Cathode-ray tube. A cathode-ray tube (CRT) is an evacuated...

Cathode-ray tube. A cathode-ray tube (CRT) is an evacuated glass tube. Electrons are produced at one end, usually by the heating of a metal. After being focused electromagnetically into a beam, they are accelerated through a potential difference, called the accelerating potential. The electrons then strike a coated screen, where they transfer their energy to the coating through collisions, causing it to glow. CRTs are found in oscilloscopes and computer monitors, as well as in earlier versions of television screens. (a) If an electron of mass $m$ and charge $-e$ is accelerated from rest through an accelerating potential $V,$ show that the speed it gains is $v=\sqrt{2 e V / m}$ . We are assuming that $V$ is small enough that the final speed is much less than the speed of light.) (b) If the accelerating potential is $95 \mathrm{V},$ how fast will the electrons be moving when they hit the screen?

Key Concepts

Conservation of Energy

This concept involves the principle that energy cannot be created or destroyed, only converted from one form to another. In the context of charged particles, the work done by an electric potential converts electrical potential energy into kinetic energy, which is why a charged particle accelerated through a potential difference gains speed.

Electric Potential Difference

Electric potential difference, or voltage, is the work per unit charge needed to move a charge between two points in an electric field. It provides a measure of the energy available to accelerate a charged particle, such as an electron, from rest, and is directly related to the kinetic energy the particle will acquire.

Non-relativistic Kinetic Energy

Non-relativistic kinetic energy refers to the classical expression for kinetic energy, ½mv², which is valid when the speed of the particle is much less than the speed of light. This approximation allows us to set the electrical work (given by the charge times the potential difference) equal to the kinetic energy without requiring relativistic corrections.

Charged Particle Acceleration

The acceleration of charged particles under the influence of an electric field demonstrates how forces act on particles with mass and charge. By understanding the relationship between electric potential energy and kinetic energy, one can derive how the speed of a charged particle depends on the accelerating potential and the particle's mass and charge.

Transcript

00:01 Okay, so this problem, we're modeling cathode ray tubes.

00:04 And so we're basically just first trying to show that if you can accelerate an electron through a potential v, that the speed is given by the formula and the problem.

00:13 So let's find the speed of an electron accelerated through a cathode ray tube.

00:19 So if we assume a constant electric field, well, actually, let me rewind.

00:26 So let's use energy conservation.

00:28 So that says that the net external work is equal to the change in kinetic energy plus the change in potential energy.

00:39 If there's no forces doing external work, if the electric one is internal or we're considering the electric is internal, then we can say change in kinetic energy is equal to minus the change in u.

00:52 Minus a change in u is minus q times a change in v changing kinetic energy is um one half m v final squared if it starts at rest we can just say that the initial kinetic energy is zero so then we obtain that the velocity is just um two m two oops two divided by um times the charge of an electron square rooted because q is minus e.

01:32 And that's exactly what they wanted us to find.

01:36 And so we want to figure out how fast they're moving if the accelerating potential is 95.

01:44 So then we can say for that one, velocity is the square root of two times one.

01:57 Point charge of an electron 1 .6 10 to the minus 19 coulomes divided by that's a c divided by the mass of an electron 9 .11 times 10 to the minus 31 kilograms so i'm actually kind of curious how this will come out so i'm going to go ahead and calculate that i'm not going to pause the video i think it'll just take me a second to do this so that's two times 1 .6 10 to the minus 19 all divided by 9 .11 times 10 to the minus 31...

Please give Ace some feedback