Impact on Mass of A Moving Charged Object – Part 1

[When Speed << c]

Till the end 19th century, when electrical science research were on peak, different pioneer came up with analysis that a charged body moving will gain mass.

J J Thomson was the first to come up with the idea of increase of mass of moving charged body when moving in space. Most electrical pioneers were mostly mathematical with very hotchpotch physical theory. Faraday tubes are cylindrical vortex filaments extending from one atom to another.

He made quantized electricity in to “physical” entity with once single concept called Faraday Tubes, which itself extends from the actual works of Faraday.

He rejected Maxwell’s mathematical theory and took a physical approach, through empirically same as that of Maxwell. But different physical meaning.

He was the first to reject magnetic field, making it as a side effect of electric field in motion. In other words he brought physical unification of electricity and magnetism, unlike Maxwell’s mathematical unification.

His visualization of mass and momentum is different than that of Newtonian. For example, it’s assumed that mass is something resides inside an object. But according to Thomson, the mass is not just inside a charged object but rather extends throughout the space. Hence its motion leads to resistance hence appears like increase in apparent mass. This so-called relativistic mass increase has it’s roots in Thomson, and Not Einstein.

He showed that magnetic field is due to motion of faraday tubes. Hence:


\begin{equation}\begin{aligned}\large\overrightarrow{H}=\overrightarrow{v} \times\overrightarrow{D}\end{aligned}\end{equation}

He treated electric lines of force as not just abstract mathematical representations rather concrete physical reality as Faraday Tube of Induction. The displacement vector D is the abstract representation of number of excessive tubes passing per unit area in space between two points. And the net charge Q is the net effective Faraday tubes attached to a charged object.
So now you cannot have infinite number of lines of forces. Hence you cannot have fractional charge too, which electrolysis also confirms. Hence electricity is quantized in this manner, which can be traced to Thomson only.

Lets have a charged sphere with radius a moving with velocity v, with surface charge Q in the origin has the electric field intensity E at any point of space in spherical coordinate system. It’s given by the expression:

\begin{equation}\begin{aligned}\large E=\frac{Q}{4\pi\epsilon r^{2}} \end{aligned}\end{equation}

Then the net magnetic field intensity becomes:

\begin{equation}\begin{aligned}\large \overrightarrow{H}& =\overrightarrow{v} \times\overrightarrow{D} \\& = vD\sin\theta \\& = v \epsilon E\sin\theta \end{aligned}\end{equation}
Hence, from eqn(2) and eqn(3), we have:

\begin{equation}\begin{aligned}\large H = \frac{v Q \sin\theta}{4\pi r^{2}}\end{aligned}\end{equation}

The kinetic energy of the moving sphere is the net magnetic energy per unit volume in the system, which is given as:

\large K.E = \frac{\mu}{2}\left| H \right|^{2}

Hence, \large K.E = \frac{\mu v^{2}Q^{2}\sin^{2}\theta}{32\pi^{2}r^{4}} \small \text{[using eqn(4))]}

Now to get the total magnetic energy, we shall integrate the above expression over spherical coordinate system, as:

\large K.E = \int_{0}^{2 \pi}\int_{0}^{\pi}\int_{a}^{\infty }\frac{\mu v^{2}Q^{2}\sin^{2}\theta}{32 \pi^{2}r^{4}}r^{2}\sin\theta \partial r \partial \theta \partial \phi
\small K.E = \int_{0}^{2 \pi}\int_{0}^{\pi}\int_{a}^{\infty }\frac{\mu v^{2}Q^{2}\sin^{2}\theta}{32 \pi^{2}r^{4}}r^{2}\sin\theta \partial r \partial \theta \partial \phi

Need to integrate over the entire space from the surface of sphere with radius a to infinity.

So,\begin{aligned}\large K.E & = \large\frac{\mu v^{2}Q^{2}}{32 \pi^{2}}\int_{0}^{2 \pi}\int_{0}^{\pi}\int_{a}^{\infty }\frac{ \sin^{3}\theta}{r^{2}} \partial r \partial \theta \partial \phi \\ & = \large \frac{\mu v^{2}Q^{2}(2\pi)}{32 \pi^{2}}\int_{0}^{\pi}\int_{a}^{\infty }\frac{ \sin^{3}\theta}{r^{2}} \partial r \partial \theta \\ & = \large \frac{\mu v^{2}Q^{2}}{8 \pi}\left[\frac{-1}{r} \right]^{\infty }_{a}\int_{0}^{\pi}\sin^{3}\theta \partial \theta \\ & = \large \frac{\mu v^{2}Q^{2}}{8 \pi a}\int_{0}^{\pi} \frac{1}{4}(\sin3\theta - 3\sin\theta)\partial \theta \end{aligned}

Hence integrating we get the final electrical(magnetic) kinetic energy of the moving charge object as:

\begin{aligned}\large K.E_{E} = \frac{\mu}{4\pi}\frac{Q^{2}v^{2}}{3a}\end{aligned}

Now if the mass of the object is m, the mechanical kinetic energy is given as:

\begin{aligned}\large K.E_{M} = \frac{1}{2}mv^{2}\end{aligned}
Now the total kinetic energy of the entire system is the sum of two energies, the mechanical and electrical.

If M is the apparent mass of the object on motion through the aether, the total kinetic energy is given by:

\begin{aligned}\large K.E = K.E_{M} + K.E_{E} \\ \large or, \frac{1}{2}Mv^{2} = \frac{1}{2}mv^{2} + \frac{\mu}{4\pi}\frac{Q^{2}}{3a} \\ \large or, M = m + \frac{\mu}{2\pi}\frac{Q^{2}}{3a} \end{aligned}
Hence the apparent mass of the object appears to increase by the factor of \frac{\mu}{2\pi}\frac{Q^{2}}{3a}This is for velocity if the speed is very less compared to that of speed of light. As we haven’t taken the electric field intensity E distortion in higher speed, just magnetic field due to motion of tubes.
In next, will do much deeper analysis considering the field distortion with potential and momentum due to motion of tubes.

Next:
https://www.parthasarathimishra.com/wordpress/index.php/2024/03/09/impact-on-mass-of-a-moving-charged-object-part-2/

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