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`Magnetic Force in Moving Metal Bar Electricity in Moving Metal Bar Electric and Magnetic Forces Relativity of Electric and Magnetic Fields Source and Reference`

# 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 𝐹net=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 𝐹21,𝑒=𝑞2𝐸1=14𝜋𝜀0𝑞2𝑞1𝑟2𝑟⇒𝐹21,𝑒=14𝜋𝜀0𝑒2𝑟2 The magnetic field due to 1 on 2 is 𝐵1=𝜇04𝜋𝑞1𝑣1𝑟2𝑟
The magnetic force on 2 due to 1 is 𝐹21,𝑚=𝑞2𝑣2×𝐵1=𝑞2𝑣2𝜇04𝜋𝑞1𝑣1𝑟2𝑟⇒𝐹21,𝑚=𝜇04𝜋𝑒2𝑣2𝑟2 The ratio of magnetic force over electric force is 𝐹21,𝑚𝐹21,𝑒=𝜇04𝜋𝑒2𝑣2𝑟2/14𝜋𝜀0𝑒2𝑟2=𝜇0𝜀0𝑣2
Since 𝜇0𝜀0=1𝑐2𝐹21,𝑚𝐹21,𝑒=𝑣2𝑐2
Therefore 𝐹𝑚≤𝐹𝑒

## Relativity of Electric and Magnetic Fields Magnetic field measurement depends on frame of reference. The Lorentz factor 𝛾=11−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 𝐿observed=𝐿/𝛾, time dilation 𝑡observed=𝛾*𝑡, mass increase 𝑚observed=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

ID: 200101202 Last Updated: 12/1/2020 Revision: 0 Home 5

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