(a) An electron is moving east in a uniform electric field...

(a) An electron is moving east in a uniform electric field of 1.50 N$/$C directed to the west. At point $A$, the velocity of the electron is 4.50 $\times 10^5$ m$/$s toward the east. What is the speed of the electron when it reaches point B, 0.375 m east of point A? (b) A proton is moving in the uniform electric field of part (a). At point $A$, the velocity of the proton is 1.90 $\times 10^4$ m$/$s, east. What is the speed of the proton at point $B$?

Key Concepts

Electric Force on Charged Particles

This concept explains that a charged particle in an electric field experiences a force given by F = qE, where q is the charge of the particle and E is the electric field. The direction of the force depends on the sign of the charge: positive charges are accelerated in the direction of the field, while negative charges are accelerated opposite to the field direction.

Work-Energy Theorem

The work-energy theorem states that the net work done on an object is equal to its change in kinetic energy. In the context of charged particles in electric fields, the work done by the electric force as the particle moves a certain distance is converted into changes in the particle's kinetic energy, allowing one to compute changes in speed as the particle moves within the field.

Uniform Electric Field

A uniform electric field is one in which the magnitude and direction of the field remain constant throughout the region of interest. In such fields, the force acting on a charged particle is constant, resulting in uniformly accelerated motion, which simplifies the analysis of the particle’s kinematics using energy methods or kinematic equations.

Transcript

00:01 So for this problem, there are two directions in what, west and east.

00:09 So let's take the west to be positive x and obviously east will then be negative x.

00:21 Now for solving both the parts of the problem, we will need the same equations.

00:29 And only the variables of those values, only the values of the variables will be different.

00:39 So first let's derive all the equations that we are going to need and then we will just plug in all the values in place of the variables and solve for the two parts.

00:52 So we know that velocity v square is equal to u square.

01:00 Plus 2a s by kinematics where v is the final velocity u is the initial velocity a is acceleration and s is the displacement now for to solve this problem there are two ways uh one is the work energy theorem and the second one is using kinematics here i'm going to use kinematics in order to solve this problem because i think that this method is relatively shorter than the work energy theorem method.

01:36 So yeah.

01:37 Now this is the equation that i wrote down using the kinematics.

01:45 Now since we are already given the electric field, we can find the acceleration by using this equation.

01:54 Now once we have acceleration, we can substitute this over here in the first equation and rest of the values to the variables are already given for both the parts.

02:08 So after that, we can plug those variables and solve for v, which will be our answer to both the parts.

02:16 Now, in the first part, we have to solve for the velocity of the electron.

02:22 Now, since, as i said, our convention is such that waste is.

02:27 In the positive x direction and east is towards the negative x direction.

02:32 So going with that convention, our initial velocity will be equal to negative 4 .5 times 10 to 5 meter per second.

02:45 S is the displacement is in the east.

02:50 So it will be negative just like the initial velocity and that is equal to 0 .375 meters.

02:58 Negative 0 .375 meter.

03:03 And the electric field is towards waste.

03:07 So this should be positive and has a value 1 .5 newton per coulum.

03:16 So now the first thing to do will be to find the acceleration.

03:20 So let me just write this to be question 1 and this to be question 2.

03:25 Now for an electron charge is equal to 1 .6 times 10 to the power minus 19 columns and mass of an electron is 9 .1 times 10 to the power minus 31 kg...

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