• Lorentz transformations for E & B fields arise from the length contraction in the charge and current density.
• Electrodynamics implies magnetism

Estimated Time: 45 minutes

Students work in groups to derive the Lorentz transformations for E & B fields using a moving capacitor.

• Length Contraction
• Familiarity with charge and current densities

• Tabletop Whiteboard with markers
• A handout for each student

A good introduction to this activity is a mini-lecture on how the charge and current density of an infinitely long wire change when viewed from a reference frame that is moving relative to the lab frame (See §11.1 of the text).

• Many students don't write $u$ in terms of $\alpha$ on part 4 and so don't see the symmetry of the E and B fields in the moving frame.
  • Once all groups have finished part 4, it is a good idea to bring the class back together and have one or more groups present their work up until that point to make sure that all groups are starting from the same place for parts 5 and 6.
• On part 6, emphasize that the goal is to get rid of $\alpha$ by writing in terms of $E_y$ and $B_z$.

Things to emphasize in the wrap-up:

• There are three different situations:
  1. stationary capacitor in the lab frame
  2. moving capacitor in the lab frame
  3. moving capacitor in frame that is moving in the opposite direction
• It is the angles that add and not the velocities.
  • Note: in 2012, one student wanted to verify that you got the same answer using Lorentz velocity addition, so he started from velocity addition and showed that $u'=c \tanh (\alpha+\beta)$.
• Show the full set of Lorentz transformations
  • They have done one piece and that the derivation for the others are in Griffiths.
  • Show the Lorentz transformations in terms of parallel and perpendicular fields.
  • Emphasize that the $E$'s and $B$'s cross.