Solution to problem 2
... Notice that the potentials are not uniquely determined; one may always perform a gauge transformation, φ → φ + ∂t Γ and
A → A − ∇Γ, and find the same electric and magnetic fields.
Four-potential. To write it in the special relativistic language, one introduces the four-potential Aµ = (φ, A). Its gau ...
... where E and B are both functions of time alone. Show that this equation
is a form a Faraday's law (from general physics) assuming that the average
magnetic flux through the circular loop is twice that on the boundary.
... “Larmor’s formula”, using a classical treatment due to J.J. Thomson and revived by Malcolm Longair. The ab initio derivation using Maxwell’s equations gives the same
We start by taking a stationary charge at rest at time t = 0. The field lines from that charge have the simple configuration s ...
... u =(z x, 0, 0). Make sure the result agrees with the divergence calculated using
Cartesian coordinates. Verify the divergence theorem for this field, with
volume V equal to the part of the cylinder x2+y2≤4 lying in the y≥0 space,
between planes z=0 and z=1.
5. (9) Consider a sphere of radius R with ...
PHY-105: Equations of Stellar Structure
... also a function or r) – see previous handout for more discussion of opacity.
These 4 equations have 7 unknowns (at a given r): P , Mr , Lr , T , ρ, ǫ, κ. So in general we require
expressions for P , κ, and ǫ in terms of ρ, T , and the compositions. These can be complicated, but for
example if we ass ...
In physics, Kaluza–Klein theory (KK theory) is a unified field theory of gravitation and electromagnetism built around the idea of a fifth dimension beyond the usual four of space and time. It is considered to be an important precursor to string theory.The five-dimensional theory was developed in three steps. The original hypothesis came from Theodor Kaluza, who sent his results to Einstein in 1919, and published them in 1921. Kaluza's theory was a purely classical extension of general relativity to five dimensions. The 5-dimensional metric has 15 components. Ten components are identified with the 4-dimensional spacetime metric, 4 components with the electromagnetic vector potential, and one component with an unidentified scalar field sometimes called the ""radion"" or the ""dilaton"". Correspondingly, the 5-dimensional Einstein equations yield the 4-dimensional Einstein field equations, the Maxwell equations for the electromagnetic field, and an equation for the scalar field. Kaluza also introduced the hypothesis known as the ""cylinder condition"", that no component of the 5-dimensional metric depends on the fifth dimension. Without this assumption, the field equations of 5-dimensional relativity are enormously more complex. Standard 4-dimensional physics seems to manifest the cylinder condition. Kaluza also set the scalar field equal to a constant, in which case standard general relativity and electrodynamics are recovered identically.In 1926, Oskar Klein gave Kaluza's classical 5-dimensional theory a quantum interpretation, to accord with the then-recent discoveries of Heisenberg and Schrödinger. Klein introduced the hypothesis that the fifth dimension was curled up and microscopic, to explain the cylinder condition. Klein also calculated a scale for the fifth dimension based on the quantum of charge.It wasn't until the 1940s that the classical theory was completed, and the full field equations including the scalar field were obtained by three independent research groups:Thiry, working in France on his dissertation under Lichnerowicz; Jordan, Ludwig, and Müller in Germany, with critical input from Pauli and Fierz; and Scherrer working alone in Switzerland. Jordan's work led to the scalar-tensor theory of Brans & Dicke; Brans and Dicke were apparently unaware of Thiry or Scherrer. The full Kaluza equations under the cylinder condition are quite complex, and most English-language reviews as well as the English translations of Thiry contain some errors. The complete Kaluza equations were recently evaluated using tensor algebra software.