Magnetic Force in Moving Metal Bar

By pulling a metal bar through a magnetic field at constant speed 𝑣, electron charges inside the metal bar are moved by magnetic force 𝐹=(-𝑒)𝑣×𝐵 to build up on one side. The electron current flows towards bottom until net force acting on electron is balanced by charge build up, that is 𝐹_net=0. 𝐹net=𝑞𝐸+𝑞𝑣×𝐵=0
⇒|𝐸|=|𝑣||𝐵| in steady state equilibrium

Electricity in Moving Metal Bar

By connecting a metal bar to a closed circuit during pulling the metal bar through a magnetic field at constant speed 𝑣, electron charges inside the metal bar are moved by magnetic force along one direction. Since the metal bar is connected to a closed circuit, the current inside the metal bar flows around the closed circuit. The work done by moving the metal bar is 𝐹𝐼=𝐼∆𝑙×𝐵=−𝐹hand⇒𝐹𝐼=𝐼𝐿𝐵 Work: 𝑊=𝐹∆𝑥=𝐼𝐿𝐵∆𝑥 Power: 𝑃=𝑊∆𝑡=𝐼𝐿𝐵∆𝑥∆𝑡⇒𝑃=𝐼𝐿𝐵𝑣  ∵emf=|𝐸|𝐿=𝑣𝐵𝐿⇒𝑃=𝐼(emf)

Electric and Magnetic Forces

The electric and magnetic forces are 𝐹=𝑞𝐸₁=𝑞𝑣×𝐵 The electric field due to 1 on 2 is 𝐸₁=14𝜋𝜀₀𝑞₁𝑟²𝑟
The electric force on 2 due to 1 is 𝐹_21,𝑒=𝑞₂𝐸₁=14𝜋𝜀₀𝑞₂𝑞₁𝑟²𝑟⇒𝐹_21,𝑒=14𝜋𝜀₀𝑒²𝑟² The magnetic field due to 1 on 2 is 𝐵₁=𝜇₀4𝜋𝑞₁𝑣₁𝑟²𝑟
The magnetic force on 2 due to 1 is 𝐹_21,𝑚=𝑞₂𝑣₂×𝐵₁=𝑞₂𝑣₂𝜇₀4𝜋𝑞₁𝑣₁𝑟²𝑟⇒𝐹_21,𝑚=𝜇₀4𝜋𝑒²𝑣²𝑟² The ratio of magnetic force over electric force is 𝐹_21,𝑚𝐹_21,𝑒=𝜇₀4𝜋𝑒²𝑣²𝑟²/14𝜋𝜀₀𝑒²𝑟²=𝜇₀𝜀₀𝑣²
Since 𝜇₀𝜀₀=1𝑐²⇒𝐹_21,𝑚𝐹_21,𝑒=𝑣²𝑐²
Therefore 𝐹_𝑚≤𝐹_𝑒

Relativity of Electric and Magnetic Fields

Magnetic field measurement depends on frame of reference. The Lorentz factor 𝛾=1√1−𝑣²/𝑐²>1
Theory of relativity is the speed of light in vacuum = constant 𝑐 = 3×10^8m/s always constant in every references frame. Space and time warp to preserve the speed of light. That is length contraction 𝐿_observed=𝐿/𝛾, time dilation 𝑡_observed=𝛾*𝑡, mass increase 𝑚_observed=𝛾*𝑚. Electric and magnetic field also adjust. Special relativity is wiggy. Since electric and magnetic fields are closely related, they are usually called electromagnetic field. The observed electromagnetic field of a rapidly moving particle is 𝐸′𝑥=𝐸𝑥 𝐸′𝑦=𝛾(𝐸𝑦−𝑣𝐵𝑧) 𝐸′𝑧=𝛾(𝐸𝑧+𝑣𝐵𝑦) 𝐵′𝑥=𝐵𝑥 𝐵′𝑦=𝛾(𝐵𝑦+𝑣𝐸𝑧/𝑐2) 𝐵′𝑧=𝛾(𝐵𝑧−𝑣𝐸𝑦/𝑐2)

Source and Reference

https://www.youtube.com/watch?v=-gLzdhG0IiE&list=PLZ6kagz8q0bvxaUKCe2RRvU_h7wtNNxxi&index=20