Particle Motion in a Homogeneous Field

The motion of a particle in a homogeneous field is shown in Fig. 20.1. There is an alternating electric field on the plates of the condenser. The equation of motion due to the field is given as follows  

There may also be a motion in x-direction, as indicated in Fig. 20.1. The first term describes the acceleration of the particle in y-direction, as the inert force. The second term contains the velocity of the particle and a friction term, i.e., the friction force rises with the velocity, counteracting the acceleration. The third term describes the electric force due to the voltage across the plates of the condenser. The friction law is similar to the law of Stokes, that is, the friction force is proportional to the velocity of the particle. / is a friction factor, in terms of the law of Stokes 

T is a dimensionless time, is the frequency of the alternating current, L is a characteristic length of the system. For example, we could adjust the distance of the condenser plate as 2L. This means further that we can place the coordinate system so that the origin is in the middle in between the plates. If the particle enters the condenser region in the middle, then the initial dimensionless T = 0. The particle touches the plates at T = 1. Eventually we arrive at the dimensionless form of Eq. (20.1) as follows  

Before we start to solve Eq. (20.2) we explain the physical meaning of the coefficients a and /3. If we insert for A we arrive at 

Inspection of Eq. (20.3) shows that a is the ratio of electric energy to kinetic energy, i.e., the energy of the harmonic motion. Similarly, Eq. (20.4) states that p is the ratio of the viscous momentum to the momentum of the harmonic motion 

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