Gauss's theorum in magnetism is the prrof for the existence of........?
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deviations have ever been found in the regime of validity of classical electrodynamics. [1] For example, a result due to Crandall [2][3] shows that any deviation to the exponent in Coulomb's law, F∝r−2+ϵF∝r−2+ϵ, must have ϵ<6×10−17ϵ<6×10−17. This extremely precise test of Coulomb's law also serves as an extremely precise test of Gauss's law.
You can derive Gauss's law if you assume the electromagnetic Lagrangian density,
L=12(E2−B2)−ρV+J⋅AL=12(E2−B2)−ρV+J⋅A
Effecting the variation gives the desired result.
In the case of Gauss's law for magnetism, if it were violated, it would imply the existence of magnetic monopoles. It seems to be the consensus that no magnetic monopoles have ever been detected, so Gauss's law for magnetism seems to be on solid ground experimentally. In the Lagrangian formalism with potentials, you get Gauss's law for magnetism "for free"---it becomes a mathematical identity. So experimental tests of Gauss's law for magnetism become instead experimental tests of the validity of the potential formulation of electrodynamics.
Gauss's law for gravity can be derived from Newton's law of gravitation. Write down Newton's law:
g(r)=−G∫ρ(r′)r−r′∥r−r′∥3d3r′g(r)=−G∫ρ(r′)r−r′‖r−r′‖3d3r′
then take the divergence of both sides,
∇⋅g=−G∫ρ(r′)∇⋅r−r′∥r−r′∥3d3r′∇⋅g=−G∫ρ(r′)∇⋅r−r′‖r−r′‖3d3r′
=−G∫4πδ(r−r′)ρ(r′)d3r=−G∫4πδ(r−r′)ρ(r′)d3r
=−4πGρ=−4πGρ
Gauss's law for gravity can also be derived from general relativity in the so-called "Newtonian limit", in which the system is nearly static, all matter are nonrelativistic [4], and energy densities are low. Under these conditions we can show [5] that particles travelling along geodesics behave as though under the influence of a Newtonian gravitational potential which satisfies
You can derive Gauss's law if you assume the electromagnetic Lagrangian density,
L=12(E2−B2)−ρV+J⋅AL=12(E2−B2)−ρV+J⋅A
Effecting the variation gives the desired result.
In the case of Gauss's law for magnetism, if it were violated, it would imply the existence of magnetic monopoles. It seems to be the consensus that no magnetic monopoles have ever been detected, so Gauss's law for magnetism seems to be on solid ground experimentally. In the Lagrangian formalism with potentials, you get Gauss's law for magnetism "for free"---it becomes a mathematical identity. So experimental tests of Gauss's law for magnetism become instead experimental tests of the validity of the potential formulation of electrodynamics.
Gauss's law for gravity can be derived from Newton's law of gravitation. Write down Newton's law:
g(r)=−G∫ρ(r′)r−r′∥r−r′∥3d3r′g(r)=−G∫ρ(r′)r−r′‖r−r′‖3d3r′
then take the divergence of both sides,
∇⋅g=−G∫ρ(r′)∇⋅r−r′∥r−r′∥3d3r′∇⋅g=−G∫ρ(r′)∇⋅r−r′‖r−r′‖3d3r′
=−G∫4πδ(r−r′)ρ(r′)d3r=−G∫4πδ(r−r′)ρ(r′)d3r
=−4πGρ=−4πGρ
Gauss's law for gravity can also be derived from general relativity in the so-called "Newtonian limit", in which the system is nearly static, all matter are nonrelativistic [4], and energy densities are low. Under these conditions we can show [5] that particles travelling along geodesics behave as though under the influence of a Newtonian gravitational potential which satisfies
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