Two horizontal metal plates, each 100 mm square, are aligned 10.0 mm apart, with one above the o

Two horizontal metal plates, each 100 mm square, are aligned...

Two horizontal metal plates, each 100 $\mathrm{mm}$ square, are aligned 10.0 $\mathrm{mm}$ apart, with one above the other. They are given equal-magnitude charges of opposite sign so that a uniform downward electric field of 2000 $\mathrm{N} / \mathrm{C}$ 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. Describe the trajectory of the particle. Which plate does it strike? Where does it strike, relative to its starting point?

Key Concepts

Projectile Motion

Projectile motion refers to the motion of an object that is launched into a field where it is subject only to constant acceleration (or forces that yield constant acceleration) in one direction while moving at constant velocity in the perpendicular direction. Typically, the motion is analyzed by decomposing it into horizontal and vertical components, which is essential in determining the trajectory of any projectile under a uniform force.

Electric Force on a Charged Particle

A charged particle in an electric field experiences an electric force given by F = qE. This force leads to an acceleration, calculated by a = F/m, that acts in the same direction as the electric field if the charge is positive and in the opposite direction if the charge is negative. This concept is crucial when replacing gravitational acceleration with an acceleration due to an electric field in projectile problems.

Component Analysis of Motion

The independent treatment of horizontal and vertical motion is a key technique for analyzing projectile trajectories. The horizontal component of the motion usually features constant velocity, while the vertical component is subject to constant acceleration (here provided by the electric field). This separation allows one to compute the time of flight, displacement, and turning points in the trajectory.

Parabolic Trajectory

When an object is subjected to a constant acceleration in one direction (vertical) and constant velocity in a perpendicular direction (horizontal), its resulting path is a parabola. This concept helps in predicting the trajectory, impact point, and overall motion of particles in both gravitational and other constant-force fields such as electric fields.

Impact Determination in Bounded Regions

Analyzing the trajectory relative to boundary constraints—such as the dimensions and positions of the confining surfaces—allows one to predict which boundary (or plate) the object will strike and where on that boundary relative to its starting position. This involves combining the results of kinematic equations with the geometry of the problem.

Transcript

00:01 In this question, we have two plates, two horizontal metal plates with an electric field of 2 ,000 neutrons per coulomb pointing downwards from the top plate to the bottom plate.

00:15 And we have a particle that is given, starting off at the center of the bottom plate, and is given an initial push of velocity of 10 to power 5 meters per second.

00:30 So for this particle, it's charged.

00:33 It's a positive 1 times 10 power minus 6 cullums.

00:41 And remember that the electric field tells us the direction that the positive charge would feel, feel a force, right, the electric force.

01:01 So if it's downwards, like what we have here, and the charge is.

01:06 Positive it means that this positive charge will feel a downward force of the 2 ,000 newtons per gloom and this will actually decelerate our particle right because our particle is initially moving upwards this is kind of like a kinematic equation kinematic question so in order to determine which plate it strikes we need to determine why is the maximum height of this trajectory and see whether it will hit the top plate.

01:49 So what we need over here first of all is to separate out the components of the velocity.

01:55 So we have a vertical component, vertical component, and we also have a horizontal component.

02:04 Horizontal component is not affected by the electric field because the force is only in the vertical axis.

02:10 So we can just look at the vertical axis which is going to call the y direction so v y pt 10 to power 5 times sine of 37 degrees we can evaluate this this is our y component velocity then using the climatic equation v square equals to u square plus 2 as where v squared is the final velocity, u squared as the initial velocity, a is the acceleration, as is the displacement.

03:43 So we want to find what is the maximum displacement in the y direction.

03:49 And we can find that by just substituting it in the final velocity as zero.

03:54 This is when it is at the turning point...

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