A proton is projected into a uniform electric field that...

A proton is projected into a uniform electric field that points vertically upward and has magnitude $E$. The initial velocity of the proton has a magnitude $v_{0}$ and is directed at an angle $\alpha$ below the horizontal. (a) Find the maximum distance $h_{\max }$ that the proton descends vertically below its initial elevation. Ignore gravitational forces. (b) After what horizontal distance $d$ does the proton return to its original elevation? (c) Sketch the trajectory of the proton. (d) Find the numerical values of $h_{\max }$ and $d$ if $E=500 \mathrm{~N} / \mathrm{C}, v_{0}=4.00 \times 10^{5} \mathrm{~m} / \mathrm{s},$ and $\alpha=30.0^{\circ}$

Key Concepts

Uniform Electric Field

A uniform electric field is one where the field strength and direction remain constant throughout the region of interest. This concept is crucial because when a charged particle such as a proton moves in this field, it experiences a constant force, leading to constant acceleration. This simplified scenario allows for straightforward application of kinematic equations similar to those used in gravitational projectile motion, but with the electric force replacing gravity.

Electric Force on Charged Particles

The electric force acting on a charged particle is given by the product of its charge and the electric field strength (F = qE). For a proton, with its positive charge, the force will be directed along the direction of the electric field. This concept is key in predicting how the velocity vector of the proton changes over time, as the force causes a constant acceleration in the direction of the field.

Decomposition of Velocity into Components

When dealing with projectile motion, the initial velocity vector is commonly separated into horizontal and vertical components. This decomposition is essential because it allows the analysis of motion independently in the horizontal direction, where there is no acceleration (in the absence of air resistance), and in the vertical direction, where the particle experiences constant acceleration due to the electric force in this context.

Kinematics Equations for Constant Acceleration

The motion of the proton under a constant force (due to the uniform electric field) can be analyzed using kinematic equations. These equations relate displacement, initial velocity, time, and acceleration. They provide the framework to determine quantities such as the maximum displacement in the vertical direction (h_max) and the time and horizontal distance required to return to the initial elevation.

Projectile Motion in a Uniform Electric Field

Projectile motion in a uniform electric field mirrors classical projectile motion under gravity but with the acceleration provided by the electric force instead of gravity. This concept helps in visualizing and analyzing the particle’s trajectory, which is a parabola. The symmetry of the path with respect to time and horizontal distance is utilized to calculate the maximum vertical descent and the horizontal distance at which the particle returns to its starting elevation.

Transcript

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00:02 The motion off a charged particle in a panel plate electric field is similar to the project.

00:08 Our motion projectile motion, the initial last e in a d direction, is given.

00:16 By then.

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00:36 It's you mex in 1/2 off the total time and it's the lost.

00:42 It will be instantaneously zero.

00:45 Then it will be begin moving in.

00:47 What little direction until reaches the original takeoff point within the same initial blast.

00:52 The compliment.

00:53 So this displacement engine max will be called to the f square divide by two a.

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02:43 So this will be prison.

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