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

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

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

carbon is found in all living things. carbon atoms move constantly through

living organisms, the oceans, the atmosphere, and the earth’s crust in what is

known as the carbon cycle. the directions taken by carbon atoms through this

cycle are very complicated and can take millions of years to make a full circle.

all animals, from humans to the dinosaurs are part of the carbon cycle. when

animals eat food, they get carbon in the form of carbohydrates and proteins.

in animals, oxygen combines with food in the cells to produce energy for daily

activity and then gives off carbon. the carbon combines with oxygen to form

carbon dioxide (co2) and is released back into the atmosphere as a waste

product when animals breathe and exhale.

from 145-65 million years ago, earth was much

hotter than today and covered with dense,

tropical swamp forests. the trees and other

plants were immense and

provided an endless supply of

food for the giant animals that

roamed the land. somewhere in

the air above one of these

forests, a lone carbon atom has joined

up with two oxygen atoms to form a molecule of co2.

the co2 molecule was sucked into the tiny holes (stomata) on the leaf of a

fern plant and joined with sunlight, chlorophyll and water to make food and

energy in the plant’s cells through photosynthesis. the oxygen (o2) from the

co2 molecule was sent back into the atmosphere; the carbon atom (c) was

detached and used to make a molecule of sugar.

if that carbon atom had been eaten instead, a totally

different situation would have developed. let’s say

that a giant plant-eating dinosaur (herbivore) ate the

fern for breakfast, and swallowed the carbon in the

fern in the form of carbohydrates and proteins. in

the process of respiration, oxygen combined with

the carbohydrates and proteins in the dinosaur’s cells

to provide energy for its daily activity.  

it is the second option.