A small object with mass m, charge q, and initial speed v0=5.00 ×10^3 m / s is projected into a

step 1 In this question, we have this setup. So there's a small mass M, charge Q, moving with velocity

A small object with mass m, charge q, and initial speed...

A small object with mass $m,$ charge $q,$ and initial speed $v_{0}=5.00 \times 10^{3} \mathrm{~m} / \mathrm{s}$ is projected into a uniform electric field between two parallel metal plates of length $26.0 \mathrm{~cm}$ (Fig. $\mathrm{P} 21.78$ ). The electric field between the plates is directed downward and has magnitude $E=800 \mathrm{~N} / \mathrm{C}$. Assume that the field is zero outside the region between the plates. The separation between the plates is large enough for the object to pass between the plates without hitting the lower plate. After passing through the field region, the object is deflected downward a vertical distance $d=1.25 \mathrm{~cm}$ from its original direction of motion and reaches a collecting plate that is $56.0 \mathrm{~cm}$ from the edge of the parallel plates. Ignore gravity and air resistance. Calculate the object's charge-to-mass ratio, $q / m$.

Key Concepts

Uniform Electric Field

A uniform electric field is one in which the electric field vectors have the same magnitude and direction at every point in the region of interest. This property means that a charged particle within this field experiences a constant force, which simplifies the analysis of its motion, as the acceleration due to the field remains uniform over time.

Electric Force on a Charged Particle

The electric force acting on a charged particle in an electric field is given by F = qE, where q is the charge of the particle and E is the magnitude of the electric field. This force causes the particle to accelerate in the direction of the field if the charge is positive, or opposite to it if the charge is negative, and is a central concept when studying the dynamics of charged particles in fields.

Projectile Motion Kinematics

Projectile motion involves the analysis of an object's motion in two dimensions, where the horizontal and vertical components are independent of each other. The horizontal motion typically involves constant velocity due to the absence of horizontal forces, while the vertical motion is subject to constant acceleration (or deceleration) due to forces like gravity or, in this case, an electric force acting in a specific direction.

Charge-to-Mass Ratio

The charge-to-mass ratio (q/m) is a fundamental parameter characterizing a particle's response to forces. It plays a significant role in determining how much acceleration a charged particle will experience in a given electric field. This ratio is particularly important in experiments and applications such as mass spectrometry and the analysis of particle trajectories under electromagnetic forces.

Separation of Motion Components

In problems involving motion under the influence of a constant force (in one direction only), it is useful to separate the motion into independent horizontal and vertical components. The horizontal component remains unaffected by the vertical force, allowing for a simplified analysis using kinematic equations. This principle is key to solving problems where a particle is deflected from its original path due to a force acting only in one direction.

Transcript

00:01 In this question, we have this setup.

00:03 So there's a small mass m, charge q, moving with velocity v0, vn is 5 ,000 meters per second.

00:18 Okay, and then it moves through a uniform field.

00:23 At first, the length is 26 cm, and then after it leaves a few, it travels 56 cm horizontally and then hits the screen.

00:38 And when it hits the screen, it's collecting plates, not a screen, but a plate.

00:47 It is at some distance d below the horizontal, okay, and the distance d is 1 .25 cm.

00:56 And then the electric fuel strength, the magnitude of the electric fuel is 800 newtons per kool -o okay so we want to find the objects charged to mass ratio of q over m okay so to solve this problem uh we need to recognize a few things okay so inside a few in the object uh undergoes projectile motion okay okay so the equation the kinematics equation for project motion will be applicable okay especially along the vertical direction because the acceleration of the object inside a few will be vertical okay will be downward okay yeah and then outside a few the object will travel outside of you the object travels with constant velocity okay yeah it has constant horizontal the component is constant, the vertical component of the velocity is constant outside the field.

02:37 So we are going to find a charge to match ratio of the object.

02:43 We will first express everything, every known quantity that we can find in terms of charge to mass.

02:50 And then we relate the vertical distance or the vertical displacement traveled by the objects inside a few and outside of few and sum them together.

03:01 And it will be equal to d.

03:03 So that's how the approach of this to solve this problem.

03:09 Okay, so first we look at inside a few, gain the acceleration will be qe over m.

03:20 That is newton second law.

03:22 Okay...

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