The electron beam emitted by an electron gun is controlled...

The electron beam emitted by an electron gun is controlled (steered) with two sets of parallel conducting plates: a horizontal set to control the vertical motion of the beam, and a vertical set to control the horizontal motion of the beam. The beam is emitted with an initial velocity of $2.00 \cdot 10^{7} \mathrm{~m} / \mathrm{s}$. The width of the plates is $d=5.00 \mathrm{~cm}$, the separation between the plates is $D=4.00 \mathrm{~cm},$ and the distance between the edge of the plates and a target screen is $L=40.0 \mathrm{~cm}$ In the absence of any applied voltage, the electron beam hits the origin of the $x y$ -coordinate system on the observation screen. What voltages need to be applied to the two sets of plates for the electron beam to hit a target placed on the observation screen at coordinates $(x, y)=(0 \mathrm{~cm}, 8.00 \mathrm{~cm}) ?$

Key Concepts

Two-Dimensional Motion Analysis

The analysis of the electron’s deflection involves decomposing its motion into two perpendicular components: one along the initial velocity (often horizontal) and one perpendicular to it (vertical) where the electric field acts. This separation allows for the independent application of kinematic equations in each direction, similar to projectile motion, to predict the overall trajectory of the charged particle.

Kinematic Equations for Constant Acceleration

When a charged particle like an electron is accelerated by a constant force, its motion can be analyzed using the kinematic equations for constant acceleration. These equations relate displacement, initial velocity, acceleration, and time, allowing the prediction of the particle’s path and eventual landing position on a target screen.

Uniform Electric Field

A uniform electric field is one in which the field strength and direction are constant in space. In the context of charged particle deflections, such as electrons between parallel plates, the assumption of a uniform electric field allows us to treat the force on the electrons as constant in magnitude and direction, leading to uniform acceleration.

Force on Charged Particles

Charged particles, like electrons, experience a force when placed in an electric field, according to the equation F = qE, where q is the charge and E is the electric field strength. This force directly influences the particle’s acceleration, altering its trajectory in the region where the field is present.

Relationship Between Voltage and Electric Field

The electric field between two parallel plates is directly related to the voltage difference applied across them and inversely related to the distance between the plates. This relationship is given by E = V/D, where V is the voltage and D is the distance between the plates. This formula is crucial for determining the electric field strength that affects the particle's motion.

Transcript

Please give Ace some feedback