the imaginary unit, ℏ is the reduced Planck constant, and ∂t/∂ represents the partial derivative with respect to time.
Implication: Describes the behavior of quantum systems, including the time evolution of wave functions, and serves as a cornerstone in quantum mechanics, facilitating the prediction of particle behavior and the interpretation of experimental results.
Redshift: z = (λobs – λemit) / λemit
Where: λobs is the wavelength of the radiation as observed by the observer, and λemit is the wavelength of the radiation as emitted by the source.
Meaning: A positive value of z indicates a redshift, meaning the source is moving away from the observer. A negative value indicates a blueshift, meaning the source is moving towards the observer.
Implication:
Doppler Effect: Redshift is a manifestation of the Doppler effect, a phenomenon where the perceived frequency of a wave changes due to relative motion between the source and the observer. In the case of light, a moving source causes a shift in wavelength.
Expansion of the Universe: One of the most significant applications of redshift is in cosmology. Astronomers have observed that the light from distant galaxies is redshifted, indicating that these galaxies are moving away from us. This observation led to the development of the Big Bang theory, which posits that the universe began as a hot, dense point and has been expanding ever since.
Measuring Distance: Redshift can also be used to estimate the distance to astronomical objects. The more redshifted the light from a galaxy, the farther away it is. This relationship is based on the Hubble Law, which states that the recessional velocity of a galaxy is proportional to its distance.
By measuring the redshift of light from distant objects, astronomers can gain insights into the universe’s age, its rate of expansion, and the distribution of matter and energy within it.