X-ray tube. An X-ray tube is similar to a cathode-ray tube....

X-ray tube. An X-ray tube is similar to a cathode-ray tube. (See previous problem.) Electrons are accelerated to high speeds at one end of the tube. If they are moving fast enough when they hit the target at the other end, they give up their energy as X-rays (a form of nonvisible light). (a) Through what potential difference should electrons be accelerated so that their speed is 1.0$\%$ of the speed of light when they hit the target? (b) What potential difference would be needed to give protons the same kinetic energy as the electrons? (c) What speed would this potential difference give to protons? Express your answer in $\mathrm{m} / \mathrm{s}$ and as a percent of the speed of light.

Key Concepts

Mass Dependence in Particle Acceleration

The acceleration and resulting speed of a charged particle under a given potential difference depend not only on its charge but also on its mass. Particles with different masses, even if they have the same charge, will acquire different velocities when subjected to the same kinetic energy input. This mass dependency is essential when comparing how different particles, such as electrons and protons, behave under similar accelerating conditions.

Relativistic Considerations

When particles approach a significant fraction of the speed of light, relativistic effects become important, and adjustments to the classical formulas are needed to accurately describe their behavior. However, at speeds around 1% of the speed of light, non-relativistic approximations are sufficiently accurate, which simplifies the analysis while still providing reliable results.

Non-relativistic Kinetic Energy

For speeds much lower than the speed of light, the kinetic energy of a particle can be accurately described by the classical formula K = (1/2)mv², where m is the mass and v is the velocity of the particle. This relationship is critical for calculating either the needed potential difference to achieve a certain speed or the speed resulting from a given potential difference, as long as relativistic effects are negligible.

Electric Potential Energy and Kinetic Energy Conversion

When a charged particle is accelerated through an electric potential difference, it gains kinetic energy equivalent to the work done by the electric field. This energy conversion is described by the relation K = q?V, where K is the kinetic energy, q is the charge, and ?V is the potential difference. This concept is central to understanding how particles are accelerated and how their energies are determined in many physical applications.

Transcript

00:01 So in this question, we have an x -ray tube that is similar to a cathode rate tube.

00:06 And this question, this is this question 26 of chapter 18, it is completely based on question 25.

00:14 So if you have not done question 25 yet, please go back and work on question 25 yet, because i will not repeat the procedure and set up of question 25, but that is essential for this question.

00:27 So i would just say from question 25, what we get is we have this expression of the velocity in terms of the potential differences in the machine.

00:44 So this is the velocity for a particle with the charge of magnitude e with the speed of...

00:52 So here, we have the same similar setup here.

00:58 And we want to know through what potential difference would electrons be accelerated, so their speed is 1 % of the speed of light.

01:07 And speed of light, c, is a constant.

01:11 It's 3 times 10 to the 8th meters per second.

01:16 So we first want to rearrange this expression that we already have from question 25, so that it becomes n v square over 2e.

01:25 Then if we want it to be 10 % of the speed of light, so that v is 0 .1c.

01:35 1%, 0 .01c.

01:37 So we have some mass of the electrical times 31 kilogram times 0 .01 times 3 times 10 to 8 meters per second square.

01:58 2 times the electron 1 .6 times 10 to the negative 19 column.

02:05 So this in total gives us 26 volts.

02:11 So that is the potential required in the x -ray machine in order for the electron to achieve the speed.

02:19 And if we look at part b, part b asks us what potential difference should be needed to give the protons the same kinetic energy as the electrons.

02:28 So now the variation we have is this one.

02:34 The mass of the proton is different from the mass of the electron...

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