Question

In a cathode-ray tube, a beam of electrons is projected horizontally with a speed of $9.6\times {10}^8\mathrm{cm}/\mathrm{s}$ into the region between a pair of horizontal plates $2.3\mathrm{cm}$ long. An electric field between the plates causes a constant downward acceleration of the electrons of magnitude $9.4\times {10}^{16}\mathrm{cm}/{\mathrm{s}}^2.$ Find $(a)$ the time required for the electrons to pass through the plates, $(b)$ the vertical displacement of the beam in passing through the plates, and (c) the horizontal and vertical components of the velocity of the beam as it emerges from the plates.

Short answer

(a) The time required for the electrons to pass through the plates is $2.4\times {10}^{-9}\mathrm{s}$, (b) the vertical displacement of the beam in passing through the plates is $27\mathrm{cm}$, and (c) the horizontal and vertical components of the velocity of the beam as it emerges from the plates are approximately $9.6\times {10}^8$ and $2.3\times {10}^8\mathrm{cm}/\mathrm{s}$ respectively.

Step-by-step solution

Determine the Time

Since the horizontal and vertical motions are independent, the time needed for the electrons to cross the plate can be calculated from horizontal motion only. The speed in the horizontal direction remains constant since there is no acceleration in the horizontal direction. So, the time $t$ can be calculated using the relation $v_0=d/t$, where $v_0=9.6\times {10}^8\mathrm{cm}/\mathrm{s}$ is the constant horizontal speed and $d=2.3\mathrm{cm}$ the length of the plate. Therefore, $t=d/v_0=2.3/9.6\times {10}^8=2.4\times {10}^{-9}\mathrm{s}$.

Determine the Vertical Displacement

Next, calculate the vertical displacement using the equation of motion $y=v_{0y}t+1/2a{t}^2$. Here $v_{0y}$ is the initial velocity in the y direction which is equal to zero since the electrons only have initial velocity in the horizontal direction. $a$ is the acceleration in the y direction and $t$ is the time calculated from step 1. Substituting the values in the equation $y=0\times t+1/2(9.4\times {10}^{16})t^2$, we get $y=1/2(9.4\times {10}^{16})(2.4\times {10}^{-9}{)}^2=27\mathrm{cm}$ .

Determine the Horizontal and Vertical Components of Velocity

The horizontal component of the velocity remains the same while the vertical component can be determined by $v_y=v_{0y}+at$. As $v_{0y}=0$ and $a=9.4\times {10}^{16}$ and $t=2.4\times {10}^{-9}$, the vertical component of the velocity would be $0+9.4\times {10}^{16}\times 2.4\times {10}^{-9}=2.3\times {10}^8\mathrm{cm}/\mathrm{s}$ .

Final Answer

The time for electrons to pass through the plates is $2.4\times {10}^{-9}\mathrm{s}$, the vertical displacement of the electron as passes through the plate is $27\mathrm{cm}$, and the velocity of the electron as it emerges from the plate has a horizontal velocity of $9.6\times {10}^8\mathrm{cm}/\mathrm{s}$ and a vertical velocity of $2.3\times {10}^8\mathrm{cm}/\mathrm{s}$.

Key concepts

Electron Beam Motion in a Cathode-Ray Tube

Understanding electron beam motion in a cathode-ray tube (CRT) is essential to grasp the basic principles of many vintage display technologies. In a CRT, an electron beam is generated at the cathode, a component serving as a source of electrons. These electrons are then accelerated through a vacuum toward a phosphorescent screen by a high voltage electric field, enabling them to move at very high speeds.

The movement of the electron beam is characterized by two components: the horizontal (x-axis) motion, and the vertical (y-axis) motion caused by the influence of electric fields within the CRT. Due to the electric field between a pair of horizontal plates in the CRT, electrons experience a constant acceleration in the vertical direction. However, in the absence of a vertical electric field, there is no force to change the beam's horizontal velocity, which remains constant.

In the case of our exercise, the horizontal motion serves as a timekeeper, allowing us to calculate the time taken for the electrons to pass through the plates. Since the electrons move at a constant horizontal speed, we can use this information, alongside the length of the plates, to determine the time of transit. It's this separation of vertical and horizontal motion that simplifies our analysis of the electron beam's journey through the CRT.

Role of Electric Fields in Physics

Electric fields are a cornerstone concept in physics, particularly when dealing with forces on charged particles such as electrons within a cathode-ray tube. An electric field is defined as a region around a charged particle or object within which a force would be exerted on other charged particles. The strength of an electric field at a point is given by the force per unit charge at that point.

In the context of our textbook exercise, the electric field existing between the horizontal plates provides a vertical force on the electrons, causing them to accelerate downward as they pass through the field. This acceleration is uniform, which means it does not change with time. Thus, the electric field interacts with the electrons to induce a motion that can be specifically quantified and predicted.

It's the vertical acceleration, provided by this electric field, that alters the path of the electrons, causing them to deflect from their initial horizontal trajectory. Understanding the way electric fields affect the motion of charged particles is crucial for several applications ranging from cathode-ray tubes to modern particle accelerators.

Kinematic Equations and Their Applications

Kinematic equations describe motion with constant acceleration. In physics problems, we can apply these equations to predict the future position and velocity of an object moving under the influence of a constant force, such as gravity or an electric field.

In our exercise, we use kinematic equations to calculate the various aspects of electron motion. Since the cathode-ray tube problem involves constant acceleration in the vertical direction, we can apply the equation $y=v_{0y}t+\frac{1}{2}a{t}^2$, where $y$ is the vertical displacement, $v_{0y}$ is the initial velocity in the vertical direction, $a$ is the vertical acceleration, and $t$ is the time.

Initially, the electrons have no vertical component of velocity ($v_{0y}=0$), and so their vertical displacement and final velocity in the y direction are entirely the result of the electric field's acceleration. By understanding these principles and applying kinematic equations, students can solve complex motion problems by breaking them down into vertical and horizontal components where each can be analyzed independently.