A typical understanding of the atom which is taught in secondary schools is the “celestial” model proposed by Bohr, whereby the electrons orbit the atom in a similar way to planets in the solar system. There are many corrections that are often noted to this incorrect/approximate model of the atom, often resorting to quantum physics. However, one of the most profound ways to notice that the Bohr model is indeed incorrect is to consider the the fact that radiation is emitted by any accelerating charge. The energy loss due to this is not negligible if the object is travelling close to the speed of light. This power radiated by a moving charge is given by an equation called the Relativistic Larmor Formula [2].
The Bohr Model of the Atom [1]
If electrons were to orbit the nucleus of atoms (where the protons and neutrons are held) in the classical fashion predicted by Bohr, they would emit radiation due to their centrifugal acceleration (the acceleration they undergo to stay in circular motion). This would lead to them collapsing into the nucleus and hence leading to all atoms being unstable.
The Relativistic Larmor Formula (RLF) also has applications for particle collider experiments, such as those conducted at CERN (which Ireland may be joining soon). For example, the RLF helps to determine which geometry of collider will have maximal of minimal or maximal radiation loss. For charges that are accelerated linearly, the power radiated is higher than for charges accelerated circularly by a factor denoted “gamma” squared, which increases rapidly with speed. This is one of the reasons why particles are accelerated in circular colliders rather than linear colliders, since the former is significantly more energy efficient. Examples of circular particle accelerators include CERN in Switzerland and DESY near Hamburg[3].
Another application of this formula on circular colliders in particular is that it tells us how energy efficiency scales with collider radius. At any given speed, a charged particle in such an accelerator will lose a given amount of energy per orbit. This energy loss decreases with radius, hence highlighting the importance of building large colliders in the interests of energy efficiency. However, the RLF also shows that at high speeds, energy loss decreases only very slowly with increased radius. This is because the gamma factor mentioned above appears to the power of 4 in the formula, whereas radius only appears to the power of 1.
The upshot of this is that, beyond a certain point, it becomes unfeasible to increase the energy efficiency of a circular particle accelerator by increasing its radius, despite the potential benefits if this were able to be realized.
In conclusion, the RLF has important applications in particle accelerator physics, and in insightful in explaining why previously proposed models, e.g. the Bohr model, of atomic theory cannot be valid. Its applications to collider physics help determine the most energy efficient types of colliders and the extent to which their efficiency can be increased by varying geometric parameters such as radius.
[1] Wikimedia Commons image in public domain https://commons.wikimedia.org/wiki/File:Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus_(white_background).jpg
[2] J.D. Jackson – Classical Electrodynamics (1999)
[3] https://www.desy.de/
