* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Section 17
Monte Carlo methods for electron transport wikipedia , lookup
Mean field particle methods wikipedia , lookup
Path integral formulation wikipedia , lookup
Canonical quantization wikipedia , lookup
Newton's theorem of revolving orbits wikipedia , lookup
Four-vector wikipedia , lookup
Renormalization group wikipedia , lookup
Laplace–Runge–Lenz vector wikipedia , lookup
Lagrangian mechanics wikipedia , lookup
Eddy current wikipedia , lookup
Rigid body dynamics wikipedia , lookup
Hunting oscillation wikipedia , lookup
Classical mechanics wikipedia , lookup
Newton's laws of motion wikipedia , lookup
Routhian mechanics wikipedia , lookup
Brownian motion wikipedia , lookup
Faraday paradox wikipedia , lookup
Work (physics) wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Analytical mechanics wikipedia , lookup
Centripetal force wikipedia , lookup
Aharonov–Bohm effect wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Equations of motion of a charge in a field Section 17 • • • • The field acts on a charge e The charge e acts on the field But if e is small, we can neglect the 2nd effect Then the field is independent of position and velocity of e. Hamilton’s principal varies the action. The condition for a minimum determines the equation of motion. r and v are the usual coordinates and velocity of the particle, But is not the usual particle momentum. It is the generalized momentum. There is an extra term due to the A.v in L. Lagrange equation of motion This gradient acts on the r dependence of A and f. (v is considered an independent variable.) HW In this problem The derivatives are taken at constant v. Equation of motion for particle in given fields Equation of motion has the total time derivative of A. The total differential of A is found using the chain rule. Change of A at fixed r. Change of A due to motion from r to r + dr at fixed time. For example, suppose Divide dA by dt Sub into the equation of motion: These two terms are the same and cancel. Equation of motion of charge in given field F = dp/dt Independent of v. Proportional and perpendicular to v. Electric field. A polar vector Magnetic field. An axial vector (Doesn’t change sign under inversion.) Relativistically correct “Lorentz force” Low velocities Rate of change of kinetic + rest energy , ekin HW Work equation Work done by the field on the particle per unit time (Power) The magnetic field does no work on the charge since the magnetic part of the force is perpendicular to dr Time reversal symmetry • Equations of mechanics are invariant under time reversal. • Relativistic equations of motion in E-M field are unchanged by the transform • Then The effect of the time-reversal transform on the potentials is If a certain motion is possible in an EM field, the reversed motion is possible in an EM field having H reversed. Motion in a magnetic field. • H F F X H F Reversed motion with reversed magnetic field • H If you don’t reverse the Hfield, the motion is not the reverse. What effect acting alone causes no change in a particle’s kinetic energy? • A spatially varying vector potential • A time dependent vector potential • A spatially varying scalar potential What effect acting alone causes no change in a particle’s kinetic energy? • A spatially varying vector potential • A time dependent vector potential • A spatially varying scalar potential