A few lines below equation 7.8 D. Tong writes
The final fact is the Lorentz transformation of the electric field: as electron moving with velocity →v in an electric field E will experience a magnetic field →B=γc2(→v×→E).
The note says that it was derived in another note but I couldn't find this expression.
Is this coefficient γ/c2 correct? Griffiths derives this to be −1/c2 and I did not find anything wrong there. See Griffiths electrodynamics, third edition, equation 12.109.
Then I looked at this book which uses Griffiths' expression in Sec. 20.5, but uses →p=m→v instead to →p=γm→v to derive the same result. Which one is correct and why?
Answer
In above Figure-01 an inertial system S′ is translated with respect to the inertial system S with constant velocity
υ=(υ1,υ2,υ3)υ=‖υ‖=√υ21+υ22+υ23∈(0,c)
The Lorentz transformation is x′=x+γ2c2(γ+1)(υ⋅x)υ−γυcctct′=γ(ct−υ⋅xc)γ=(1−υ2c2)−12
For the Lorentz transformation (03a)-(03b), the vectors E and B of the electromagnetic field are transformed as follows E′=γE−γ2c2(γ+1)(E⋅υ)υ+γ(υ×B)B′=γB−γ2c2(γ+1)(B⋅υ)υ−γc2(υ×E)
From equations (05a)-(05b) we have B′=−γc2(υ×E)=−1c2(υ×γE)=−1c2(υ×[γE−γ2c2(γ+1)(E⋅υ)υ])=−1c2(υ×E′)
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ADDENDUM
If in system S we have E=0, then from (04a)-(04b) E′=γ(υ×B)B′=γB−γ2c2(γ+1)(B⋅υ)υ
−−E=v×−−B.abab
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