Many particle accelerators, including the cyclotron and the...

Many particle accelerators, including the cyclotron and the synchrotron (Sections $17.11$ and $18.11$ ), hold charged particles in a circular orbit using a suitable magnetic field. The centripetal acceleration, $a=v^{2} / r$ can be very large and can lead to serious energy loss by radiation, in accordance with Eq. (11.1). (a) Consider a $10-\mathrm{MeV}$ proton in a cyclotron of radius $0.5 \mathrm{~m}$. Use the formula (11.1) to calculate the rate of energy loss in $\mathrm{eV} / \mathrm{s}$ due to radiation. (b) Suppose that we tried to produce electrons with the same kinetic energy in a circular machine of the same radius. In this case the motion would be relativistic and formula (11.1) is modified by an extra factor ${ }^{\text {* }}$ of $\gamma^{4}$ : $$ P=\frac{2 k q^{2} a^{2} \gamma^{4}}{3 c^{3}} $$ Find the rate of energy loss of the electron, and compare with that for a proton. (Your answer for the electron should be enormously larger than for the proton. This explains why most electron accelerators are linear, not circular, since the acceleration in a linear accelerator - once $v \sim c-$ is far smaller than the centripetal acceleration considered here.)

Many particle accelerators, including the cyclotron and the synchrotron (Sections $17.11$ and $18.11$ ), hold charged particles in a circular orbit using a suitable magnetic field. The centripetal acceleration, $a=v^{2} / r$ can be very large and can lead to serious energy loss by radiation, in accordance with Eq. (11.1). (a) Consider a $10-\mathrm{MeV}$ proton in a cyclotron of radius $0.5 \mathrm{~m}$. Use the formula (11.1) to calculate the rate of energy loss in $\mathrm{eV} / \mathrm{s}$ due to radiation. (b) Suppose that we tried to produce electrons with the same kinetic energy in a circular machine of the same radius. In this case the motion would be relativistic and formula (11.1) is modified by an extra factor ${ }^{\text {* }}$ of $\gamma^{4}$ :
$$
P=\frac{2 k q^{2} a^{2} \gamma^{4}}{3 c^{3}}
$$
Find the rate of energy loss of the electron, and compare with that for a proton. (Your answer for the electron should be enormously larger than for the proton. This explains why most electron accelerators are linear, not circular, since the acceleration in a linear accelerator - once $v \sim c-$ is far smaller than the centripetal acceleration considered here.)

Key Concepts

Larmor Formula

The Larmor formula gives the power radiated by an accelerating charged particle in classical electrodynamics. It relates the energy loss per unit time to the square of the acceleration and the square of the charge, providing a fundamental tool to estimate radiation losses in various contexts including particle accelerators.

Centripetal Acceleration

Centripetal acceleration describes the acceleration of an object moving in a circular path, directed towards the center of the circle. In the context of particle accelerators, this acceleration is crucial because it causes charged particles to emit radiation when they are forced to travel along curved trajectories.

Relativistic Corrections

In high-energy accelerators, particles often travel at speeds close to the speed of light, necessitating relativistic corrections. These corrections introduce factors, such as the Lorentz gamma factor, which can significantly enhance the radiated power compared to nonrelativistic predictions and are essential for accurately modeling the behavior of light particles like electrons.

Synchrotron Radiation

Synchrotron radiation refers to the electromagnetic radiation emitted by charged particles when they are accelerated radially as in circular motion. This type of radiation becomes especially significant for relativistic particles in circular accelerators and impacts the design and operation of these machines due to its strong energy dependence on the particle mass and acceleration.

Particle Mass Effects

The mass of a particle plays a critical role in determining radiation losses. Lighter particles such as electrons experience much greater energy loss due to radiation compared to heavier particles like protons under similar acceleration conditions. This mass dependence explains why accelerator designs differ for electrons and protons, with electron machines often opting for linear acceleration to minimize energy losses.

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