# How to derive the Galilean transformation for magnetic field B without taking help of special theory of relativity?

I have seen in some texts that the Galilean transformation of the magnetic field across inertial reference frames is given by: $$\vec B'= \vec B - (1/c^2)(\vec v\times \vec E)$$. Every where it is stated that it can be derived using Galilean transformation. However I have never found it derived using only Galilean transform. (i.e. without taking help of Lorentz transformations). The other equation concerning Galilean transformation of electric field: $$\vec E'=\vec E + (\vec v \times \vec B)$$ has been seen derived by only Galilean transformation (i.e. by invariance of Lorentz force) very elegantly in many texts. Is it possible to derive this equation $$\vec B'= \vec B - (1/c^2)(\vec v\times \vec E)$$ by only using Galilean transformation?

• See related question: What is the Galilean transformation of the EM field? – Thomas Fritsch Jul 11 at 12:41
• I have seen in some texts Please give titles and authors. This claim just sounds wrong to me. For a treatment of this topic, see Marc De Montigny, Germain Rousseaux, "On the electrodynamics of moving bodies at low velocities," arxiv.org/abs/physics/0512200 – Ben Crowell Jul 11 at 13:39
• The subtlety is that the Galilean transform isn't unique; there are two different possibilities, in the so-called electric and magnetic limits. That's why, to avoid confusion, it's almost always best to start from the Lorentz transformations and take either limit. – knzhou Jul 11 at 14:42