Two horizontal metal plates, each 10.0 cm square, are...

Two horizontal metal plates, each 10.0 cm square, are aligned 1.00 cm apart with one above the other. They are given equal-magnitude charges of opposite sign so that a uniform downward electric field of $2.00 \times 10^{3} \mathrm{N} / \mathrm{C}$ exists in the region between them. A particle of mass $2.00 \times$ $10^{-16} \mathrm{kg}$ and with a positive charge of $1.00 \times 10^{-6} \mathrm{C}$ leaves the center of the bottom negative plate with an initial speed of $1.00 \times 10^{5} \mathrm{m} / \mathrm{s}$ at an angle of $37.0^{\circ}$ above the horizontal. (a) Describe the trajectory of the particle. (b) Which plate does it strike? (c) Where does it strike, relative to its starting point?

Key Concepts

Trajectory Deflection by Electric Force

The concept of trajectory deflection explains how a constant force, such as that from a uniform electric field, alters the path of a charged particle. The particle’s initial velocity is modified by the added acceleration from the force, resulting in a curved or parabolic trajectory relative to the initial motion. Understanding this helps in predicting where the particle will impact a boundary.

Projectile Motion

Projectile motion describes the two-dimensional movement of an object under the influence of a constant acceleration, typically gravity, or in analogous cases, a uniform electric force. In projectile motion, the horizontal component of velocity remains constant while the vertical component is uniformly accelerated. This separation allows the use of kinematic equations independently in each direction.

Kinematics with Constant Acceleration

Kinematics with constant acceleration involves equations that link displacement, initial velocity, time, and acceleration. These equations allow one to predict the trajectory and final position of moving objects. In contexts where a charged particle experiences a constant force, similar techniques used in gravitational motion can be applied to determine its path.

Electric Field

An electric field is a region in space where a charged particle experiences a force, given by F = qE. In a uniform electric field, this force has constant magnitude and direction, which simplifies the analysis of the motion of charged particles. This concept is crucial as it underpins how charged objects interact with their surroundings, causing acceleration in the field’s direction.

Force on a Charged Particle

When a charged particle is placed in an electric field, it experiences a force directly proportional to its charge and the strength of the field. The force is given by F = qE, and it determines the acceleration of the particle via Newton’s second law. This concept is central for predicting how particles deviate from their initial trajectory under the influence of the field.

Transcript

00:01 Okay, so this is the graph of these questions.

00:03 So there's a positive particle lying in the middle of the negative plates, okay? and the electric field has a direction from positive plates to the negative plate, which is point downward.

00:17 So therefore, the trajectory of the particle should be a parabola, okay, because its initial speed was at an angle, which is about 37 degrees above the horizontal.

00:29 So it will eventually fall down due to the electric field from this graph.

00:36 So eventually it will fall down on the bottom place, which is the negative place here.

00:43 And this expands the question b.

00:47 And well, we know that since the particle was having an initial speed that was at an angle of 37 degree above the horizontal.

00:59 So that means the particle will move along the horizontal direction and the vertical direction simultaneously, which means that, so if this speed or the velocity of the particle, let's say, v, so we can decompose the velocity here with x component and y components, okay? and you move along the y -axis and x -axis simultaneously, okay? so therefore, the time of the motion along either direction should be the same, okay? so now let's try to determine the acceleration that was act on the particle first.

01:44 So, well, first we know the force, the net force, is equal to the electric force from this question, which is equal to charge times the electric field, which can be also equal to mass times acceleration.

02:00 Therefore, we can determine acceleration a is just simply equal to qe over m.

02:07 Okay? and charge q, which is the charge of particles, given as 1 .00 times 10 to the power of negative 6, column.

02:21 And the electric fuel is given as 2 .00 times 10 to the power of 3, new term of coolon.

02:29 And we know the mass, the wrong question was given as 2 .00 times 10 to the power of negative 16.

02:41 Therefore, the acceleration a should be equal to 1 times 10 to the power of 3.

02:53 I'm sorry, not 3, 13 meter per second square.

02:59 Okay? and if we take look at the free body diagram on the positive charge particle, so when it's in the air, it should look something like this.

03:10 So if this black dye here is the positive charged particle, so you only experienced a force that was point downward, which is the electric force, okay? which is equal to qe.

03:25 But remember, along the vertical directions, the particle is moving upward since it has an initial speed about 1 times 10 to the power of 5 meter per second at an angle 37 degree above the horizontal.

03:40 Okay? so it should move like this along the y -axis.

03:44 Okay, so this is a speed of a particle, which means that the acceleration on the particle is in the opposite direction when compared with the speed of a positive particle along the y -i direction.

03:58 Okay, let's say here is vy since it's a long vertical direction, okay? and this is acceleration a...

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