At one instant a proton

PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Relativity & Gravitation · Relativistic dynamics and fields in flat spacetime

The Geometry of Gravity, Ch Minkowski spacetime and relativistic dynamics, completed

Problem

At one instant a proton, charge e=1.602×10−19 Ce=1.602\times10^{-19}\ \mathrm{C}, moves with velocity v⃗=(0.60c, 0, 0)\vec v=(0.60c,\,0,\,0) through an electric field E⃗=(3.0, 4.0, 0)×105 V m−1\vec E=(3.0,\,4.0,\,0)\times10^5\ \mathrm{V\,m^{-1}} and a magnetic field B⃗=(0, 0, 2.0×10−3) T\vec B=(0,\,0,\,2.0\times10^{-3})\ \mathrm{T}. Take c=3.00×108 m s−1c=3.00\times10^8\ \mathrm{m\,s^{-1}}. ((a)) Find the three-force F⃗=e(E⃗+v⃗×B⃗)\vec F=e(\vec E+\vec v\times\vec B) at this instant. ((b)) Construct the four-force fμf^\mu from F⃗\vec F and v⃗\vec v. ((c)) Verify fμuμ=0f^\mu u_\mu=0 by writing out the sum, find the rate dE/dtdE/dt at which the proton gains energy, and say which of the two fields does that work.

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