[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 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}

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:

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

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

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

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

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

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

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

[When Speed → c]

In previous post we have seen how the mass increase for a moving charged object and calculated the value when it’s speed is very less compared to that of light.
https://www.parthasarathimishra.com/wordpress/index.php/2024/02/27/impact-on-mass-of-a-moving-charged-object/

Now we will take a much deeper and detailed analysis for a charged body on how it’s electric field distorts based on speed as first devised by J J Thomson.

The magnetic energy per unit volume of the system is given as:

The electrostatic energy per unit volume is given as:

Total energy of the system shall remain constant, hence:

Taking derivative with respect to E:

As, as kinetic energy is increasing, hence potential energy shall decrease. So the rate of change of potential energy shall be negative.

To find the derivative of v x E, lets write each vector in its component form and calculate the cross product.

We have:

This is the electric field intensity due to motional charge. Let’s call it E₁ . Let’s E₂ be the electric field in space due to the electric scalar potential Φ. Such that:

The scalar potential can be zero in closed loop, but not in our case.

The total electric field intensity in space for the system is given as (E₁ + E₂):

Now lets say a charged object of surface charge Q and radius a moving with a velocity v in the direction of z-axis, for the sake of simplicity.

The velocity vector v becomes\large \overrightarrow{v}=v_{z} \hat{k}

Now the electric field intensity E becomes:

Hence\large \begin{align*}\frac{\overrightarrow{D}}{\epsilon} &= \mu(v_{z}^{2}D_{x}\hat{i}+v_{z}^{2}D_{y}\hat{j})- \overrightarrow{\nabla} \phi\end{align*}

Or,\large \begin{align*}\overrightarrow{D} &= \mu \epsilon(v_{z}^{2}D_{x}\hat{i}+v_{z}^{2}D_{y}\hat{j})- \epsilon\overrightarrow{\nabla} \phi \\ &= \frac{v_{z}^{2}}{c^{2}}(D_{x}\hat{i}+D_{y}\hat{j})- \epsilon\overrightarrow{\nabla} \phi \:\:\:\:\:\small[\mu \epsilon =\frac{1}{c^{2}}] \end{align*}

Splitting the above equation for D into components we get:

Putting the vector components together we get:

Now taking divergent both sides:

As we are dealing with field in space, hence no charge is established there, so the divergence of D should be 0. Hence:

Now to solve this equation, we have to do coordinate transform of Z axis.

Lets put:

Using chain rule:\large \frac{\partial\phi}{\partial z'} = \frac{\partial\phi}{\partial z}\frac{\partial z}{\partial z'} \\ \begin{align*}or,\:\large \frac{\partial^{2}\phi}{\partial z'^{2}} &= \frac{\partial\phi}{\partial z}\frac{\partial z}{\partial z'}\frac{\partial z}{\partial z'} \\ &= \frac{\partial^{2}\phi}{\partial z^{2}}(\frac{\partial z}{\partial z'})^{2} \\ &=(\frac{{c^{2}}-v_{z}^{2}}{c^{2}})\frac{\partial^{2}\phi}{\partial z^{2}}\end{align*}

Now eqn(7) becomes:\large \frac{\partial^{2} \phi }{\partial x^{2}}+\frac{\partial^{2} \phi }{\partial y^{2}}+\frac{\partial^{2} \phi }{\partial z'^{2}} = 0

Now this is Laplace equation for potential. Following is the standard solution for electric potential:

To find the constant A, we need to first find gradient of the scalar potential Φ and find the Displacement D, and then integrate the Displacement over entire surface of the sphere and equate with the net charge Q as per Gauss law.

The gradient of scalar potential, ∇.Φ is:

Now putting the value of components of gradient of potential in eqn(4), (5) and (6), we get:

Now according to Gauss law of electrostatics, the total electric displacement across a surface of a charged object is always the amount of charge contained.

