Electrons are accelerated through a potential difference of 750 kV, so that their kinetic energy

step 1 This is chapter 37 problem number 36. We have electrons accelerated under a potential difference

Electrons are accelerated through a potential difference of...

Key Concepts

Electron Acceleration by Electric Potential

When an electron is accelerated by a potential difference, it gains kinetic energy equal to the product of its charge and the voltage (K = eV). This conversion from electric potential energy to kinetic energy is fundamental in understanding how charged particles are manipulated in electric fields, and it establishes a direct relationship between the applied voltage and the energy acquired by the electron.

Classical Kinetic Energy

In classical mechanics, the kinetic energy of a particle is given by the formula K = ½ mv², where m is the mass of the particle and v is its velocity. This relationship allows one to compute the speed of a particle if its kinetic energy is known, under the assumption that relativistic effects are negligible, which is valid when the speed is significantly less than the speed of light.

Relativistic Kinetic Energy

At high speeds approaching the speed of light, classical mechanics fails to accurately describe a particle's behavior. In relativistic mechanics, the kinetic energy is given by K = (? - 1)mc², where ? (the Lorentz factor) accounts for time dilation and other relativistic effects. This formula is crucial when dealing with particles accelerated through large potential differences, as their speeds can be a significant fraction of the speed of light.

Dimensionless Velocity (Beta)

The ratio of a particle’s speed to the speed of light, denoted by ? = v/c, is a dimensionless quantity that is especially useful in relativistic physics. This concept helps in understanding and comparing the magnitude of the velocity relative to the universal speed limit set by the speed of light, thus indicating whether relativistic effects must be taken into account for accurate calculations.

Transcript

00:01 This is chapter 37 problem number 36.

00:03 We have electrons accelerated under a potential difference, and their kinetic energy is also given to us as 7 .5 times center per 5 electron volts.

00:13 Let's convert this to joules 0 .5 times centrifugal of 5 times the charge of a single electron is going to give us the kinetic energy in joules.

00:27 I'm converting it right away because in the equation.

00:30 You know that we need to use the energy in terms of tools.

00:34 Now, going back to the problem, what's asking the problem is the ratio of the velocity of the electron over the speed of light.

00:42 So we're basically looking at b over c, which, as you know, this is called the beta, and this is embedded in the gamma factor, the lawrence factor.

00:52 Lawrence factor is 1 over square to 1 minus v squared over c squared.

00:56 As you can see, we have v over c here.

01:00 We can rewrite the lawrence factor as 1 square to 1 minus beta square.

01:07 So how are we going to get there? well, in the kinetic energy equation for the relativistic equation, gamma minus 1 times mc squared gives us the kinetic energy.

01:18 So in this equation, we know everything other than the gamma factor here, right? so i'm going to try and isolate this gamma -factor action with the ultimate goal of going to the beta, right? so if we do gamma minus 1, then, is going to be equal to kinetic energy over mc squared, right? if you divide both sides by mc squared.

01:39 From here, gamma then is kinetic energy over mc squared plus 1.

01:45 Now we know that gamma equals to 1 over square of 1 minus beta squared.

01:51 So now what we need to do is to find out what beta is.

01:56 In order to do that, i'm going to have to take square of both sides, right, so that i can get rid of this square root here.

02:04 Then, on a new page, kinetic energy over mc squared plus 1 squared, is going to be equal to 1 over 1 minus beta squared.

02:19 Now let's do the cross product.

02:21 I mean, multiplying the one in the denominator on the right hand side by the term we have on the left -hand side here...

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