Question

1. Show that a Lagrangian of the form $L = T - V = frac{1}{2}mv^2 - qV + qvec{A} cdot vec{v}$ for a particle of mass m, charge q, and velocity $vec{v}$ inside a general, time-dependent electromagnetic field described by electric potential $V(vec{x}, t)$ and magnetic vector potential $vec{A}(vec{x}, t)$ produces the correct form $vec{F} = q(vec{E} + vec{v} imes vec{B})$ of the Lorentz force.

          1. Show that a Lagrangian of the form $L = T - V = frac{1}{2}mv^2 - qV + qvec{A} cdot vec{v}$ for a particle of mass m, charge q, and velocity $vec{v}$ inside a general, time-dependent electromagnetic field described by electric potential $V(vec{x}, t)$ and magnetic vector potential $vec{A}(vec{x}, t)$ produces the correct form $vec{F} = q(vec{E} + vec{v} 	imes vec{B})$ of the Lorentz force.

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$1. Show that a Lagrangian of the form L = T - V = frac12mv^2 - qV + qvecA cdot vecv for a particle of mass m, charge q, and velocity vecv inside a general, time-dependent electromagnetic field described by electric potential V(vecx, t) and magnetic vector potential vecA(vecx, t) produces the correct form vecF = q(vecE + vecv imes vecB) of the Lorentz force.$

Added by Gema W.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Sri K

08/31/2023

Step 1

We can do this by using the Euler-Lagrange equation: d/dt (∂L/∂v) - ∂L/∂x = 0 where v is the velocity of the particle and x is its position. Taking the partial derivatives of the Lagrangian with respect to v and x, we get: ∂L/∂v = 2mv - qA ∂L/∂x = -∂V/∂x + Show more…

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Show that a Lagrangian of the form L = T - V = 1/2mv^2 - qV + qA · v for a particle of mass m, charge q, and velocity v inside a general, time-dependent electromagnetic field described by electric potential V(x,t) and magnetic vector potential A(x,t) produces the correct form F = q(E + v x B) of the Lorentz force.

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