Uniform Alternate Electric Field

Suppose a particle moves under a uniform (i.e. independent of space co-ordinates) but alternating electric field E given by,

E = E₀ sin ω t = ( E₀ i) sin ω t                                     (i)

where, E₀ is amplitude and ω is angular frequency of oscillation. We define the direction of electric field as X-axis.

The equation of motion of the particle of mass m and charge q in this field is,

 

Integrating, we find

  

where constant of integration c₁ is set by initial condition: at t = 0, when E = 0, let v = v₀. Hence, we get

 

Integrating once again, we find

  

where constant of integration r₀ is the initial position of the particle; at t = 0, r = r₀.

We can always put r₀ = 0; let us also take v₀ = 0. Further, if E₀ = E₀ i, we have

 

The particle moves along X-axis with time dependent speed v_x (t). The average speed of the particle during a cycle is,

 

v_x is called the drift velocity of the particle. The displacement of the particle during one complete cycle, i.e. in time T,

 

The graph between x and t is obtained by superposing the sine curve and the straight line,

x = x₁ + x₂

 

If v₀ ≠ 0, the particle may continue to move along y and z directions with constant initial velocities v_oy and v_oz. Simultaneously, it is drifted towards X-axis by a fixed distance per cycle, viz.