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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

image 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×𝐡 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=π‘žπΈ+π‘žπ‘£Γ—π΅=0
β‡’|𝐸|=|𝑣||𝐡| in steady state equilibrium

Electricity in Moving Metal Bar

image 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=πΌβˆ†π‘™Γ—π΅=βˆ’πΉhand⇒𝐹𝐼=𝐼𝐿𝐡 Work: π‘Š=πΉβˆ†π‘₯=πΌπΏπ΅βˆ†π‘₯ Power: 𝑃=π‘Šβˆ†π‘‘=πΌπΏπ΅βˆ†π‘₯βˆ†π‘‘β‡’π‘ƒ=𝐼𝐿𝐡𝑣 β€‰βˆ΅emf=|𝐸|𝐿=𝑣𝐡𝐿⇒𝑃=𝐼(emf)

Electric and Magnetic Forces

The electric and magnetic forces are 𝐹=π‘žπΈ1=π‘žπ‘£Γ—π΅ image The electric field due to 1 on 2 is 𝐸1=14πœ‹πœ€0π‘ž1π‘Ÿ2π‘Ÿ
The electric force on 2 due to 1 is 𝐹21,𝑒=π‘ž2𝐸1=14πœ‹πœ€0π‘ž2π‘ž1π‘Ÿ2π‘Ÿβ‡’πΉ21,𝑒=14πœ‹πœ€0𝑒2π‘Ÿ2 image The magnetic field due to 1 on 2 is 𝐡1=πœ‡04πœ‹π‘ž1𝑣1π‘Ÿ2π‘Ÿ
The magnetic force on 2 due to 1 is 𝐹21,π‘š=π‘ž2𝑣2×𝐡1=π‘ž2𝑣2πœ‡04πœ‹π‘ž1𝑣1π‘Ÿ2π‘Ÿβ‡’πΉ21,π‘š=πœ‡04πœ‹π‘’2𝑣2π‘Ÿ2 image The ratio of magnetic force over electric force is 𝐹21,π‘šπΉ21,𝑒=πœ‡04πœ‹π‘’2𝑣2π‘Ÿ2/14πœ‹πœ€0𝑒2π‘Ÿ2=πœ‡0πœ€0𝑣2
Since πœ‡0πœ€0=1𝑐2⇒𝐹21,π‘šπΉ21,𝑒=𝑣2𝑐2
Therefore πΉπ‘šβ‰€πΉπ‘’

Relativity of Electric and Magnetic Fields

image Magnetic field measurement depends on frame of reference. The Lorentz factor 𝛾=1√1βˆ’π‘£2/𝑐2>1
Theory of relativity is the speed of light in vacuum = constant 𝑐 = 3Γ—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 𝑑observed=𝛾*𝑑, mass increase π‘šobserved=𝛾*π‘š. Electric and magnetic field also adjust. Special relativity is wiggy. Since electric and magnetic fields are closely related, they are usually called electromagnetic field. image The observed electromagnetic field of a rapidly moving particle is 𝐸′π‘₯=𝐸π‘₯ 𝐸′𝑦=𝛾(πΈπ‘¦βˆ’π‘£π΅π‘§) 𝐸′𝑧=𝛾(𝐸𝑧+𝑣𝐡𝑦) 𝐡′π‘₯=𝐡π‘₯ 𝐡′𝑦=𝛾(𝐡𝑦+𝑣𝐸𝑧/𝑐2) 𝐡′𝑧=𝛾(π΅π‘§βˆ’π‘£πΈπ‘¦/𝑐2)

Source and Reference


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


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