Deflection between condenser plates. A particle of charge q...

Deflection between condenser plates. A particle of charge $q$ and mass $M$ with an initial velocity $v_{0} \hat{x}$ enters an electric field $-E \hat{y}$ (see Fig. 3.26). We assume $\mathbf{E}$ is uniform, i.e., its value is constant at all points in the region between plates of length $L$ (except for small variations near the edges of the plates, which we shall neglect). (a) What forces act in the $x$ and $y$ directions, respectively? Ans. $F_{z}=0 ; F_{y}=-q E \hat{y} .$ (b) Will a force in the $y$ direction influence the $x$ component of the velocity? (c) Solve for $v_{x}$ and $v_{y}$ as functions of time, and write the complete vector equation for $\mathbf{v}(t)$ Ans. $v_{0} \hat{\mathbf{x}}-(q E / M) t \hat{\mathbf{y}} .$ (d) Choose the origin at the point of entry, and write the complete vector equation for the position of the partiele as a function of time while the particle is between the plates.

Deflection between condenser plates. A particle of charge $q$ and mass $M$ with an initial velocity $v_{0} \hat{x}$ enters an electric field $-E \hat{y}$ (see Fig. 3.26). We assume $\mathbf{E}$ is uniform, i.e., its value is constant at all points in the region between plates of length $L$ (except for small variations near the edges of the plates, which we shall neglect).
(a) What forces act in the $x$ and $y$ directions, respectively? Ans. $F_{z}=0 ; F_{y}=-q E \hat{y} .$
(b) Will a force in the $y$ direction influence the $x$ component of the velocity?
(c) Solve for $v_{x}$ and $v_{y}$ as functions of time, and write the complete vector equation for $\mathbf{v}(t)$ Ans. $v_{0} \hat{\mathbf{x}}-(q E / M) t \hat{\mathbf{y}} .$
(d) Choose the origin at the point of entry, and write the complete vector equation for the position of the partiele as a function of time while the particle is between the plates.

Key Concepts

Uniform Electric Field

A uniform electric field has the same magnitude and direction at every point in space, which simplifies the analysis of charged particle motion. In this context, it allows for the assumption that the force experienced by the particle is constant over the region of interest.

Electric Force on a Charged Particle

When a charged particle is placed in an electric field, it experiences a force that is proportional to both the magnitude of the charge and the strength of the electric field. This force is given by the formula F = qE, and it determines the particle's acceleration in the direction of the field.

Newton's Second Law

Newton's second law, expressed as F = ma, is fundamental in linking the force acting on a particle to its acceleration. In problems involving constant forces, such as those in a uniform electric field, this law enables the calculation of the particle's acceleration and subsequent velocity and displacement over time.

Independence of Motion in Orthogonal Directions

In classical mechanics, the motion of a particle in perpendicular directions is independent of each other. This means that forces and accelerations affecting one component (for example, the vertical direction) have no impact on the motion in the perpendicular component (for example, the horizontal direction), which simplifies the analysis of two-dimensional motion.

Kinematic Equations for Constant Acceleration

The kinematic equations describe the relationships between displacement, velocity, acceleration, and time for uniformly accelerated motion. These equations are essential for determining the time evolution of both position and velocity vectors when a constant acceleration, such as that caused by an electric field, acts on a particle.

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