Galilean non-invariance of classical electromagnetism

In order to check whether Maxwell's equation is invariant under Galilean transformation, we have to check how the electric and magnetic field transforms under Galilean transformation.Let a charged particle/s or body is moving at a velocity ${\displaystyle {\vec {v}}}$ with respect to S frame.
So, we know that ${\displaystyle {\vec {F}}=q({\vec {E}}+{\vec {v}}\times {\vec {B}})}$ in ${\displaystyle S}$ frame and ${\displaystyle {\vec {F}}\prime =q\prime ({\vec {E}}\prime +({\vec {v}}-{\vec {v_{0}}})\times {\vec {B}}\prime )}$ in${\displaystyle S\prime }$ frame from Lorentz Force.
Now, we assume that Galilean invariance holds. That is, ${\displaystyle {\vec {F}}={\vec {F}}\prime }$ and ${\displaystyle q=q\prime }$(from observation).
${\displaystyle \implies q({\vec {E}}+{\vec {v}}\times {\vec {B}})=q\prime ({\vec {E}}\prime +({\vec {v}}-{\vec {v_{0}}})\times {\vec {B}}\prime )}$

${\displaystyle \implies {\vec {E}}+{\vec {v}}\times {\vec {B}}={\vec {E}}\prime +({\vec {v}}-{\vec {v_{0}}})\times {\vec {B}}\prime }$

(1)

${\displaystyle \implies {\vec {E}}\prime ={\vec {E}}+({\vec {v}}\times {\vec {B}})+({\vec {v_{0}}}\times {\vec {B}}\prime )-({\vec {v}}\times {\vec {B}}\prime )}$
This equation is valid for all ${\displaystyle {\vec {v}}}$.
Let, ${\displaystyle {\vec {v}}=0}$

${\displaystyle \therefore {\vec {E}}\prime ={\vec {E}}+({\vec {v_{0}}}\times {\vec {B}}\prime )}$

(a)

By using equation (a) in (1), we get

${\displaystyle {\vec {B}}\prime ={\vec {B}}}$

(b)

Now, we have to find the transformation(if any) of charge and current densities under Galilean transformation.
Let, ${\displaystyle \rho }$ and ${\displaystyle {\vec {J}}}$ be charge and current densities with respective to S frame respectively. Then, ${\displaystyle \rho \prime }$ and ${\displaystyle {\vec {J}}\prime }$ be the charge and current densities in ${\displaystyle S\prime }$ frame respectively.
We know, ${\displaystyle {\vec {J}}=\rho {\vec {v}}}$
Again, we know that ${\displaystyle \rho =\rho \prime }$
Thus, ${\displaystyle {\vec {J}}\prime =\rho \prime {\vec {v}}\prime =\rho ({\vec {v}}-{\vec {v_{0}}})}$
${\displaystyle \implies {\vec {J}}\prime ={\vec {J}}-{\vec {J_{0}}}here,{\vec {J_{0}}}=\rho {\vec {v_{0}}}}$
Thus, we have

${\displaystyle \rho =\rho \prime }$

(c)

and

${\displaystyle {\vec {J}}\prime ={\vec {J}}-{\vec {J_{0}}}}$

(d)

We know that ${\displaystyle \mu _{0}=N/A^{2}}$. Here, ${\displaystyle A={\frac {q}{t}}}$. Since q'=q, ${\displaystyle {\vec {F}}\prime ={\vec {F}}}$ and t'=t(Galilean principle), we get

${\displaystyle \mu _{0}\prime =\mu _{0}}$

(e)

Now, Let ${\displaystyle f(x,y,z,t),\therefore f(x',y',z',t')}$
t'=t
${\displaystyle {\vec {r}}\prime ={\vec {r}}-{\vec {v_{0}}}t}$
${\displaystyle \therefore {\frac {\partial f}{\partial x}}={\frac {\partial f}{\partial x'}}}$
As, ${\displaystyle {\frac {\partial x'}{\partial x}}=1,{\frac {\partial y'}{\partial x}}=0,{\frac {\partial z'}{\partial x}}=0,{\frac {\partial t'}{\partial x}}=0}$
Similarly, ${\displaystyle {\frac {\partial f}{\partial y}}={\frac {\partial f}{\partial y'}},{\frac {\partial f}{\partial z}}={\frac {\partial f}{\partial z'}}}$ and ${\displaystyle {\frac {\partial }{\partial t'}}={\frac {\partial }{\partial t}}+{\vec {v_{0}}}.{\vec {\nabla }}}$
Thus, we get

${\displaystyle \nabla \prime =\nabla }$

(f)

${\displaystyle {\frac {\partial }{\partial t'}}={\frac {\partial }{\partial t}}+{\vec {v_{0}}}.{\vec {\nabla }}}$

(g)

Now by using equations (a) to (g) we can easily see that Gauss's law and Ampère's circuital law doesn't preserve its form. That is, it non-invariant under Galilean transformation. Whereas, Gauss's law for magnetism and Faraday's law preserve its form under Galilean transformation. Thus, we can see that Maxwell's equation does not preserve its form under Galilean transformation, i.e., it is not invariant under Galilean transformation.

• https://m.youtube.com/watch?feature=youtu.be&v=YdtHAIh-kas
• https://iopscience.iop.org/article/10.1088/0143-0807/30/2/017
• https://www.researchgate.net/publication/228648161_On_the_Galilean_non-invariance_of_classical_electromagnetism
• https://link.springer.com/article/10.1007/s11229-021-03155-y