Electromagnetic radiation is emitted by accelerating charges. The rate at which energy is emitte

Electromagnetic radiation is emitted by accelerating...

Key Concepts

Energy Loss Fraction in Accelerators

This concept involves calculating the fraction of a particle's kinetic energy that is dissipated as electromagnetic radiation per unit time. In accelerator physics, understanding the rate at which a particle loses energy due to its circular motion is crucial for designing systems that maintain high energies. This consideration is especially important for electrons and protons, which can have significantly different radiation losses due to their differing masses and resulting accelerations.

Circular Motion and Centripetal Acceleration

When a particle moves in a circular path, it experiences centripetal acceleration that is directed towards the center of the circle. This acceleration arises from the constant change in the direction of the velocity vector and is given by the square of the speed divided by the radius of the circle. In the context of radiating charges, this acceleration is the source of electromagnetic energy loss, making it a key factor when analyzing particles in circular accelerators.

Larmor Radiation Formula

The Larmor formula quantifies the power radiated by a non-relativistic accelerating charge. It shows that the radiated power is proportional to the square of both the charge and the acceleration, and includes fundamental constants that ensure the correct units and magnitude. This formula is essential in understanding how energy is lost by charged particles under acceleration in various physical settings.

Dimensional Analysis

Dimensional analysis is a technique used to verify that the units on both sides of a physical equation are consistent. It provides a check against algebraic or conceptual errors by ensuring that all terms in an equation have the appropriate base dimensions (such as length, mass, time, etc.). This step is critical in confirming that derived equations conform to the fundamental laws of physics.

Electromagnetic Radiation from Accelerating Charges

This concept is based on the principle that any charged particle which undergoes acceleration will emit electromagnetic radiation. This fundamental idea is rooted in classical electrodynamics and explains phenomena such as synchrotron radiation observed in particle accelerators and other astrophysical processes. The emitted radiation carries energy away from the particle, which can affect its dynamics and energy balance.

Transcript

00:01 Hello everyone.

00:02 In this problem, we're going to be learning about bramstrallon, which is the radiation emitted during the acceleration of a charge.

00:11 So we're told that the loss of energy due to having an accelerated charge is equal to q squared a squared over 6 pi epsilon zero times c cubed.

00:23 And in the first part of the question, we're asked to see or to show that dimensionally this makes sense and indeed this kind of combination has units of energy or joules per seconds.

00:36 So we're gonna determine the dimensions of each of these quantities as we go along.

00:42 So one by one, q has dimensions of coulomb, a has dimensions of mass or, sorry, meters per second squared.

00:49 Epsilon zero, you can write epsilon zero's units as coulomb squared divided by neutrin's meters squared.

00:57 And c is a speed, right? so it has units of meters per second.

01:01 And then de has units of joules or munum meters, and dt has units of seconds, so units of time.

01:10 So in this combination of q squared, a squared, over 6x0, c cubed, we know that 6 pi is just a number, right? so it's dimensionless.

01:18 So everything is coming from q squared, a squared, and epsilon 0 and c cubed.

01:24 So the combination of q squared and a squared gives you coulomb squared times meter squared per seconds to the four.

01:31 Then the 1 over epsilon 0 gives you newton meters squared per coulums squared, and then the 1 over c cubed factor gives you dimensions of seconds cubed per meters cubed.

01:45 And so what you can notice here is that the seconds cubed on the top cancels all but one factor of seconds of the denominator.

01:54 The coulum squared cancels.

01:56 And again, you have meters cubed in the bottom, so that cancel 1 meter squared and 1 factor of the other meter squared.

02:05 This you're left with mutinence times meters per seconds, which is actually just joules per seconds.

02:11 So this is nothing else but joules per seconds.

02:14 And that is exactly the same as what you have for the dimensionality of de by dt.

02:20 So again, here you have a joules per seconds.

02:23 Okay, so this is, sorry, let's use green.

02:27 So this is what you have for the right -hand side, and this is what you have for the left side.

02:32 And we see that these two are the same.

02:34 And so dimensionally, this expression for the loss of energy does make sense.

02:40 Okay, now in part b, we're told that there is a proton that's orbiting in a, that's moving around in a circular accelerator at a radius of 0 .750 meters.

02:53 It has a charge, 1 .6 times 10 to minus 19 quillums, which is just a standard, electric charge for a charged particle, and it has a mass of 1 .67 times 10 to the minus 27 kilograms, and it has a kinetic energy of 6 mega electron volts, which you can convert to joules, and it turns out to be 9 .61 times 10 to minus 13 joules.

03:19 So the question is, what fraction of its energy is this particle, is this proton, radiating way, d .t.

03:28 Acceleration or branch problem.

03:30 Okay, so first for that, we have to find out what de by dt is.

03:36 And for that, since this is proportional to the acceleration, we need to find what the acceleration is.

03:42 Well, we're told that the charge is moving around in a circular orbit, so its acceleration is going to be the centripetal acceleration, which is v squared over r, where v is its velocity.

03:54 Okay, but how do we find a velocity? okay, so the kinetic energy of the particle we are told, right? so we know that the kinetic energy is a half times mp times v squared.

04:05 If our mp is the mass of the proton.

04:07 And so we can rearrange this for v squared.

04:10 And the reason i'm rearranging for b squared and not v is because i'm going to be needing v squared to determine the acceleration.

04:16 So i would eventually square root this and then square it back again.

04:20 So it's just easier to find v squared from the get -go.

04:23 So v squared we can find by rearranging this.

04:25 Equation here to get that b squared is equal to two times the kinetic energy of the proton divided by the mass of the proton...

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