The normal unit vector n for the sphere with radius a is defined as:

Now integral (11) becomes:

Now for proper evaluation, we need to convert this to spherical coordinate.
So we have:\large x = rcos\phi sin\theta \\ \large y = rsin\phi sin\theta \\ \large z = rcos\theta \\ \large dS = r^{2}sin\theta d\theta d\phi

The variable r should be set to radius a as we are not integrating over volume.

So now putting the above expression into the integral we get:

Now we solve this integral by substituting:\frac{v_{z}}{\sqrt{c^{2}-v_{z}^{2}}}cos\theta = tan\psi \\ Hence,\: \frac{-v_{z}}{\sqrt{c^{2}-v_{z}^{2}}}sin\theta\: \partial\theta = sec^{2}\psi \: \partial \psi

Now A becomes:\large A = \frac{Q}{4 \pi \epsilon}\frac{\sqrt{c^2-v^2_{z}}}{c}

You can see when the speed approaches speed of light the factor v_z/c becomes prevalent and the potential becomes undefined at that speed.

The potential can be expressed in spherical coordinate as:

Now putting the value of A in eqn(8), (9), (10) we get D as:

The above equation shows that the the faraday tubes are radial and the resultant polarization D varies inversely as\large D \propto \frac{1}{r^{2}(sin^{2}\theta+\frac{1}{1-\frac{v_{z}^2}{c^2}}cos^{2}\theta)^{\frac{3}{2}}}

To plot the potential and displacement, we can convert eqn(12) of potential into two dimensional equation for the sake of simplicity in visualization. We can replace z as x as we can visualize moving in x direction and remove the actual x part. Also the equation is with respect to moving frame. To get the proper field picture, we can transform the x-axis into the stationary from using x  → x – vt. Then we can ploy y as:

Following are the plot of potential (in gray) and displacement (in blue) for the the charged object when the v =0 and when v = 0.8c.

You can clearly see when the speed approach that of light, the field starts to distort as they don’t get enough time to realign, hence compressing the aether. Following is a short visualization with animation.

Converting the kinetic energy per unit volume into spherical coordinate we get:\large \frac{\mu Q^2}{32 \pi^{2}}\frac{v_{z}^{2}}{(1-\frac{v_{z}^{2}}{c^{2}})}\frac{sin^2\theta}{ r^{4}(sin^{2}\theta+\frac{1}{1-\frac{v_{z}^2}{c^2}}cos^{2}\theta)^{3}}

To solve the above integral, let’s set:cos\theta = u \\ Hence,\: sin\theta\: \partial\theta = - \partial u

Now again set:\begin{align*}\large \frac{v_{z}}{\sqrt{c^{2}-v_{z}^{2}}}u = tan \psi \\ so, \partial u = \frac{\sqrt{c^{2}-v_{z}^{2}}}{v_{z}}sec^{2}\psi \partial \psi \end{align*}

By substituting:\large tan^{-1}\frac{v_{z}}{\sqrt{c^{2}-v_{z}^{2}}} = v_{2}the above equation reduced to:

By evaluating the above extra mass quantity by converting factors sin2v₂ and cos2v₂ in terms of tan and evaluate it using the value of v₂, we get:

You can see if velocity approaches to that of speed of light (v_z = c), the mass tends to increase to infinity. So it’s velocity will remain constant. So its impossible to increase the velocity of a charged boy more than that of the speed of light in this scenario.

This is the same factor which we obtained in our pervious analysis without considering any field distortion.

So does this prove that speed of light is the maximum limit to any charged bodies? And what about neutral bodies?

Will write about it in future post.

Why you believe he derived only 4 equations? Just because your textbook and consensus says? Why not read the original papers and books of Maxwell itself. They are all rotting in archive.org.

For your kind information the so-called Maxwell’s equation used today are actually Heaviside equations.

