Suppose E and B represent the electric and magnetic fields in a...

The Lorentz transformations for the transverse components of electric and magnetic fields in SI units, for a frame $S'$ moving with velocity $v$ along the x-axis relative to $S$, are given by:

$E_y' = \gamma (E_y - v B_z)$

B_y' = \gamma \left(B_y +\dfrac{v}{c^2}E_z\right)

where $\gamma =$\dfrac{1}{\sqrt{1 - v^2/c^2}}.

For non-relativistic speeds ($v \ll c$), $\gamma \approx 1$, and the equations approximate to:

$E_y' \approx E_y - v B_z$

$B_y' \approx B_y +$\dfrac{v}{c^2}$E_z$

This can also be verified using dimensional analysis. In SI units, the dimensions of electric field $E$ and magnetic field $B$ are related by $[E] = [vB]$.

Thus, the term $v B_z$ has the dimensions of an electric field, making equation (1) dimensionally correct.

Similarly, the term \dfrac{v E_z}{c^2} has the dimensions of \dfrac{[v][vB]}{[v^2]} = [$B]$, making equation (4) dimensionally correct.

Equations (2) and (3) are dimensionally inconsistent because \dfrac{v B_z}{c^2} does not have the dimensions of an electric field, and $v E_z does not have the dimensions of a magnetic field.<br><br>Therefore, equations (2) and (3) are incorrect.$