Electric Field Charged Particle
Let us consider a particle of mass m and charge q in a uniform constant electric field E. By uniform, we mean E is same (both in magnitude and direction) everywhere in space, i.e. E = (Ex, Ey, Ez) are not functions of position co-ordinates (x, y, z); by constant, we mean that E does not change with time too. The equation of motion of the particle, therefore, is
Integrating it, we find
where v0 is constant of integration, v = v0 at t = 0. Further integration gives,
where constant of integer r0 is initial position of the particle. We can always choose r0 = 0, i.e. fix the origin of frame S at the initial position of particle. Eq. (iii) is the well known equation of motion of a particle moving with constant acceleration,
The electric field is a conservative force. The work done by an electric field is equal to negative change of potential energy of the particle:
The work done is always equal to the change in kinetic energy of the particle.
The energy conservation in electrical interaction arises from the fact that Coulomb’s law has no explicit time dependence. Same charge distribution produces same interaction, whether today or tomorrow. That is, electric force is conservative because it respects homogeneity of time.
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