Maxwell compiled 8 sets of quaternion equations form the experimental findings of Faraday, Gauss, Ampere, Ohm, Weber, Helmholtz, Kohlrausch etc. and converted them into symbolic calculation methods, in his paper A Dynamic Theory of Electromagnetic Field.

Maxwell’s only original contribution is displacement current.

If you expand the vector into component form, there will be total twenty equations involving twenty variable quantities, as you can see in below.

As vector notation were not there at his time, hence all vector equations are written in individual component form.

He thought that certain phenomena in electricity and magnetism lead to the same conclusion as those of optics, namely, that there is an aethereal medium pervading all bodies, and modified only in degree by their presence; that the parts of this medium are capable of being set in motion by electric currents and magnets; that this motion is communicated from one part of the medium to another by forces arising from the connections of those parts; that under the action of these forces there is a certain yielding depending on the elasticity of these connections; and that therefore energy in two different forms may exist in the medium, the one form being the actual energy of motion of its parts, and the other being the potential energy stored up in the connections, in virtue of their elasticity.

We know that when an electric current is established in a conducting circuit, the neighboring part of field is characterized by certain magnetic properties, and that if two circuits are in the field, the magnetic properties of the field due to the two currents are combined. Thus each part of the field is in connection with both currents, and the two currents are put in connection with each other in virtue of their connection with the magnetization of the field.

The first result of this connection that I propose to examine, is the induction of one current by another, and by the motion of conductors in the field.

The second result, which is deduced from this, is the mechanical action between conductors carrying currents. This phenomenon was deduced by Helmholtz and W Thomson. Maxwell followed the reverse order and deduced the mechanical action from the law of induction.

He then applied the phenomena of induction and attraction of currents to the exploration of the electromagnetic field, and the laying down system of lines of magnetic force which indicate its magnetic properties.

He used 8 sets of question A-H as I mentioned at the beginning. They represents following:
(A) The relation between electric displacement, true condition, and total current, compounded of both.
(B) The relation between the lines of magnetic force and the inductive coefficients of a circuit, as already deducted from the laws of induction.
(C) The relation between the strength of a current and its magnetic effects, according to the electromagnetic system of measurement.
(D) The value of the electromotive force in a body, as arising from the motion of the body in the field, the alteration of the field itself, and the variation of electric potential from one part of the field to another.
(E) The relation between an electric displacement, and electromotive force which produced it.
(F) The relation between an electric current, and the electromotive force which produces it.
(G) The relation between the amount of free electricity at any point, and the electric displacements in the neighborhood.
(H) The relation between the increase or diminution of free electricity and the electric currents in the neighborhood.

He determined the mechanical force acting 1st, on a movable conductor carrying an electric current; 2ndly, on a magnetic pole; 3rdly, on an electrified body.

The last result, namely, the mechanical force acting on an electrified body, gives rise to an independent method of electrical measurement founded on its electrostatic effects. The relation between the units employed in the two methods is shown to depend on what he called the “electric elasticity” (now reciprocal of permittivity) of the medium, and to be a velocity, which has been experimentally determined by Weber and Kohlrausch.

The displacement current leads to the possibility of wave propagation in free space. This also lead to the possibility of electric lines of force forming closed loops, independent of any charge. This evolves the way of primitive thinking that current is not something inside the conductor like water in a pipe, rather all around the space surrounding it. Unfortunately today this thinking has taken a backseat.

It’s Oliver Heaviside who invented his vector calculus to downgrade and simplify Maxwell equations into 4 vector form which we “incorrectly” address as Maxwell’s equations.

Heaviside gave birth to practical electrical engineering by solving transatlantic cable problem. He wanted to establish telephone lines. So he removed the scalar and vector potential and force equation from Maxwell original sets of equation to make it more practical, also limited.

Potentials can lead to realization of longitudinal waves also which would be considered impossible as per Heaviside equation.

Unfortunately, due to academic laziness, these truncated equations are still dominating in industry and claimed as Maxwell’s.