Each of the electrons in a particle beam has a kinetic energy of 1.60 ×10^-17 J. (a) What is th

Each of the electrons in a particle beam has a kinetic...

Key Concepts

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion. In the context of charged particles like electrons, it quantifies the energy that must be dissipated or transformed (for example, by doing work against an electric force) in order for the particles to come to a stop.

Uniform Electric Field

A uniform electric field is a region in space where the electric force experienced by a charge is constant in magnitude and direction. This concept is vital because it allows the use of simple relationships between force, acceleration, and displacement to analyze how charged particles, such as electrons, respond over a distance.

Work-Energy Principle

The work-energy principle states that the work done on an object is equal to its change in kinetic energy. In problems involving a stopping distance, this principle is used to relate the work done by the electric force over a given distance to the initial kinetic energy of the particle, allowing calculation of fields and forces required for stopping.

Force on a Charged Particle

A charged particle in an electric field experiences a force that is directly proportional to the field strength and the magnitude of the charge (F = qE). This relationship is crucial for determining the acceleration (or deceleration) of a particle and for applying kinematics and energy conservation principles.

Kinematics of Constant Acceleration

Kinematic equations for constant acceleration provide tools to relate displacement, initial velocity, acceleration, and time. When an electron decelerates in a constant electric field, these equations allow one to compute both the time taken to come to a stop and how its motion changes over the stopping distance.

Transcript

00:01 If you want to stop an electron, you must apply an electric field in the direction of motion of the electron.

00:08 So remember that electrons feel a force that's opposite the electric field in the amount f equals q times electric field.

00:23 And that force is opposite e, direction of e, for a negative charge.

00:31 And electrons, of course, are negative charges.

00:42 So here what we would like to do is find the electric field necessary to stop, and we'll use the magnitude.

00:53 Find the magnitude of the electric field needed to stop the electron in a certain stopping distance.

01:03 So the 10 centimeters is stopping the electron having initial.

01:14 Kinetic energy equal to 1 .6 times 10 to the minus 17 joules of energy.

01:23 And for this, it's probably good to start off and use conservation of energy.

01:29 So there's no time involved, at least not yet, conservation of energy.

01:36 We're going to be using the work energy theorem, that the work done by the electric field is equal to the change in the kinetic energy.

01:47 The work done by the electric field is going to be negative.

01:51 It's going to take energy out and will be in the amount force of the electric field times the stopping distance is equal to zero minus one -half mass of the electron times its initial speed square.

02:13 But we can just all lump this together and call it initial kinetic energy.

02:20 And of course, the electric field has magnitude e.

02:26 Sorry, the electric force has magnitude e times the magnitude of the electric field.

02:34 Then we have the stopping distance, and all that has to equal to the minus initial kinetic energy.

02:43 So we can certainly solve this for the.

02:46 The electric field strength.

02:54 So we'll take our initial kinetic energy, divide by the fundamental unit of charge in coolums, 1 .6 times 10 to the minus 19, and also divide by the stopping distance.

03:15 And the unit should be newton's per coolum.

03:20 We have joules in the numerator and force divided by, yeah, the units will work out.

03:39 I feel confident of that.

03:44 And let's see.

03:45 That gives us, the electric field is 1 ,000, newton's per coulum, nice, simple answer.

04:01 That is the magnitude.

04:06 Now, if we wanted to find the time that it took to stop, we would need some kinematics.

04:12 So remember, energy has no time in it.

04:18 But to figure out the stopping time, we'll need some kinematics, and it's good to start with the constant acceleration.

04:33 If the electric field is constant, we have the electric force is equal to mass times acceleration, mass of the electron...

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