Momentum of Charged Particle in EM Field Explained

[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.)

 

[PJC01]

I can confirm that the equation p = qA represents the momentum of a charged particle in an electromagnetic field. This equation is derived from the Lorentz force law, which describes the force experienced by a charged particle in an electromagnetic field.

The momentum of a charged particle is a vector quantity, meaning it has both magnitude and direction. In an electromagnetic field, the particle experiences a force due to the electric field and the magnetic field. The electric force is given by qE, where q is the charge of the particle and E is the electric field. The magnetic force is given by qv x B, where v is the velocity of the particle and B is the magnetic field.

The vector magnetic potential A is defined as the curl of the vector potential, A = ∇ x A. This potential is related to the magnetic field by B = ∇ x A. Using the definition of the electric and magnetic forces, we can rewrite the equation for momentum as p = qE + qv x B. Substituting in the expression for B in terms of A, we get p = qE + qv x (∇ x A).

Now, using vector calculus identities, we can rewrite the second term as ∇(v·A) - (v·∇)A. The first term can be recognized as the gradient of a scalar potential, which we can denote as φ. The second term can be rewritten as (v·∇)A = (v x ∇)A. Combining these terms, we get p = qE + q(∇φ - v x ∇)A.

Finally, using the definition of the vector potential, we can rewrite this as p = qE + q∇(φ - v·A). This expression can also be written as p = qE + q∇(φ - v·∇t)A, where t is the time. This is the final form of the equation, p = qE + q∇(φ - v·∇t)A, which is equivalent to p = qA in the case of a time-varying electromagnetic field.

In summary, the equation p = qA represents the momentum of a charged particle in an electromagnetic field, taking into account both the electric and magnetic forces. It is derived from the Lorentz force law and the definition of the vector potential. I hope this explanation helps to