Question

An electric charge q of mass m in an oscillating electric field Eosinot experiences force q Eosinot. Suppose it starts from rest at time to, and position x-0. a) What is its velocity as a function of time? b) What is its kinetic energy as a function of time? c) What is the particle's time averaged kinetic energy (averaged over an integer # of complete cycles)? d) The vector potential A is defined so that the electric field is E at and the canonical momentum is then given by Pcan P can (t). mv + qA. Find A(t) and

Answer 1

speed of the charge electric is v = - (Eo q/m) cos t

Step-by-step explanation:

The electric charge has a very small mass so it follows the oscillations of the electric field. We force ourselves on the load,

F = q Eo sint

a) To find the velocity of the particle, let's use Newton's second law to find the acceleration and of this by integration the velocity

F = ma

q Eo sint = ma

a = Eo q / m sint

a = dv / dt

dv = adt

∫ dv = ∫ a dt

v-vo = I (Eoq / m) sin t dt

v- vo = Eo q / m (-cos t)

We evaluate the integral from the initial point, as the particle starts from rest Vo = 0, for t = 0

v = - (Eo q / m) cos t

b) Kinetic energy

K = ½ m v2

K = ½ m (Eoq / m)²2 (sint)²

K = ¹/₂ Eo² q² / m sin² t

c) The average kinetic energy over a period

K = ½ m v2

<v2> = (Eoq / m) 2 <cos2 t>

The average of cos2 t = ½, substitute and calculate

K = ½ m (Eoq / m)² ½

K = ¼ Eo² q² / m