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lugita15
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A "paradox" in electromagnetic theory
In Volume II Chapter 28 of his Lectures on Physics, Feynman describes a fundamental inconsistency in classical electromagnetic theory, concerning electromagnetic mass:
Consider an electron in which all of the charge q is uniformly distributed on the surface of a sphere of radius a. The magnitude E of the electric field at a distance r from the center of the electron is [tex]E=\frac{q}{4\pi\epsilon_{0}r^{2}}[/tex]and the density u of its field energy is given by
[tex]u=\frac{\epsilon_{0}E^{2}}{2}=\frac{q^{2}}{32\pi^{2}\epsilon_{0}r^{4}}[/tex]
In order to find the total energy in the electric field produced by the electron, we must integrate this density over all space:
[tex]U=\int^{\infty}_{a}\frac{q^{2}}{8\pi\epsilon_{0}r^{2}}dr=\frac{1}{2}\frac{q^{2}}{4\pi\epsilon_{0}}\frac{1}{a}=\frac{1}{2}\frac{e^{2}}{a}[/tex]
where [tex]e^{2}=\frac{q^{2}}{4\pi\epsilon_{0}}[/tex]
Now according to the theory of relativity, the energy U of a particle is always [tex]mc^{2}[/tex]. So according to relativity, the mass of an electron due to its own electric field is
[tex]m=\frac{1}{2}\frac{e^{2}}{ac^{2}}[/tex]
However, the mass of an electron due to its own field can also be calculated another way. Consider an electron moving at a speed v<<c. The momentum density [tex]\vec{g}[/tex] of its electric field is given by
[tex]\vec{g}=\epsilon_{0}\vec{E}\times\vec{B}[/tex] where [tex]\vec{B}=\vec{v}\times\vec{E}/c^{2}[/tex].
In order to find the total momentum in the electric field produced by the electron, we must integrate this density over all of space. This integration is quite messy, but the result is [tex]\vec{p}=\frac{2}{3}\frac{e^{2}}{ac^{2}}\vec{v}[/tex].
At low velocities, [tex]\vec{p}=m\vec{v}[/tex]. So according to electromagnetic theory, the mass of an electron due to its own electric field is
[tex]m=\frac{2}{3}\frac{e^{2}}{ac^{2}}[/tex]
So electromagnetic theory and special relativity give contradictory results for the mass of an electron due to its own electron field. What is the resolution to this apparent paradox?
Any help would be greatly appreciated.
Thank You in Advance.
In Volume II Chapter 28 of his Lectures on Physics, Feynman describes a fundamental inconsistency in classical electromagnetic theory, concerning electromagnetic mass:
Consider an electron in which all of the charge q is uniformly distributed on the surface of a sphere of radius a. The magnitude E of the electric field at a distance r from the center of the electron is [tex]E=\frac{q}{4\pi\epsilon_{0}r^{2}}[/tex]and the density u of its field energy is given by
[tex]u=\frac{\epsilon_{0}E^{2}}{2}=\frac{q^{2}}{32\pi^{2}\epsilon_{0}r^{4}}[/tex]
In order to find the total energy in the electric field produced by the electron, we must integrate this density over all space:
[tex]U=\int^{\infty}_{a}\frac{q^{2}}{8\pi\epsilon_{0}r^{2}}dr=\frac{1}{2}\frac{q^{2}}{4\pi\epsilon_{0}}\frac{1}{a}=\frac{1}{2}\frac{e^{2}}{a}[/tex]
where [tex]e^{2}=\frac{q^{2}}{4\pi\epsilon_{0}}[/tex]
Now according to the theory of relativity, the energy U of a particle is always [tex]mc^{2}[/tex]. So according to relativity, the mass of an electron due to its own electric field is
[tex]m=\frac{1}{2}\frac{e^{2}}{ac^{2}}[/tex]
However, the mass of an electron due to its own field can also be calculated another way. Consider an electron moving at a speed v<<c. The momentum density [tex]\vec{g}[/tex] of its electric field is given by
[tex]\vec{g}=\epsilon_{0}\vec{E}\times\vec{B}[/tex] where [tex]\vec{B}=\vec{v}\times\vec{E}/c^{2}[/tex].
In order to find the total momentum in the electric field produced by the electron, we must integrate this density over all of space. This integration is quite messy, but the result is [tex]\vec{p}=\frac{2}{3}\frac{e^{2}}{ac^{2}}\vec{v}[/tex].
At low velocities, [tex]\vec{p}=m\vec{v}[/tex]. So according to electromagnetic theory, the mass of an electron due to its own electric field is
[tex]m=\frac{2}{3}\frac{e^{2}}{ac^{2}}[/tex]
So electromagnetic theory and special relativity give contradictory results for the mass of an electron due to its own electron field. What is the resolution to this apparent paradox?
Any help would be greatly appreciated.
Thank You in Advance.
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