- #1
johne1618
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Imagine an electron of mass [itex]m_e[/itex] inside a hollow dielectric sphere.
Assume that the electron is traveling with constant velocity [itex]v[/itex] relative to the sphere.
The momentum of the system [itex]p[/itex] comprises just the momentum of the electron
[itex] p = m_e v [/itex]
There is also a circulating magnetic field [itex]B[/itex] around the electron due to its moving charge [itex]e[/itex].
Now let us charge up the dielectric sphere to +V volts.
There is now a radial electric field [itex]E[/itex] outside the sphere due to its surface charge as well as the circulating magnetic field [itex]B[/itex] from the electron.
Thus there is a momentum density [itex]g[/itex] at every point in the electromagnetic field outside the sphere given by:
[itex] g = \epsilon_0 E \times B [/itex]
When this momentum density is integrated over all space outside the sphere one finds a resultant momentum given by
[itex]p_{field} = -\frac{2}{3} \frac{e V}{c^2} v[/itex]
In order for the total momentum of the system to be conserved we must have
[itex] p = p_e + p_{field} [/itex]
Therefore
[itex] p_e = m_e v + \frac{2}{3} \frac{e V}{c^2} v [/itex]
or
[itex] p_e = (m_e + \frac{2}{3} \frac{e V}{c^2}) v [/itex]
There are no forces acting on the electron inside the sphere to change its velocity so the increase of its momentum must be solely due to an increase in its inertia.
Is this reasoning correct?
Assume that the electron is traveling with constant velocity [itex]v[/itex] relative to the sphere.
The momentum of the system [itex]p[/itex] comprises just the momentum of the electron
[itex] p = m_e v [/itex]
There is also a circulating magnetic field [itex]B[/itex] around the electron due to its moving charge [itex]e[/itex].
Now let us charge up the dielectric sphere to +V volts.
There is now a radial electric field [itex]E[/itex] outside the sphere due to its surface charge as well as the circulating magnetic field [itex]B[/itex] from the electron.
Thus there is a momentum density [itex]g[/itex] at every point in the electromagnetic field outside the sphere given by:
[itex] g = \epsilon_0 E \times B [/itex]
When this momentum density is integrated over all space outside the sphere one finds a resultant momentum given by
[itex]p_{field} = -\frac{2}{3} \frac{e V}{c^2} v[/itex]
In order for the total momentum of the system to be conserved we must have
[itex] p = p_e + p_{field} [/itex]
Therefore
[itex] p_e = m_e v + \frac{2}{3} \frac{e V}{c^2} v [/itex]
or
[itex] p_e = (m_e + \frac{2}{3} \frac{e V}{c^2}) v [/itex]
There are no forces acting on the electron inside the sphere to change its velocity so the increase of its momentum must be solely due to an increase in its inertia.
Is this reasoning correct?