Lorentz Force Law in Potential Form

The force on a charge particle in an electromagnetic field is given by
            F = dp/dt q[E + v x B]                            … (1)

In terms of electromagnetic potentials (A and V) the field vectors E and B are given by
            B = ∇ x A and E = - ∇V - ∂A/∂t.                        … (2)
So. Eq. (1) becomes

            F = dp/dt = q[(- ∇V - ∂A/∂t) + v x (V x A)]
                = q[ -∇V - ∂A/∂t + ∇(v.A) –(v.∇)A]
            Dt/dt = - q[∂A/∂t + (v.∇)A + ∇(v-v.A)]                    … (3)
    [∂A/∂t + (v.∇)A] is called the convective derivative of A and written dA/dt. It represents the time rate of charge of A at the location of the moving particle.
    Suppose that at time t the particle is at point r where the potential is A(r + v dt, t + dt).
The change in A is
        dA = A(r + v dt, t + dt) – A(r,t)
        = (∂A/∂x)(vx dt) + (∂A/∂y)(vy dt) + ((∂A/∂z)(vzdt) + (∂A/∂t) dt

. :        dA/dt = ∂A/∂t + (v.∇)A

Eq.(3) becomes
        dp/dt = - q[dA/dt + ∇(V – v.A)]

. :        d(p + qA)/dt = - ∇[q(V – v.A)

Eq. (4) is the Lorentz force formula in terms of Electromagnetic potentials.


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