Canonical momentum of a charged particle in an EM field

[Master J]

Can someone demonstrate how the momentum of a charged particle in a time-varying electromagnetic field is given by

p - qA

where A is the vector magnetic potential?

I've always wondered :-)

Cheers!

 

[chickenz]

If you are looking for a mathematical answer based off Lagrangian/Hamiltonian mechanics, here you go:
http://galileo.phys.virginia.edu/classes/752.mf1i.spring03/ParticleMagneticField.htm

 

[Master J]

Thanks for that. Is there not a simpler derivation ? Surely there must be?

 

[Bill_K]

Sure, just use special relativity.

1) The quantities P_μ = (p, E/c) form a 4-vector
2) The quantities A_μ = (A, Φ) form a 4-vector
3) The total energy of the particle is E' = E + q Φ
4) E' is the 4th component of a 4-vector P'_μ = (P', E')
5) It must be that P'_μ = P_μ + q/c A_μ
6) Hence the other three components are P' = P + q/c A

(If you're wondering about the sign in front of q/c A, note that P' is the canonical momentum and P is the mechanical momentum.
P' = P + q/c A, but P = P' - q/c A.)