Electrons in a television's CRT are accelerated from rest by...

Electrons in a television's CRT are accelerated from rest by an electric field through a potential difference of $2.5 \mathrm{kV}$. In contrast to an oscilloscope, where the electron beam is deflected by an electric field, the beam is deflected by a magnetic field. (a) What is the speed of the electrons? (b) The beam is deflected by a perpendicular magnetic field of magnitude $0.80 \mathrm{~T}$. What is the acceleration of the electrons while in the field? (c) What is the speed of the electrons after they travel $4.0 \mathrm{~mm}$ through the magnetic field? (d) What strength electric field would give the electrons the same magnitude acceleration as in (b)? (e) Why do we have to use an electric field in the first place to get the electrons up to speed? Why not use the large acceleration due to a magnetic field for that purpose?

Key Concepts

Lorentz Force

The Lorentz force is the force experienced by a moving charged particle in the presence of electric and magnetic fields. In the context of a magnetic field, the force is given by the cross product of the particle's velocity and the magnetic field, meaning the force is perpendicular to the velocity. This perpendicular force deflects the particle without changing its speed, which is a key consideration for understanding its trajectory in devices like CRTs.

Centripetal Acceleration in Uniform Circular Motion

When a charged particle moves in a magnetic field and experiences a force perpendicular to its velocity, it undergoes centripetal acceleration, resulting in circular motion. This acceleration is determined by the magnitude of the magnetic Lorentz force divided by the mass of the particle, and it does not change the speed of the particle, only its direction.

Electric Potential Difference and Kinetic Energy Conversion

This concept involves the idea that when a charged particle is accelerated through an electric potential difference, its electric potential energy is converted into kinetic energy. The relationship is expressed in the energy balance equation, where the charge multiplied by the potential difference equals the kinetic energy gained, allowing us to calculate the speed of the particle.

Comparison of Electric and Magnetic Acceleration

While both electric and magnetic fields can accelerate charged particles, their impacts are different. An electric field accelerates particles in the direction of the field by doing work, thereby increasing their speed. In contrast, a magnetic field exerts a force perpendicular to the particle's velocity, which changes the direction of motion but does no work, so it does not increase the particle's speed. This distinction is crucial in understanding why CRTs use an accelerating electric field to bring electrons up to speed before magnetic deflection is applied.

Transcript

00:01 All right, we are going to be accelerating an electron, which you can easily google or look up the mass of the electron, which is 9 .11 times 10 to the negative 31 kilograms and the charge of the electron, a negative 1 .1 .1906 times 10 to the negative 19.

00:35 Coolomes.

00:36 We are going to use an electric field with a potential difference of 2 .5 kilowatts, a potential difference of 2 .5 kilovolts, 2 ,500 volts, to accelerate the electron.

00:56 And then we're going to put it in a magnetic field and deflect it.

01:00 All right.

01:00 So to begin with, we want to find out what the final velocity is, what is the velocity of the electron after it goes through the potential difference.

01:11 To calculate that, you take the square root of negative 2, charge, potential difference, divided by the mass.

01:21 All right, make sure when you type that into the calculator, you do it correctly, multiply the three things up there, negative 2 times the charge times the potential difference, hit enter, divide by the mass hit enter take the square root if you do that you should get the three times 10 to the seventh meters per second so now this is how fast the electron is moving when we put it into the magnetic field oh the magnetic field has a magnitude of 0 .8 teslas.

02:01 All right.

02:03 So that's capital b.

02:05 0 .80 teslas.

02:08 We are looking at two significant figures.

02:10 So up here, you should probably have 3 .0, by the way.

02:14 All right.

02:14 We would like to know what the acceleration of this electron is when it's being deflected.

02:24 So we've got, and there it is, we've got force.

02:30 Is equal to the charge times the velocity cross product of the magnetic field.

02:38 Now force is mass times acceleration.

02:41 So to get the acceleration, you just divide by the mass.

02:45 Everything is perpendicular, so you don't have to worry about the whole cross product thing.

02:52 And throwing in an angle.

02:53 All right, so that's all you need to do.

02:55 Take the charge.

02:56 The electrons charge in coulones, multiply it by its velocity.

03:00 This is the velocity that's a small one.

03:03 So 3 times 10 to the seventh multiply it by the magnetic field, which is 0 .8, and divided by the mass.

03:11 And you should get the acceleration, which is 4 .2 times 10 to the 18 meters per second squared.

03:25 All right.

03:26 Now c would like to know what is the speed after 4.

03:32 0 .00 millimeters distance is 4 .0 millimeters.

03:39 Deci centimilli move it three times so we get 0 .04 meters.

03:48 So we should be able to use in old -fashioned motion equation there from the first half of the semester.

03:55 The final speed is equal to the initial speed squared...

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