$B$-fields can be considered as a relativistic correction to $E$-fields to achieve this we combine:

• Coulomb’s Law
• Relativistic invariance of charge
• Special relativity

Properties

• Test charge moving $\parallel$ to neutral current carrying wire feels no electrostatic force
• Lorentz transforming into inertial frame of charge results in length contraction of distance between electrons and ions in wire
  • Degree of length contractions are different
• Therefore a current carrying wire appears to have a net charge
• Net charge gives rise to $E$-field
• Force on test charge due to $E-$field in this frame
• Interpret as force due to magnetic field in lab